Research ArticleHeat and Mass Transfer in a Thin Liquid Film over an UnsteadyStretching Surface in the Presence of Thermosolutal Capillarityand Variable Magnetic Field

Yan Zhang Min Zhang and Shujuan Qi

School of Science Beijing University of Civil Engineering and Architecture Beijing 100044 China

Correspondence should be addressed to Yan Zhang zhangbiceasohucom

Received 20 January 2016 Accepted 30 June 2016

Academic Editor Mohamed Abd El Aziz

Copyright copy 2016 Yan Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The heat and mass transfer characteristics of a liquid film which contain thermosolutal capillarity and a variable magnetic fieldover an unsteady stretching sheet have been investigated The governing equations for momentum energy and concentrationare established and transformed to a set of coupled ordinary equations with the aid of similarity transformation The analyticalsolutions are obtained using the double-parameter transformation perturbation expansion method The effects of various relevantparameters such as unsteady parameter Prandtl number Schmidt number thermocapillary number and solutal capillary numberon the velocity temperature and concentration fields are discussed and presented graphically Results show that increasing valuesof thermocapillary number and solutal capillary number both lead to a decrease in the temperature and concentration fieldsFurthermore the influences of thermocapillary number on various fields aremore remarkable in comparison to the solutal capillarynumber

1 Introduction

In the recent years researches on the flow and heat transferof a liquid film on an unsteady stretching sheet have got moreand more attentions for its wide applications For exampleduringmechanical forming processes such as polymer extru-sion melt spinning process the process of shaping by forcingthrough a die wire and fiber coating and food stuff processthe flow of a liquid film on an unsteady stretching sheet willbe met

In 1970 Crane [1] studied the analytic solutions of two-dimensional boundary layer flow due to a stretching flat elas-tic sheet Munawar et al [2] considered time-dependent flowand heat transfer over a stretching cylinder Shehzad et al [3]investigated thermally radiative flow with internal heat gen-eration and magnetic field Furthermore many scholars dis-cussed other effects on the flow over a stretching sheet suchas three-dimensional flow [4 5] heat andmass transfer [6ndash9]MHD [10] first-order chemical reaction [11] non-Newtonianfluids [12 13] or different possible combinations of these

above effects [14ndash16] All of the above studies mainly focuson infinite fluid In fact the flow and heat transfer of a finitefilm are more suitable to describe industrial engineeringThehydrodynamics of the thin liquid film over a stretching sheetwere first considered byWang [17] who reduced the unsteadyNavier-Stokes equations to the coupled nonlinear ordinarydifferential equations by similarity transformation and solvedthe problem using a kind of multiple shooting method(see Roberts and Shipman [18]) Subsequently Wang [19]obtained the analytical solutions of a liquid thin film and con-firmed the validity of the homotopy analysis method On thebasis of Wangrsquos work [17] several authors [20ndash28] exploredfinite fluid domain of both Newtonian and non-Newtonianfluids using various velocity and thermal boundary con-ditions The combined effect of viscous dissipation andmagnetic field on the flow and heat transfer in a liquid filmover an unsteady stretching surface was presented by Abel etal [29] The thermocapillary effect in finite fluid domain wasfirst discussed by Dandapat et al [30] Noor and Hashim [31]extended the flow problem to hydromagnetic case

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8521580 12 pageshttpdxdoiorg10115520168521580

2 Mathematical Problems in Engineering

B

x

y

Slot

u T C

Stretching sheet

Free surface

h(t)

Us Ts Cs

Figure 1 Physics model with interface condition and coordinate system

Marangoni convection is caused by surface-tension gra-dient at a free liquid-gas or liquid-liquid interface that occursdue to gradient of temperature or concentration in the courseof heat or mass transfer The surface-tension variation on thefree liquid surface resulting from the temperature gradientor concentration gradient can induce motion within the fluidcalled thermocapillary flow or solutal capillary flow (thermalMarangoni convection or solutal Marangoni convection)Pop et al [32] investigated thermosolutal Marangoni forcedconvection boundary layers On the flow field of power-law fluid Lin et al [33] analyzed the effect of radiation onMarangoni convection flow and heat transfer in the fluidswith variable thermal conductivity Then Lin et al [34]dealt with thermosolutal Marangoni convection flow in thepresence of internal heat generation The surface tensionplays an important role on the free liquid surface Otherstudies about thermocapillary effect on a thin film can befound in [35ndash38]

The main objective of our study is to extend previousresearch to the solutal Marangoni effect and mass diffusionBy means of an exact similarity transformation governingPDEs are reduced into coupled nonlinear ODEs And theanalytical solutions are obtained using the double-parametertransformation perturbation expansion method [11] Theinfluences of various relevant parameters such as unsteadi-ness parameter 119878 HartmannnumberMa the Prandtl numberPr the Schmidt number Sc the thermocapillary number119872

1

and the solutal capillary number 1198722on the flow field are

elucidated through graphs and tables

2 Mathematical Formulation

21 Governing Equations and Boundary Conditions Con-sider the thin elastic sheet that emerges from a narrowslit at origin of the Cartesian coordinate system shown inFigure 1 A variable magnetic field 119861 = 119861

0(1 minus 120572119905)

12

normal to the stretching sheet is applied where 120572 is a positiveconstant And in the abovemodel we take concentration intoconsideration A thin liquid film with uniform thickness ℎ(119905)rests on the horizontal sheet By applying the boundary layerassumptions [39] the governing time-dependent equationsfor mass momentum energy and concentration are given by

120597119906

120597119909+120597V120597119910

= 0 (1)

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910= 120592

1205972119906

1205971199102minus1198612

120588119906

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910= 120581

1205972119879

1205971199102

120597119862

120597119905+ 119906

120597119862

120597119909+ V

120597119862

120597119910= 119863

1205972119862

1205971199102

(2)

The boundary conditions are

119906 = 119880119904

V = 0

119879 = 119879119904

119862 = 119862119904

at 119910 = 0

120583120597119906

120597119910=120597120590

120597119909

120597119879

120597119910= 0

120597119862

120597119910= 0

V =119889ℎ

119889119905

at 119910 = ℎ (119905)

(3)

where 119906 and V are the velocity components of the fluid in the119909- and 119910-directions 119905 is the time 120592 is the kinematic viscosity is the electrical conductivity 119861 is the magnetic field 120588 is thedensity 119879 is the temperature 120581 is the thermal diffusivity 119862is the concentration 119863 is the mass diffusivity 120583 is dynamicviscosity and ℎ(119905) is the uniform thickness of the liquid filmThe dependence of surface tension on the temperature andconcentration can be expressed as [40]

120590 = 1205900[1 minus 120575

119879(119879 minus 119879

0) minus 120575119862(119862 minus 119862

0)] (4)

where 120575119879= minus(1120590

0)(120597120590120597119879)|

119862and 120575

119862= minus(1120590

0)(120597120590120597119862)|

119879

The fluidmotionwithin liquid film resulted fromnot only theviscous shear arising from the stretching of the elastic sheet

Mathematical Problems in Engineering 3

but also the surface-tension gradient The stretching velocity119880119904(119909 119905) is assumed to be of the same form as that considered

by Wang [17]

119880119904=

119887119909

1 minus 120572119905 (5)

where 119887 is a positive constant and denotes the initial stretch-ing rate With unsteady stretching (ie 120572 = 0) however120572minus1 becomes the representative time scale of the resulting

unsteady boundary layer problem The adopted formulationof the velocity sheet 119880

119904(119909 119905) in (5) is valid only for times 119905 lt

120572minus1 unless 120572 = 0 Also the temperature and the concentration

of the surface of the elastic sheet are assumed to vary bothalong the sheet and with time respectively as [22 30]

119879119904= 1198790minus 119879ref

1198871199092

2120592(1 minus 120572119905)

minus32

119862119904= 1198620minus 119862ref

1198871199092

2120592(1 minus 120572119905)

minus32

(6)

where 1198790and 119862

0are the temperature and the concentration

at the slit respectively and 119879ref and 119862ref are the referencetemperature and the reference concentration in the case of119905 lt 120572

minus1 respectively Equation (6) represents a situationin which the sheet temperature and concentration decreasefrom 119879

0and 119862

0at the slot in proportion to 1199092 The nonuni-

form distributions of temperature and concentration causesurface-tension gradient which leads to the fluid flow fromlower surface tension to higher surface tension Accordingto the temperature and concentration boundary layer con-ditions (6) one can conclude that thermosolutal Marangoniconvection flow is in accordwith the flowdirection of the thinfilm

22 Similarity Transformation The special forms of thestretching velocity surface temperature and fluid concentra-tion in (5) and (6) respectively are chosen to allow (2) tobe converted into a set of ordinary differential equations bymeans of similarity transformations

119906 =119887119909

1 minus 1205721199051198911015840(120578)

V = minus(120592119887

1 minus 120572119905)

12

119891 (120578) 120573

119879 = 1198790minus 119879ref

1198871199092

2120592(1 minus 120572119905)

minus32120579 (120578)

119862 = 1198620minus 119862ref

1198871199092

2120592(1 minus 120572119905)

minus32120601 (120578)

(7)

where the similarity variable 120578 is given by Wang [19]

120578 = (119887

120592)

12

(1 minus 120572119905)minus12

119910120573minus1 (8)

and120573 is yet an unknown constant denoting the dimensionlessfilm thickness defined by

120573 = (119887

120592)

12

(1 minus 120572119905)minus12

ℎ (119905) (9)

Substituting (7)ndash(9) into (1)ndash(3) (1)ndash(3) can be trans-formed into the following nonlinear ordinary differentialequations

119891101584010158401015840+ 120574 [119891119891

10158401015840minus 11989110158402minus1

212057811987811989110158401015840minus (119878 +Ma) 1198911015840] = 0

Prminus112057910158401015840 + 120574 (1198911205791015840 minus 3

2119878120579 minus

1

21198781205791015840120578 minus 2119891

1015840120579) = 0

Scminus112060110158401015840 + 120574 (1198911206011015840 minus 3

2119878120601 minus

1

21198781206011015840120578 minus 2119891

1015840120601) = 0

(10)

subject to the boundary conditions

119891 (0) = 0

1198911015840(0) = 1

120579 (0) = 1

120601 (0) = 1

at 120578 = 0

(11)

119891 (1) =119878

2

11989110158401015840(1) = 119872

1120579 (1) + 119872

2120601 (1)

1205791015840(1) = 0

1206011015840(1) = 0

at 120578 = 1

(12)

where Ma = 1198610(120588119887) is Hartmann number 119878 = 120572119887 is

the dimensionless unsteadiness parameter Pr = 120592120581 is thePrandtl number Sc = 120592119863 is the Schmidt number 120574 = 120573

2

is the dimensionless film thickness and the thermocapillarynumber119872

1and the solutal capillary number119872

2are defined

as 1198721= 1205751198791205900119879ref120573120583radic119887120592 and 119872

2= 1205751198621205900119862ref120573120583radic119887120592

respectivelyThe skin friction coefficient 119862

119891and the Nusselt number

Nu119909are defined as

119862119891=

120591119908

120588119880119904

22

Nu119909=119902119908119909

119896119879ref

(13)

where the wall skin friction 120591119908and the sheet heat flux 119902

119908are

120591119908= 120583(

120597119906

120597119910)

119910=0

119902119908= minus119896(

120597119879

120597119910)

119910=0

(14)

4 Mathematical Problems in Engineering

Table 1 Comparison of values of film thickness 120573 and skin frictioncoefficient minus11989110158401015840(0) with Ma = 0

119878Wang [19] Present results

120573 minus11989110158401015840(0) 120573 minus119891

10158401015840(0)

08 2151990 2680940 2152839 268192010 1543620 1972385 1543620 197238512 1127781 1442631 1127781 144262514 0821032 1012784 0821032 1012780

Substituting (14) into (13) then we get

119862119891=2

12057311989110158401015840(0)Re

119909

minus12

Nu119909=

1

2120573 (1 minus 120572119905)1205791015840(0)Re

119909

32

(15)

where Re119909= 119880119904119909120592 is the local Reynolds number

23 Solution Approach In order to solve (10) We used thedouble-parameter transformation perturbation expansionmethod that confirmed the validity by Zhang and Zheng [11]We introduced an artificial small parameter 120576 the equationscan be expanded to the form of power series We can obtainanalytical solutions in the form of series by comparing thecoefficients of same power and solving the equation sets

We transform the dependent variable and independentvariable as follows

119891 (120578) = 120576119892 (120585) + 120578 +1205721

21205782

= 120576119892 (120585) + 12057613120585 +

1205721

2120576231205852

120579 (120578) = 12057623ℎ (120585) + 1 + 120572

2120578 = 12057623ℎ (120585) + 1 + 120572

212057613120585

120601 (120578) = 12057623120603 (120585) + 1 + 120572

3120578 = 12057623120603 (120585) + 1 + 120572

312057613120585

(16)

where 120585 = 120576minus13

120578 and 120576 is an artificial small parameter Weassume 11989110158401015840(0) = 120572

1 1205791015840(0) = 120572

2 and 1206011015840(0) = 120572

3 here 120572

1 1205722

and 1205723are constants the boundary conditions become

119892 (0) = 0

1198921015840(0) = 0

11989210158401015840(0) = 0

ℎ (0) = 0

ℎ1015840(0) = 0

120603 (0) = 0

1206031015840(0) = 0

(17)

Equation (10) reduces to

119892101584010158401015840(120585) + 120574 ((120576119892 (120585) + 120576

13120585 +

1205721

2120576231205852) (1205761311989210158401015840+ 1205721)

minus1

211987812057613120585 (1205761311989210158401015840+ 1205721) minus (120576

231198921015840+ 1 + 120572

112057613120585)2

minus (119878 +Ma) (120576231198921015840 + 1 + 120572112057613120585)) = 0

ℎ10158401015840(120585)

Pr+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (12057613ℎ1015840(120585) + 120572

2) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623ℎ (120585) + 1 + 120572

212057613120585) minus

1

2

sdot 11987812057613120585 (12057613ℎ1015840(120585) + 120572

2)

minus3119878

2(12057623ℎ (120585) + 1 + 120572

212057613120585)) = 0

12060310158401015840(120585)

Sc+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (120576131206031015840(120585) + 120572

3) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623120603 (120585) + 1 + 120572

312057613120585) minus

1

2

sdot 11987812057613120585 (120576131206031015840(120585) + 120572

3)

minus3119878

2(12057623120603 (120585) + 1 + 120572

312057613120585)) = 0

(18)

And we can assume

119892 (120585) = 1198920(120585) + 120576

131198921(120585) + 120576

231198922(120585) + 120576119892

3(120585)

+ 120576431198924(120585) + sdot sdot sdot

ℎ (120585) = ℎ0(120585) + 120576

13ℎ1(120585) + 120576

23ℎ2(120585) + 120576ℎ

3(120585)

+ 12057643ℎ4(120585) + sdot sdot sdot

120603 (120585) = 1206030(120585) + 120576

131206031(120585) + 120576

231206032(120585) + 120576120603

3(120585)

+ 120576431206034(120585) + sdot sdot sdot

(19)

with the boundary conditions

119892119899(0) = 0

1198921015840

119899(0) = 0

11989210158401015840

119899(0) = 0

119892101584010158401015840

119899(0) = 0

119899 = 0 1 2

Mathematical Problems in Engineering 5

Table 2 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 and S is varied

119878 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

08 1706139 0245858 0190409 0079435 2791406 2738673 383558410 1340872 0387092 0249442 0118379 2180379 2252363 316099212 1088400 0541510 0309461 0161276 1714381 1897617 2678301

Table 3 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 and Ma is varied

Ma 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1955885 0228035 0131483 0047797 2776933 321537 446398815 1629020 0252600 0211365 0092719 2802801 2590747 364089530 1370781 0283080 0299672 0154063 2866680 2089107 2983500

Table 4 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with S = 08 Ma = 11198722= 1 Pr = 1 and Sc = 18 and119872

1is varied

1198721

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1895449 0296754 0143799 0054085 2978572 3080157 42856355 1987330 0335046 0123865 0044928 3120442 324009 44987058 2089998 0380759 0106127 0036553 3279882 3415187 4733832

Table 5 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1 Pr = 1 and Sc = 18 and119872

2is varied

1198722

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1823807 0265936 0162696 0062564 2870952 295405 41202615 1870447 0284213 0148046 0056928 2942314 303796 42296068 1927131 030837 0135444 0050747 3028503 3136665 4360637

Table 6 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Sc = 18 and Pr is varied

Pr 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844352 170801 0216269 0065849 0079270 2701304 4083229 38517395 1675911 0203976 0009869 0084604 2654055 6569892 377576610 1671722 020159 0000697 0085353 2648593 9447705 3766435

Table 7 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Pr = 1 and Sc is varied

Sc 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1884207 0289866 0144404 0313734 2963305 3062276 20119231 1810615 0259498 0163349 0160799 2852006 2932347 293350315 177745 0245319 0170494 0093966 280298 2874491 36259918 1765648 0242329 0178412 0070413 2783885 2850602 3984435

ℎ119899(0) = 0

ℎ1015840

119899(0) = 0

ℎ10158401015840

119899(0) = 0

119899 = 0 1 2

120603119899(0) = 0

1206031015840

119899(0) = 0

12060310158401015840

119899(0) = 0

119899 = 0 1 2

(20)

6 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

f998400 (120578)

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120579(120578)

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120601(120578)

(c)

Figure 2 (a) Velocity profiles for different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of 119878 with Ma = 12

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

Let 119878 = 08 and Ma = 1 and we can obtain

119891 (120578) = 120578 +1

212057211205782+7

151205741205783+2

1512057211205741205784

+ (1

1201205721

2120574 +

91

15001205742) 1205785+1

60120572112057421205786

+ (23

252001205721

21205742+

77

150001205743) 1205787

+ (589

50400012057211205743minus

77

403201205721

31205742) 1205788

+ (383

90720001205721

21205743+

3871

162000001205744) 1205789

+ (1

4320001205721

31205743+

209

756000012057211205744) 12057810+ sdot sdot sdot

1198911015840(120578) = 1 + 120572

1120578 +

7

451205741205782+1

3012057211205741205783

+ (1

6001205721

2120574 +

91

45001205742) 1205784+

1

360120572112057421205785

+ (23

1764001205721

21205742+

11

150001205743) 1205786

+ (589

467200012057211205743minus

77

3225601205721

31205742) 1205787

+ (383

816480001205721

21205743+

3871

1458000001205744) 1205788

+ (1

43200001205721

31205743+

209

7560000012057211205744) 1205789+ sdot sdot sdot

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

B

x

y

Slot

u T C

Stretching sheet

Free surface

h(t)

Us Ts Cs

Figure 1 Physics model with interface condition and coordinate system

Marangoni convection is caused by surface-tension gra-dient at a free liquid-gas or liquid-liquid interface that occursdue to gradient of temperature or concentration in the courseof heat or mass transfer The surface-tension variation on thefree liquid surface resulting from the temperature gradientor concentration gradient can induce motion within the fluidcalled thermocapillary flow or solutal capillary flow (thermalMarangoni convection or solutal Marangoni convection)Pop et al [32] investigated thermosolutal Marangoni forcedconvection boundary layers On the flow field of power-law fluid Lin et al [33] analyzed the effect of radiation onMarangoni convection flow and heat transfer in the fluidswith variable thermal conductivity Then Lin et al [34]dealt with thermosolutal Marangoni convection flow in thepresence of internal heat generation The surface tensionplays an important role on the free liquid surface Otherstudies about thermocapillary effect on a thin film can befound in [35ndash38]

The main objective of our study is to extend previousresearch to the solutal Marangoni effect and mass diffusionBy means of an exact similarity transformation governingPDEs are reduced into coupled nonlinear ODEs And theanalytical solutions are obtained using the double-parametertransformation perturbation expansion method [11] Theinfluences of various relevant parameters such as unsteadi-ness parameter 119878 HartmannnumberMa the Prandtl numberPr the Schmidt number Sc the thermocapillary number119872

1

and the solutal capillary number 1198722on the flow field are

elucidated through graphs and tables

2 Mathematical Formulation

21 Governing Equations and Boundary Conditions Con-sider the thin elastic sheet that emerges from a narrowslit at origin of the Cartesian coordinate system shown inFigure 1 A variable magnetic field 119861 = 119861

0(1 minus 120572119905)

12

normal to the stretching sheet is applied where 120572 is a positiveconstant And in the abovemodel we take concentration intoconsideration A thin liquid film with uniform thickness ℎ(119905)rests on the horizontal sheet By applying the boundary layerassumptions [39] the governing time-dependent equationsfor mass momentum energy and concentration are given by

120597119906

120597119909+120597V120597119910

= 0 (1)

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910= 120592

1205972119906

1205971199102minus1198612

120588119906

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910= 120581

1205972119879

1205971199102

120597119862

120597119905+ 119906

120597119862

120597119909+ V

120597119862

120597119910= 119863

1205972119862

1205971199102

(2)

The boundary conditions are

119906 = 119880119904

V = 0

119879 = 119879119904

119862 = 119862119904

at 119910 = 0

120583120597119906

120597119910=120597120590

120597119909

120597119879

120597119910= 0

120597119862

120597119910= 0

V =119889ℎ

119889119905

at 119910 = ℎ (119905)

(3)

where 119906 and V are the velocity components of the fluid in the119909- and 119910-directions 119905 is the time 120592 is the kinematic viscosity is the electrical conductivity 119861 is the magnetic field 120588 is thedensity 119879 is the temperature 120581 is the thermal diffusivity 119862is the concentration 119863 is the mass diffusivity 120583 is dynamicviscosity and ℎ(119905) is the uniform thickness of the liquid filmThe dependence of surface tension on the temperature andconcentration can be expressed as [40]

120590 = 1205900[1 minus 120575

119879(119879 minus 119879

0) minus 120575119862(119862 minus 119862

0)] (4)

where 120575119879= minus(1120590

0)(120597120590120597119879)|

119862and 120575

119862= minus(1120590

0)(120597120590120597119862)|

119879

The fluidmotionwithin liquid film resulted fromnot only theviscous shear arising from the stretching of the elastic sheet

Mathematical Problems in Engineering 3

but also the surface-tension gradient The stretching velocity119880119904(119909 119905) is assumed to be of the same form as that considered

by Wang [17]

119880119904=

119887119909

1 minus 120572119905 (5)

where 119887 is a positive constant and denotes the initial stretch-ing rate With unsteady stretching (ie 120572 = 0) however120572minus1 becomes the representative time scale of the resulting

unsteady boundary layer problem The adopted formulationof the velocity sheet 119880

119904(119909 119905) in (5) is valid only for times 119905 lt

120572minus1 unless 120572 = 0 Also the temperature and the concentration

of the surface of the elastic sheet are assumed to vary bothalong the sheet and with time respectively as [22 30]

119879119904= 1198790minus 119879ref

1198871199092

2120592(1 minus 120572119905)

minus32

119862119904= 1198620minus 119862ref

1198871199092

2120592(1 minus 120572119905)

minus32

(6)

where 1198790and 119862

0are the temperature and the concentration

at the slit respectively and 119879ref and 119862ref are the referencetemperature and the reference concentration in the case of119905 lt 120572

minus1 respectively Equation (6) represents a situationin which the sheet temperature and concentration decreasefrom 119879

0and 119862

0at the slot in proportion to 1199092 The nonuni-

form distributions of temperature and concentration causesurface-tension gradient which leads to the fluid flow fromlower surface tension to higher surface tension Accordingto the temperature and concentration boundary layer con-ditions (6) one can conclude that thermosolutal Marangoniconvection flow is in accordwith the flowdirection of the thinfilm

22 Similarity Transformation The special forms of thestretching velocity surface temperature and fluid concentra-tion in (5) and (6) respectively are chosen to allow (2) tobe converted into a set of ordinary differential equations bymeans of similarity transformations

119906 =119887119909

1 minus 1205721199051198911015840(120578)

V = minus(120592119887

1 minus 120572119905)

12

119891 (120578) 120573

119879 = 1198790minus 119879ref

1198871199092

2120592(1 minus 120572119905)

minus32120579 (120578)

119862 = 1198620minus 119862ref

1198871199092

2120592(1 minus 120572119905)

minus32120601 (120578)

(7)

where the similarity variable 120578 is given by Wang [19]

120578 = (119887

120592)

12

(1 minus 120572119905)minus12

119910120573minus1 (8)

and120573 is yet an unknown constant denoting the dimensionlessfilm thickness defined by

120573 = (119887

120592)

12

(1 minus 120572119905)minus12

ℎ (119905) (9)

Substituting (7)ndash(9) into (1)ndash(3) (1)ndash(3) can be trans-formed into the following nonlinear ordinary differentialequations

119891101584010158401015840+ 120574 [119891119891

10158401015840minus 11989110158402minus1

212057811987811989110158401015840minus (119878 +Ma) 1198911015840] = 0

Prminus112057910158401015840 + 120574 (1198911205791015840 minus 3

2119878120579 minus

1

21198781205791015840120578 minus 2119891

1015840120579) = 0

Scminus112060110158401015840 + 120574 (1198911206011015840 minus 3

2119878120601 minus

1

21198781206011015840120578 minus 2119891

1015840120601) = 0

(10)

subject to the boundary conditions

119891 (0) = 0

1198911015840(0) = 1

120579 (0) = 1

120601 (0) = 1

at 120578 = 0

(11)

119891 (1) =119878

2

11989110158401015840(1) = 119872

1120579 (1) + 119872

2120601 (1)

1205791015840(1) = 0

1206011015840(1) = 0

at 120578 = 1

(12)

where Ma = 1198610(120588119887) is Hartmann number 119878 = 120572119887 is

the dimensionless unsteadiness parameter Pr = 120592120581 is thePrandtl number Sc = 120592119863 is the Schmidt number 120574 = 120573

2

is the dimensionless film thickness and the thermocapillarynumber119872

1and the solutal capillary number119872

2are defined

as 1198721= 1205751198791205900119879ref120573120583radic119887120592 and 119872

2= 1205751198621205900119862ref120573120583radic119887120592

respectivelyThe skin friction coefficient 119862

119891and the Nusselt number

Nu119909are defined as

119862119891=

120591119908

120588119880119904

22

Nu119909=119902119908119909

119896119879ref

(13)

where the wall skin friction 120591119908and the sheet heat flux 119902

119908are

120591119908= 120583(

120597119906

120597119910)

119910=0

119902119908= minus119896(

120597119879

120597119910)

119910=0

(14)

4 Mathematical Problems in Engineering

Table 1 Comparison of values of film thickness 120573 and skin frictioncoefficient minus11989110158401015840(0) with Ma = 0

119878Wang [19] Present results

120573 minus11989110158401015840(0) 120573 minus119891

10158401015840(0)

08 2151990 2680940 2152839 268192010 1543620 1972385 1543620 197238512 1127781 1442631 1127781 144262514 0821032 1012784 0821032 1012780

Substituting (14) into (13) then we get

119862119891=2

12057311989110158401015840(0)Re

119909

minus12

Nu119909=

1

2120573 (1 minus 120572119905)1205791015840(0)Re

119909

32

(15)

where Re119909= 119880119904119909120592 is the local Reynolds number

23 Solution Approach In order to solve (10) We used thedouble-parameter transformation perturbation expansionmethod that confirmed the validity by Zhang and Zheng [11]We introduced an artificial small parameter 120576 the equationscan be expanded to the form of power series We can obtainanalytical solutions in the form of series by comparing thecoefficients of same power and solving the equation sets

We transform the dependent variable and independentvariable as follows

119891 (120578) = 120576119892 (120585) + 120578 +1205721

21205782

= 120576119892 (120585) + 12057613120585 +

1205721

2120576231205852

120579 (120578) = 12057623ℎ (120585) + 1 + 120572

2120578 = 12057623ℎ (120585) + 1 + 120572

212057613120585

120601 (120578) = 12057623120603 (120585) + 1 + 120572

3120578 = 12057623120603 (120585) + 1 + 120572

312057613120585

(16)

where 120585 = 120576minus13

120578 and 120576 is an artificial small parameter Weassume 11989110158401015840(0) = 120572

1 1205791015840(0) = 120572

2 and 1206011015840(0) = 120572

3 here 120572

1 1205722

and 1205723are constants the boundary conditions become

119892 (0) = 0

1198921015840(0) = 0

11989210158401015840(0) = 0

ℎ (0) = 0

ℎ1015840(0) = 0

120603 (0) = 0

1206031015840(0) = 0

(17)

Equation (10) reduces to

119892101584010158401015840(120585) + 120574 ((120576119892 (120585) + 120576

13120585 +

1205721

2120576231205852) (1205761311989210158401015840+ 1205721)

minus1

211987812057613120585 (1205761311989210158401015840+ 1205721) minus (120576

231198921015840+ 1 + 120572

112057613120585)2

minus (119878 +Ma) (120576231198921015840 + 1 + 120572112057613120585)) = 0

ℎ10158401015840(120585)

Pr+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (12057613ℎ1015840(120585) + 120572

2) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623ℎ (120585) + 1 + 120572

212057613120585) minus

1

2

sdot 11987812057613120585 (12057613ℎ1015840(120585) + 120572

2)

minus3119878

2(12057623ℎ (120585) + 1 + 120572

212057613120585)) = 0

12060310158401015840(120585)

Sc+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (120576131206031015840(120585) + 120572

3) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623120603 (120585) + 1 + 120572

312057613120585) minus

1

2

sdot 11987812057613120585 (120576131206031015840(120585) + 120572

3)

minus3119878

2(12057623120603 (120585) + 1 + 120572

312057613120585)) = 0

(18)

And we can assume

119892 (120585) = 1198920(120585) + 120576

131198921(120585) + 120576

231198922(120585) + 120576119892

3(120585)

+ 120576431198924(120585) + sdot sdot sdot

ℎ (120585) = ℎ0(120585) + 120576

13ℎ1(120585) + 120576

23ℎ2(120585) + 120576ℎ

3(120585)

+ 12057643ℎ4(120585) + sdot sdot sdot

120603 (120585) = 1206030(120585) + 120576

131206031(120585) + 120576

231206032(120585) + 120576120603

3(120585)

+ 120576431206034(120585) + sdot sdot sdot

(19)

with the boundary conditions

119892119899(0) = 0

1198921015840

119899(0) = 0

11989210158401015840

119899(0) = 0

119892101584010158401015840

119899(0) = 0

119899 = 0 1 2

Mathematical Problems in Engineering 5

Table 2 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 and S is varied

119878 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

08 1706139 0245858 0190409 0079435 2791406 2738673 383558410 1340872 0387092 0249442 0118379 2180379 2252363 316099212 1088400 0541510 0309461 0161276 1714381 1897617 2678301

Table 3 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 and Ma is varied

Ma 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1955885 0228035 0131483 0047797 2776933 321537 446398815 1629020 0252600 0211365 0092719 2802801 2590747 364089530 1370781 0283080 0299672 0154063 2866680 2089107 2983500

Table 4 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with S = 08 Ma = 11198722= 1 Pr = 1 and Sc = 18 and119872

1is varied

1198721

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1895449 0296754 0143799 0054085 2978572 3080157 42856355 1987330 0335046 0123865 0044928 3120442 324009 44987058 2089998 0380759 0106127 0036553 3279882 3415187 4733832

Table 5 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1 Pr = 1 and Sc = 18 and119872

2is varied

1198722

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1823807 0265936 0162696 0062564 2870952 295405 41202615 1870447 0284213 0148046 0056928 2942314 303796 42296068 1927131 030837 0135444 0050747 3028503 3136665 4360637

Table 6 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Sc = 18 and Pr is varied

Pr 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844352 170801 0216269 0065849 0079270 2701304 4083229 38517395 1675911 0203976 0009869 0084604 2654055 6569892 377576610 1671722 020159 0000697 0085353 2648593 9447705 3766435

Table 7 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Pr = 1 and Sc is varied

Sc 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1884207 0289866 0144404 0313734 2963305 3062276 20119231 1810615 0259498 0163349 0160799 2852006 2932347 293350315 177745 0245319 0170494 0093966 280298 2874491 36259918 1765648 0242329 0178412 0070413 2783885 2850602 3984435

ℎ119899(0) = 0

ℎ1015840

119899(0) = 0

ℎ10158401015840

119899(0) = 0

119899 = 0 1 2

120603119899(0) = 0

1206031015840

119899(0) = 0

12060310158401015840

119899(0) = 0

119899 = 0 1 2

(20)

6 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

f998400 (120578)

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120579(120578)

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120601(120578)

(c)

Figure 2 (a) Velocity profiles for different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of 119878 with Ma = 12

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

Let 119878 = 08 and Ma = 1 and we can obtain

119891 (120578) = 120578 +1

212057211205782+7

151205741205783+2

1512057211205741205784

+ (1

1201205721

2120574 +

91

15001205742) 1205785+1

60120572112057421205786

+ (23

252001205721

21205742+

77

150001205743) 1205787

+ (589

50400012057211205743minus

77

403201205721

31205742) 1205788

+ (383

90720001205721

21205743+

3871

162000001205744) 1205789

+ (1

4320001205721

31205743+

209

756000012057211205744) 12057810+ sdot sdot sdot

1198911015840(120578) = 1 + 120572

1120578 +

7

451205741205782+1

3012057211205741205783

+ (1

6001205721

2120574 +

91

45001205742) 1205784+

1

360120572112057421205785

+ (23

1764001205721

21205742+

11

150001205743) 1205786

+ (589

467200012057211205743minus

77

3225601205721

31205742) 1205787

+ (383

816480001205721

21205743+

3871

1458000001205744) 1205788

+ (1

43200001205721

31205743+

209

7560000012057211205744) 1205789+ sdot sdot sdot

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

but also the surface-tension gradient The stretching velocity119880119904(119909 119905) is assumed to be of the same form as that considered

by Wang [17]

119880119904=

119887119909

1 minus 120572119905 (5)

where 119887 is a positive constant and denotes the initial stretch-ing rate With unsteady stretching (ie 120572 = 0) however120572minus1 becomes the representative time scale of the resulting

unsteady boundary layer problem The adopted formulationof the velocity sheet 119880

119904(119909 119905) in (5) is valid only for times 119905 lt

120572minus1 unless 120572 = 0 Also the temperature and the concentration

of the surface of the elastic sheet are assumed to vary bothalong the sheet and with time respectively as [22 30]

119879119904= 1198790minus 119879ref

1198871199092

2120592(1 minus 120572119905)

minus32

119862119904= 1198620minus 119862ref

1198871199092

2120592(1 minus 120572119905)

minus32

(6)

where 1198790and 119862

0are the temperature and the concentration

at the slit respectively and 119879ref and 119862ref are the referencetemperature and the reference concentration in the case of119905 lt 120572

minus1 respectively Equation (6) represents a situationin which the sheet temperature and concentration decreasefrom 119879

0and 119862

0at the slot in proportion to 1199092 The nonuni-

form distributions of temperature and concentration causesurface-tension gradient which leads to the fluid flow fromlower surface tension to higher surface tension Accordingto the temperature and concentration boundary layer con-ditions (6) one can conclude that thermosolutal Marangoniconvection flow is in accordwith the flowdirection of the thinfilm

22 Similarity Transformation The special forms of thestretching velocity surface temperature and fluid concentra-tion in (5) and (6) respectively are chosen to allow (2) tobe converted into a set of ordinary differential equations bymeans of similarity transformations

119906 =119887119909

1 minus 1205721199051198911015840(120578)

V = minus(120592119887

1 minus 120572119905)

12

119891 (120578) 120573

119879 = 1198790minus 119879ref

1198871199092

2120592(1 minus 120572119905)

minus32120579 (120578)

119862 = 1198620minus 119862ref

1198871199092

2120592(1 minus 120572119905)

minus32120601 (120578)

(7)

where the similarity variable 120578 is given by Wang [19]

120578 = (119887

120592)

12

(1 minus 120572119905)minus12

119910120573minus1 (8)

and120573 is yet an unknown constant denoting the dimensionlessfilm thickness defined by

120573 = (119887

120592)

12

(1 minus 120572119905)minus12

ℎ (119905) (9)

Substituting (7)ndash(9) into (1)ndash(3) (1)ndash(3) can be trans-formed into the following nonlinear ordinary differentialequations

119891101584010158401015840+ 120574 [119891119891

10158401015840minus 11989110158402minus1

212057811987811989110158401015840minus (119878 +Ma) 1198911015840] = 0

Prminus112057910158401015840 + 120574 (1198911205791015840 minus 3

2119878120579 minus

1

21198781205791015840120578 minus 2119891

1015840120579) = 0

Scminus112060110158401015840 + 120574 (1198911206011015840 minus 3

2119878120601 minus

1

21198781206011015840120578 minus 2119891

1015840120601) = 0

(10)

subject to the boundary conditions

119891 (0) = 0

1198911015840(0) = 1

120579 (0) = 1

120601 (0) = 1

at 120578 = 0

(11)

119891 (1) =119878

2

11989110158401015840(1) = 119872

1120579 (1) + 119872

2120601 (1)

1205791015840(1) = 0

1206011015840(1) = 0

at 120578 = 1

(12)

where Ma = 1198610(120588119887) is Hartmann number 119878 = 120572119887 is

the dimensionless unsteadiness parameter Pr = 120592120581 is thePrandtl number Sc = 120592119863 is the Schmidt number 120574 = 120573

2

is the dimensionless film thickness and the thermocapillarynumber119872

1and the solutal capillary number119872

2are defined

as 1198721= 1205751198791205900119879ref120573120583radic119887120592 and 119872

2= 1205751198621205900119862ref120573120583radic119887120592

respectivelyThe skin friction coefficient 119862

119891and the Nusselt number

Nu119909are defined as

119862119891=

120591119908

120588119880119904

22

Nu119909=119902119908119909

119896119879ref

(13)

where the wall skin friction 120591119908and the sheet heat flux 119902

119908are

120591119908= 120583(

120597119906

120597119910)

119910=0

119902119908= minus119896(

120597119879

120597119910)

119910=0

(14)

4 Mathematical Problems in Engineering

Table 1 Comparison of values of film thickness 120573 and skin frictioncoefficient minus11989110158401015840(0) with Ma = 0

119878Wang [19] Present results

120573 minus11989110158401015840(0) 120573 minus119891

10158401015840(0)

08 2151990 2680940 2152839 268192010 1543620 1972385 1543620 197238512 1127781 1442631 1127781 144262514 0821032 1012784 0821032 1012780

Substituting (14) into (13) then we get

119862119891=2

12057311989110158401015840(0)Re

119909

minus12

Nu119909=

1

2120573 (1 minus 120572119905)1205791015840(0)Re

119909

32

(15)

where Re119909= 119880119904119909120592 is the local Reynolds number

23 Solution Approach In order to solve (10) We used thedouble-parameter transformation perturbation expansionmethod that confirmed the validity by Zhang and Zheng [11]We introduced an artificial small parameter 120576 the equationscan be expanded to the form of power series We can obtainanalytical solutions in the form of series by comparing thecoefficients of same power and solving the equation sets

We transform the dependent variable and independentvariable as follows

119891 (120578) = 120576119892 (120585) + 120578 +1205721

21205782

= 120576119892 (120585) + 12057613120585 +

1205721

2120576231205852

120579 (120578) = 12057623ℎ (120585) + 1 + 120572

2120578 = 12057623ℎ (120585) + 1 + 120572

212057613120585

120601 (120578) = 12057623120603 (120585) + 1 + 120572

3120578 = 12057623120603 (120585) + 1 + 120572

312057613120585

(16)

where 120585 = 120576minus13

120578 and 120576 is an artificial small parameter Weassume 11989110158401015840(0) = 120572

1 1205791015840(0) = 120572

2 and 1206011015840(0) = 120572

3 here 120572

1 1205722

and 1205723are constants the boundary conditions become

119892 (0) = 0

1198921015840(0) = 0

11989210158401015840(0) = 0

ℎ (0) = 0

ℎ1015840(0) = 0

120603 (0) = 0

1206031015840(0) = 0

(17)

Equation (10) reduces to

119892101584010158401015840(120585) + 120574 ((120576119892 (120585) + 120576

13120585 +

1205721

2120576231205852) (1205761311989210158401015840+ 1205721)

minus1

211987812057613120585 (1205761311989210158401015840+ 1205721) minus (120576

231198921015840+ 1 + 120572

112057613120585)2

minus (119878 +Ma) (120576231198921015840 + 1 + 120572112057613120585)) = 0

ℎ10158401015840(120585)

Pr+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (12057613ℎ1015840(120585) + 120572

2) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623ℎ (120585) + 1 + 120572

212057613120585) minus

1

2

sdot 11987812057613120585 (12057613ℎ1015840(120585) + 120572

2)

minus3119878

2(12057623ℎ (120585) + 1 + 120572

212057613120585)) = 0

12060310158401015840(120585)

Sc+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (120576131206031015840(120585) + 120572

3) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623120603 (120585) + 1 + 120572

312057613120585) minus

1

2

sdot 11987812057613120585 (120576131206031015840(120585) + 120572

3)

minus3119878

2(12057623120603 (120585) + 1 + 120572

312057613120585)) = 0

(18)

And we can assume

119892 (120585) = 1198920(120585) + 120576

131198921(120585) + 120576

231198922(120585) + 120576119892

3(120585)

+ 120576431198924(120585) + sdot sdot sdot

ℎ (120585) = ℎ0(120585) + 120576

13ℎ1(120585) + 120576

23ℎ2(120585) + 120576ℎ

3(120585)

+ 12057643ℎ4(120585) + sdot sdot sdot

120603 (120585) = 1206030(120585) + 120576

131206031(120585) + 120576

231206032(120585) + 120576120603

3(120585)

+ 120576431206034(120585) + sdot sdot sdot

(19)

with the boundary conditions

119892119899(0) = 0

1198921015840

119899(0) = 0

11989210158401015840

119899(0) = 0

119892101584010158401015840

119899(0) = 0

119899 = 0 1 2

Mathematical Problems in Engineering 5

Table 2 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 and S is varied

119878 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

08 1706139 0245858 0190409 0079435 2791406 2738673 383558410 1340872 0387092 0249442 0118379 2180379 2252363 316099212 1088400 0541510 0309461 0161276 1714381 1897617 2678301

Table 3 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 and Ma is varied

Ma 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1955885 0228035 0131483 0047797 2776933 321537 446398815 1629020 0252600 0211365 0092719 2802801 2590747 364089530 1370781 0283080 0299672 0154063 2866680 2089107 2983500

Table 4 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with S = 08 Ma = 11198722= 1 Pr = 1 and Sc = 18 and119872

1is varied

1198721

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1895449 0296754 0143799 0054085 2978572 3080157 42856355 1987330 0335046 0123865 0044928 3120442 324009 44987058 2089998 0380759 0106127 0036553 3279882 3415187 4733832

Table 5 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1 Pr = 1 and Sc = 18 and119872

2is varied

1198722

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1823807 0265936 0162696 0062564 2870952 295405 41202615 1870447 0284213 0148046 0056928 2942314 303796 42296068 1927131 030837 0135444 0050747 3028503 3136665 4360637

Table 6 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Sc = 18 and Pr is varied

Pr 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844352 170801 0216269 0065849 0079270 2701304 4083229 38517395 1675911 0203976 0009869 0084604 2654055 6569892 377576610 1671722 020159 0000697 0085353 2648593 9447705 3766435

Table 7 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Pr = 1 and Sc is varied

Sc 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1884207 0289866 0144404 0313734 2963305 3062276 20119231 1810615 0259498 0163349 0160799 2852006 2932347 293350315 177745 0245319 0170494 0093966 280298 2874491 36259918 1765648 0242329 0178412 0070413 2783885 2850602 3984435

ℎ119899(0) = 0

ℎ1015840

119899(0) = 0

ℎ10158401015840

119899(0) = 0

119899 = 0 1 2

120603119899(0) = 0

1206031015840

119899(0) = 0

12060310158401015840

119899(0) = 0

119899 = 0 1 2

(20)

6 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

f998400 (120578)

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120579(120578)

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120601(120578)

(c)

Figure 2 (a) Velocity profiles for different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of 119878 with Ma = 12

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

Let 119878 = 08 and Ma = 1 and we can obtain

119891 (120578) = 120578 +1

212057211205782+7

151205741205783+2

1512057211205741205784

+ (1

1201205721

2120574 +

91

15001205742) 1205785+1

60120572112057421205786

+ (23

252001205721

21205742+

77

150001205743) 1205787

+ (589

50400012057211205743minus

77

403201205721

31205742) 1205788

+ (383

90720001205721

21205743+

3871

162000001205744) 1205789

+ (1

4320001205721

31205743+

209

756000012057211205744) 12057810+ sdot sdot sdot

1198911015840(120578) = 1 + 120572

1120578 +

7

451205741205782+1

3012057211205741205783

+ (1

6001205721

2120574 +

91

45001205742) 1205784+

1

360120572112057421205785

+ (23

1764001205721

21205742+

11

150001205743) 1205786

+ (589

467200012057211205743minus

77

3225601205721

31205742) 1205787

+ (383

816480001205721

21205743+

3871

1458000001205744) 1205788

+ (1

43200001205721

31205743+

209

7560000012057211205744) 1205789+ sdot sdot sdot

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 1 Comparison of values of film thickness 120573 and skin frictioncoefficient minus11989110158401015840(0) with Ma = 0

119878Wang [19] Present results

120573 minus11989110158401015840(0) 120573 minus119891

10158401015840(0)

08 2151990 2680940 2152839 268192010 1543620 1972385 1543620 197238512 1127781 1442631 1127781 144262514 0821032 1012784 0821032 1012780

Substituting (14) into (13) then we get

119862119891=2

12057311989110158401015840(0)Re

119909

minus12

Nu119909=

1

2120573 (1 minus 120572119905)1205791015840(0)Re

119909

32

(15)

where Re119909= 119880119904119909120592 is the local Reynolds number

23 Solution Approach In order to solve (10) We used thedouble-parameter transformation perturbation expansionmethod that confirmed the validity by Zhang and Zheng [11]We introduced an artificial small parameter 120576 the equationscan be expanded to the form of power series We can obtainanalytical solutions in the form of series by comparing thecoefficients of same power and solving the equation sets

We transform the dependent variable and independentvariable as follows

119891 (120578) = 120576119892 (120585) + 120578 +1205721

21205782

= 120576119892 (120585) + 12057613120585 +

1205721

2120576231205852

120579 (120578) = 12057623ℎ (120585) + 1 + 120572

2120578 = 12057623ℎ (120585) + 1 + 120572

212057613120585

120601 (120578) = 12057623120603 (120585) + 1 + 120572

3120578 = 12057623120603 (120585) + 1 + 120572

312057613120585

(16)

where 120585 = 120576minus13

120578 and 120576 is an artificial small parameter Weassume 11989110158401015840(0) = 120572

1 1205791015840(0) = 120572

2 and 1206011015840(0) = 120572

3 here 120572

1 1205722

and 1205723are constants the boundary conditions become

119892 (0) = 0

1198921015840(0) = 0

11989210158401015840(0) = 0

ℎ (0) = 0

ℎ1015840(0) = 0

120603 (0) = 0

1206031015840(0) = 0

(17)

Equation (10) reduces to

119892101584010158401015840(120585) + 120574 ((120576119892 (120585) + 120576

13120585 +

1205721

2120576231205852) (1205761311989210158401015840+ 1205721)

minus1

211987812057613120585 (1205761311989210158401015840+ 1205721) minus (120576

231198921015840+ 1 + 120572

112057613120585)2

minus (119878 +Ma) (120576231198921015840 + 1 + 120572112057613120585)) = 0

ℎ10158401015840(120585)

Pr+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (12057613ℎ1015840(120585) + 120572

2) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623ℎ (120585) + 1 + 120572

212057613120585) minus

1

2

sdot 11987812057613120585 (12057613ℎ1015840(120585) + 120572

2)

minus3119878

2(12057623ℎ (120585) + 1 + 120572

212057613120585)) = 0

12060310158401015840(120585)

Sc+ 120574 ((120576119892

0(120585) + 120576

13120585 +

1205721

2120576231205852)

sdot (120576131206031015840(120585) + 120572

3) minus (2 (120576

231198921015840(120585) + 1 + 120572

112057613120585))

sdot (12057623120603 (120585) + 1 + 120572

312057613120585) minus

1

2

sdot 11987812057613120585 (120576131206031015840(120585) + 120572

3)

minus3119878

2(12057623120603 (120585) + 1 + 120572

312057613120585)) = 0

(18)

And we can assume

119892 (120585) = 1198920(120585) + 120576

131198921(120585) + 120576

231198922(120585) + 120576119892

3(120585)

+ 120576431198924(120585) + sdot sdot sdot

ℎ (120585) = ℎ0(120585) + 120576

13ℎ1(120585) + 120576

23ℎ2(120585) + 120576ℎ

3(120585)

+ 12057643ℎ4(120585) + sdot sdot sdot

120603 (120585) = 1206030(120585) + 120576

131206031(120585) + 120576

231206032(120585) + 120576120603

3(120585)

+ 120576431206034(120585) + sdot sdot sdot

(19)

with the boundary conditions

119892119899(0) = 0

1198921015840

119899(0) = 0

11989210158401015840

119899(0) = 0

119892101584010158401015840

119899(0) = 0

119899 = 0 1 2

Mathematical Problems in Engineering 5

Table 2 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 and S is varied

119878 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

08 1706139 0245858 0190409 0079435 2791406 2738673 383558410 1340872 0387092 0249442 0118379 2180379 2252363 316099212 1088400 0541510 0309461 0161276 1714381 1897617 2678301

Table 3 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 and Ma is varied

Ma 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1955885 0228035 0131483 0047797 2776933 321537 446398815 1629020 0252600 0211365 0092719 2802801 2590747 364089530 1370781 0283080 0299672 0154063 2866680 2089107 2983500

Table 4 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with S = 08 Ma = 11198722= 1 Pr = 1 and Sc = 18 and119872

1is varied

1198721

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1895449 0296754 0143799 0054085 2978572 3080157 42856355 1987330 0335046 0123865 0044928 3120442 324009 44987058 2089998 0380759 0106127 0036553 3279882 3415187 4733832

Table 5 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1 Pr = 1 and Sc = 18 and119872

2is varied

1198722

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1823807 0265936 0162696 0062564 2870952 295405 41202615 1870447 0284213 0148046 0056928 2942314 303796 42296068 1927131 030837 0135444 0050747 3028503 3136665 4360637

Table 6 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Sc = 18 and Pr is varied

Pr 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844352 170801 0216269 0065849 0079270 2701304 4083229 38517395 1675911 0203976 0009869 0084604 2654055 6569892 377576610 1671722 020159 0000697 0085353 2648593 9447705 3766435

Table 7 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Pr = 1 and Sc is varied

Sc 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1884207 0289866 0144404 0313734 2963305 3062276 20119231 1810615 0259498 0163349 0160799 2852006 2932347 293350315 177745 0245319 0170494 0093966 280298 2874491 36259918 1765648 0242329 0178412 0070413 2783885 2850602 3984435

ℎ119899(0) = 0

ℎ1015840

119899(0) = 0

ℎ10158401015840

119899(0) = 0

119899 = 0 1 2

120603119899(0) = 0

1206031015840

119899(0) = 0

12060310158401015840

119899(0) = 0

119899 = 0 1 2

(20)

6 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

f998400 (120578)

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120579(120578)

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120601(120578)

(c)

Figure 2 (a) Velocity profiles for different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of 119878 with Ma = 12

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

Let 119878 = 08 and Ma = 1 and we can obtain

119891 (120578) = 120578 +1

212057211205782+7

151205741205783+2

1512057211205741205784

+ (1

1201205721

2120574 +

91

15001205742) 1205785+1

60120572112057421205786

+ (23

252001205721

21205742+

77

150001205743) 1205787

+ (589

50400012057211205743minus

77

403201205721

31205742) 1205788

+ (383

90720001205721

21205743+

3871

162000001205744) 1205789

+ (1

4320001205721

31205743+

209

756000012057211205744) 12057810+ sdot sdot sdot

1198911015840(120578) = 1 + 120572

1120578 +

7

451205741205782+1

3012057211205741205783

+ (1

6001205721

2120574 +

91

45001205742) 1205784+

1

360120572112057421205785

+ (23

1764001205721

21205742+

11

150001205743) 1205786

+ (589

467200012057211205743minus

77

3225601205721

31205742) 1205787

+ (383

816480001205721

21205743+

3871

1458000001205744) 1205788

+ (1

43200001205721

31205743+

209

7560000012057211205744) 1205789+ sdot sdot sdot

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 2 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 and S is varied

119878 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

08 1706139 0245858 0190409 0079435 2791406 2738673 383558410 1340872 0387092 0249442 0118379 2180379 2252363 316099212 1088400 0541510 0309461 0161276 1714381 1897617 2678301

Table 3 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 and Ma is varied

Ma 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1955885 0228035 0131483 0047797 2776933 321537 446398815 1629020 0252600 0211365 0092719 2802801 2590747 364089530 1370781 0283080 0299672 0154063 2866680 2089107 2983500

Table 4 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with S = 08 Ma = 11198722= 1 Pr = 1 and Sc = 18 and119872

1is varied

1198721

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1895449 0296754 0143799 0054085 2978572 3080157 42856355 1987330 0335046 0123865 0044928 3120442 324009 44987058 2089998 0380759 0106127 0036553 3279882 3415187 4733832

Table 5 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1 Pr = 1 and Sc = 18 and119872

2is varied

1198722

120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844353 1823807 0265936 0162696 0062564 2870952 295405 41202615 1870447 0284213 0148046 0056928 2942314 303796 42296068 1927131 030837 0135444 0050747 3028503 3136665 4360637

Table 6 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Sc = 18 and Pr is varied

Pr 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

1 1765648 0242329 0178412 0070413 2783885 2850602 39844352 170801 0216269 0065849 0079270 2701304 4083229 38517395 1675911 0203976 0009869 0084604 2654055 6569892 377576610 1671722 020159 0000697 0085353 2648593 9447705 3766435

Table 7 Values of 120573 1198911015840(1) 120579(1) and 120601(1) minus11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 119878 = 08 Ma = 11198721= 1119872

2= 1 and Pr = 1 and Sc is varied

Sc 120573 1198911015840(1) 120579(1) 120601(1) minus119891

10158401015840(0) minus120579

1015840(0) minus120601

1015840(0)

05 1884207 0289866 0144404 0313734 2963305 3062276 20119231 1810615 0259498 0163349 0160799 2852006 2932347 293350315 177745 0245319 0170494 0093966 280298 2874491 36259918 1765648 0242329 0178412 0070413 2783885 2850602 3984435

ℎ119899(0) = 0

ℎ1015840

119899(0) = 0

ℎ10158401015840

119899(0) = 0

119899 = 0 1 2

120603119899(0) = 0

1206031015840

119899(0) = 0

12060310158401015840

119899(0) = 0

119899 = 0 1 2

(20)

6 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

f998400 (120578)

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120579(120578)

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120601(120578)

(c)

Figure 2 (a) Velocity profiles for different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of 119878 with Ma = 12

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

Let 119878 = 08 and Ma = 1 and we can obtain

119891 (120578) = 120578 +1

212057211205782+7

151205741205783+2

1512057211205741205784

+ (1

1201205721

2120574 +

91

15001205742) 1205785+1

60120572112057421205786

+ (23

252001205721

21205742+

77

150001205743) 1205787

+ (589

50400012057211205743minus

77

403201205721

31205742) 1205788

+ (383

90720001205721

21205743+

3871

162000001205744) 1205789

+ (1

4320001205721

31205743+

209

756000012057211205744) 12057810+ sdot sdot sdot

1198911015840(120578) = 1 + 120572

1120578 +

7

451205741205782+1

3012057211205741205783

+ (1

6001205721

2120574 +

91

45001205742) 1205784+

1

360120572112057421205785

+ (23

1764001205721

21205742+

11

150001205743) 1205786

+ (589

467200012057211205743minus

77

3225601205721

31205742) 1205787

+ (383

816480001205721

21205743+

3871

1458000001205744) 1205788

+ (1

43200001205721

31205743+

209

7560000012057211205744) 1205789+ sdot sdot sdot

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

f998400 (120578)

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120579(120578)

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

S = 08

S = 10

S = 12

120601(120578)

(c)

Figure 2 (a) Velocity profiles for different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of 119878 with Ma = 121198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of 119878 with Ma = 12

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

Let 119878 = 08 and Ma = 1 and we can obtain

119891 (120578) = 120578 +1

212057211205782+7

151205741205783+2

1512057211205741205784

+ (1

1201205721

2120574 +

91

15001205742) 1205785+1

60120572112057421205786

+ (23

252001205721

21205742+

77

150001205743) 1205787

+ (589

50400012057211205743minus

77

403201205721

31205742) 1205788

+ (383

90720001205721

21205743+

3871

162000001205744) 1205789

+ (1

4320001205721

31205743+

209

756000012057211205744) 12057810+ sdot sdot sdot

1198911015840(120578) = 1 + 120572

1120578 +

7

451205741205782+1

3012057211205741205783

+ (1

6001205721

2120574 +

91

45001205742) 1205784+

1

360120572112057421205785

+ (23

1764001205721

21205742+

11

150001205743) 1205786

+ (589

467200012057211205743minus

77

3225601205721

31205742) 1205787

+ (383

816480001205721

21205743+

3871

1458000001205744) 1205788

+ (1

43200001205721

31205743+

209

7560000012057211205744) 1205789+ sdot sdot sdot

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

Ma = 05

Ma = 15

Ma = 3

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Ma = 05

Ma = 15

Ma = 3

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Ma = 05

Ma = 15

Ma = 3

(c)

Figure 3 (a) Velocity profiles for different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of Ma with 119878 = 081198721= 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of Ma with 119878 = 08

1198721= 1119872

2= 1 Pr = 1 and Sc = 18

11989110158401015840(120578) = 120572

1+7

90120574120578 +

1

9012057211205741205782

+ (1

24001205721

2120574 +

91

180001205742) 1205783+

1

1800120572112057421205784

+ (23

10584001205721

21205742+

11

900001205743) 1205785

+ (589

3480400012057211205743minus

77

22579201205721

31205742) 1205786

+ (383

6531840001205721

21205743+

3871

1164000001205744) 1205787

+ (1

388800001205721

31205743+

209

68040000012057211205744) 1205788+ sdot sdot sdot

120579 (120578) = 1205722120578 +

8

51205741205782+ (

13

301205741205722+1

31205721120574) 1205783

+ (1

21205742+1

812057412057221205721) 1205784

+ (147

100012057421205722+47

30012057211205742) 1205785

+ (1

12012057421205721

2+599

45001205743+1

24120574212057211205722) 1205786

+ (989

2100012057211205743+2533

900012057431205722+

1

560120574212057221205721

2) 1205787

+ sdot sdot sdot

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M1 = 1

M1 = 3

M1 = 5

M1 = 8

(c)

Figure 4 (a) Velocity profiles for different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198721with 119878 = 08 Ma = 1119872

2= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

1with 119878 = 08

Ma = 11198722= 1 Pr = 1 and Sc = 18

1205791015840(120578) = 120572

2+4

5120574120578 + (

13

901205741205722+1

91205721120574) 1205782

+ (1

81205742+1

3212057412057211205722) 1205783

+ (147

500012057421205722+

47

150012057211205742) 1205784

+ (1

72012057421205721

2+

599

270001205743+

1

144120574212057211205722) 1205785

+ (989

14700012057211205743+2533

6300012057431205722+

1

3920120574212057221205721

2) 1205786

+ sdot sdot sdot

120601 (120578) = 1 + 1205723120578 +

16

51205741205782+ (

13

151205741205723+2

31205721120574) 1205783

+ (23

151205742+1

412057412057211205723) 1205784

+ (133

37512057421205723+13

2512057421205721) 1205785

+ (1171

22501205743+1

3612057421205721

2+47

450120574212057211205723) 1205786

+ (1

280120574212057231205721

2+1424

787512057431205721+1961

2250012057431205723) 1205787

+ sdot sdot sdot

1206011015840(120578) = 120572

3+8

5120574120578 + (

13

451205741205723+2

91205721120574) 1205782

+ (23

601205742+1

1612057412057211205723) 1205783

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

f998400 (120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(a)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(b)

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

M2 = 1

M2 = 3

M2 = 5

M2 = 8

(c)

Figure 5 (a) Velocity profiles for different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (b) Temperature profiles for

different values of1198722with 119878 = 08 Ma = 1119872

1= 1 Pr = 1 and Sc = 18 (c) Concentration profiles for different values of119872

2with 119878 = 08

Ma = 11198721= 1 Pr = 1 and Sc = 18

+ (133

187512057421205723+13

12512057421205721) 1205784

+ (1171

135001205743+

1

21612057421205721

2+

47

2700120574212057211205723) 1205785

+ (1

1960120574212057231205721

2+1424

2212512057431205721+

1961

15750012057431205723) 1205786

+ sdot sdot sdot

(21)

Substituting (21) into (12) we can get four equations involvingthe four variables such as 120572

1 1205722 1205723 and 120574 Analytical

solutions of (10)ndash(12) in the form of series can be obtainedwith this method

3 Results and Discussion

The effects of physical parameters on velocity temperatureand concentration fields are presented in Tables 1ndash7 respec-tively Based on Table 1 the values of dimensionless filmthickness 120573 and skin friction coefficient minus11989110158401015840(0) obtainedare compared with the results by Wang [19] for the caseMa = 0 and good agreement can be seen From Table 2it can be observed that increasing the value of unsteadinessparameter 119878 will decrease the film thickness 120573 the skinfriction coefficient minus11989110158401015840(0) the heat flux minus120579

1015840(0) and the

diffusion flux minus1206011015840(0) but increase the free velocity 1198911015840(1) freetemperature 120579(1) and free concentration 120601(1) In Table 3as the magnetic parameter Ma increases the hydrodynamicsbehavior is the same as that in Table 2 except for the skinfriction coefficient minus11989110158401015840(0)

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120579(120578)

Pr = 1

Pr = 2

Pr = 5

Pr = 10

Figure 6 Temperature profiles for different values of Pr with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Sc = 18

120578

1

09

08

07

06

05

04

03

02

01

010908070605040302010

120601(120578)

Sc = 05

Sc = 1

Sc = 15

Sc = 18

Figure 7 Concentration profiles for different values of Sc with 119878 =08 Ma = 1119872

1= 1119872

2= 1 and Pr = 1

From Tables 4 and 5 it can be seen that1198721and119872

2have

the same effects on velocity temperature and concentrationfields Increasing the value of the thermocapillary number119872

1

or solutal capillary number1198722can cause a descent in the free

temperature 120579(1) and free concentration 120601(1) but a rise in thefilm thickness 120573 the skin friction coefficient minus11989110158401015840(0) the heatflux minus1205791015840(0) and the free velocity 1198911015840(1) Table 6 shows thatthe values of the free concentration 120601(1) and the heat fluxminus1205791015840(0) increase while the film thickness 120573 the skin friction

coefficient minus11989110158401015840(0) the diffusion flux minus1206011015840(0) the free velocity1198911015840(0) and free temperature 120579(1) decrease with an increment

in the Prandtl number Pr Table 7 shows the effect of Schmidtnumber Sc on velocity temperature and concentration fields

The velocity temperature and concentration profiles forthe hydromagnetic flow in a thin film over an unsteadystretching sheet with thermocapillary are presented graph-ically in Figures 2ndash7 Figures 2(a)ndash2(c) show the impacts

of unsteadiness parameter on the velocity temperature andconcentration respectively We can find the enhancement inthe velocity is more remarkable compared with the temper-ature and concentration when increment of 119878 is the sameAnd unsteadiness parameter has the similar effects on thetemperature and concentration fields Figures 3(a)ndash3(c) showthe effects of Hartmann number on the velocity temperatureand concentration fields which are the same as Table 3 FromFigure 3(a) it can be seen that when the Hartmann numberMa increases at the beginning the velocity decreases slightlydue to the fact that magnetic field produces a drag and thenthe velocity near the surface of the thin film rises up graduallyunder theMarangoni effect which explains the emergence ofthe intersection Meanwhile the temperature and concentra-tion rise significantly andmonotonously as Ma increases (seeFigures 3(b) and 3(c)) It can be proved that themagnetic fieldpromotes heat and mass transfer in a thin liquid film

Figures 4(a) and 5(a) show the velocity profiles for differ-ent values of thermocapillary number 119872

1and solutal capil-

lary number1198722 Similar to Figure 3(a) with increasing value

of thermocapillary number or solutal capillary number onecan see that the velocity decreases initially mainly due to themagnetic field effect then the velocity near the surface of thethin film obviously rises up mainly because of theMarangonieffect furthermore the two factors are balanced at theintersection The effects of thermocapillary number 119872

1on

the temperature and concentration field are shown in Figures4(b) and 4(c) Results demonstrate that the temperature andconcentration both reducewhen the thermocapillary numberincreases Figures 5(b) and 5(c) illustrate that the fluid tem-perature and concentration are the decreasing functions ofsolutal capillary number The lower value of thermocapillarynumber has higher temperature and concentration Further-more decrement in the temperature and concentration pro-files due to thermocapillary number119872

1is more remarkable

in comparison to the solutal capillary number1198722

The effect of Pr on temperature is presented in Figure 6It can be observed that heat transfer behaviors are stronglydependent on the value of the Prandtl number Figure 6 showsthat the temperature of the fluid decreases monotonouslywith the increasing of Pr That is to say Pr plays an oppositeeffect on the temperature field compared with 119878 and Ma Inaddition the effect of Schmidt number on the concentrationdistribution is displayed in Figure 7 Increasing the Schmidtnumber causes a decrease in the concentration profile and itis clear that weakermass transfer ratio is obtained with biggerSc values

4 Conclusions

In this study a similarity analysis for thermosolutal capillarityand a magnetic field in a thin liquid film on an unsteadyelastic stretching sheet has been studied The effects ofvarious relevant parameters on the velocity temperature andconcentration fields have been discussed The main points ofpresented analysis are listed below

(1) The thermocapillary number and solutal capillarynumber both restrict the heat and mass transfer

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Furthermore the effects of the thermocapillary num-ber on the temperature and concentration are moreremarkable than solutal capillary number

(2) The velocity temperature and concentration allincrease for the higher values of unsteadiness param-eter

(3) Increasing Prandtl number can reduce the fluid tem-perature and increasing Schmidt number causes adescent in the concentration

(4) Marangoni convection caused by surface-tension gra-dient creates a promotion in flow near the surfaceregion of the film

Nomenclature

b Positive constant119905 Time (s)119906 119909-component of velocity (msminus1)V 119910-component of velocity (msminus1)119909 Horizontal coordinates (m)119910 Vertical coordinate (m)119891 Velocity similarity variable119863 Mass diffusivity (m2sminus1)119861 Magnetic field (T)1198610 Initial magnetic field (T)

Ma Hartmann number1198721 Thermocapillary number

1198722 Solutal capillary number

Pr Prandtl number119878 Dimensionless unsteadiness parameter119880119904 Stretching surface velocity

119879 Temperature (K)1198790 Temperature at slit (K)

119879119904 Temperature of slit (K)

119879ref Reference temperature (K)119862 Concentration (kgmminus3)1198620 Concentration at slit (kgmminus3)

119862119904 Concentration of slit (kgmminus3)

119862ref Reference concentration (kgmminus3)

Greek Symbols

120572 1205721 1205722 1205723 Constants

120573 Dimensionless film thickness120581 Thermal diffusivity (m2sminus1) Electrical conductivity (s (cm)minus1)120578 Similarity variable120583 Dynamic viscosity (Nsmminus2)120592 Kinematic viscosity (m2sminus1)120574 Dimensionless film thickness120579 Temperature similarity variable120601 Concentration similarity variable120576 Artificial small parameter120588 Density (kgmminus3)120590 Surface tension (Nmminus1)

Subscripts

119879 Thermal quantity119862 Solutal quantity0 At the slit119904 At the elastic sheet

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (nos 21206009 and 21576023) and FundingProject for Academic Human Resources Development inBeijing University of Civil Engineering and Architecture (no21221214111)

References

[1] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[2] S Munawar A Mehmood and A Ali ldquoTime-dependent flowand heat transfer over a stretching cylinderrdquo Chinese Journal ofPhysics vol 50 no 5 pp 828ndash848 2012

[3] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism amp Magnetic Materials vol 397 pp 108ndash114 2016

[4] C Y Wang ldquoThe threeminusdimensional flow due to a stretchingflat surfacerdquo The Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[5] S Munawar A Mehmood and A Ali ldquoThree-dimensionalsqueezing flow in a rotating channel of lower stretching porouswallrdquo Computers amp Mathematics with Applications vol 64 no6 pp 1575ndash1586 2012

[6] M E Ali and EMagyari ldquoUnsteady fluid and heat flow inducedby a submerged stretching surface while its steady motion isslowed down graduallyrdquo International Journal of Heat and MassTransfer vol 50 no 1-2 pp 188ndash195 2007

[7] M M Rashidi M Ali B Rostami P Rostami and G-N XieldquoHeat and mass transfer for MHD viscoelastic fluid flow overa vertical stretching sheet with considering Soret and DufoureffectsrdquoMathematical Problems in Engineering vol 2015 ArticleID 861065 12 pages 2015

[8] P V S Narayana and D H Babu ldquoNumerical study of MHDheat and mass transfer of a Jeffrey fluid over a stretching sheetwith chemical reaction and thermal radiationrdquo Journal of theTaiwan Institute of Chemical Engineers vol 59 pp 18ndash25 2016

[9] A Mehmood S Munawar and A Ali ldquoCooling of a hotstretching surface in the presence of across mass transferphenomenon in a channel flowrdquo Zeitschrift fur NaturforschungA vol 69 no 1-2 pp 34ndash42 2014

[10] S A Shehzad T Hayat and A Alsaedi ldquoThree-dimensionalMHD flow of Casson fluid in porous medium with heat

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

generationrdquo Journal of Applied Fluid Mechanics vol 9 no 1 pp215ndash223 2016

[11] Y Zhang and L C Zheng ldquoAnalysis of MHD thermosolutalMarangoni convection with the heat generation and a first-order chemical reactionrdquo Chemical Engineering Science vol 69no 1 pp 449ndash455 2012

[12] T Hayat A Safdar M Awais and S Mesloub ldquoSoret andDufour effects for three-dimensional flow in a viscoelastic fluidover a stretching surfacerdquo International Journal of Heat andMass Transfer vol 55 no 7-8 pp 2129ndash2136 2012

[13] S A Shehzad Z Abdullah F M Abbasi T Hayat and AAlsaedi ldquoMagnetic field effect in three-dimensional flow ofan Oldroyd-B nanofluid over a radiative surfacerdquo Journal ofMagnetism ampMagnetic Materials vol 399 pp 97ndash108 2016

[14] AMastroberardino ldquoAccurate solutions for viscoelastic bound-ary layer flow and heat transfer over stretching sheetrdquo AppliedMathematics and Mechanics vol 35 no 2 pp 133ndash142 2014

[15] J A Khan M Mustafa T Hayat and A Alsaedi ldquoThree-dimensional flow of nanofluid over a non-linearly stretchingsheet an application to solar energyrdquo International Journal ofHeat and Mass Transfer vol 86 pp 158ndash164 2015

[16] S A Shehzad F M Abbasi T Hayat F Alsaadi and G MousaldquoPeristalsis in a curved channel with slip condition and radialmagnetic fieldrdquo International Journal of Heat andMass Transfervol 91 pp 562ndash569 2015

[17] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[18] S M Roberts and J S Shipman Two Point Boundary ValueProblems ShootingMethods Elsevier NewYork NYUSA 1972

[19] C Wang ldquoAnalytic solutions for a liquid film on an unsteadystretching surfacerdquo Heat and Mass Transfer vol 42 no 8 pp759ndash766 2006

[20] M M Nandeppanavar K Vajravelu M Subhas Abel S Raviand H Jyoti ldquoHeat transfer in a liquid film over an unsteadystretching sheetrdquo International Journal of Heat and Mass Trans-fer vol 55 no 4 pp 1316ndash1324 2012

[21] Y Zhang M Zhang and Y Bai ldquoFlow and heat transfer ofan Oldroyd-B nanofluid thin film over an unsteady stretchingsheetrdquo Journal of Molecular Liquids vol 220 pp 665ndash670 2016

[22] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[23] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[24] B S Dandapat B Santra and K Vajravelu ldquoThe effects ofvariable fluid properties and thermocapillarity on the flow of athin film on an unsteady stretching sheetrdquo International Journalof Heat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[25] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[26] C Wang and I Pop ldquoAnalysis of the flow of a power-law fluidfilm on an unsteady stretching surface by means of homotopyanalysis methodrdquo Journal of Non-Newtonian Fluid Mechanicsvol 138 no 2-3 pp 161ndash172 2006

[27] C-H Chen ldquoEffect of viscous dissipation on heat transfer in anon-Newtonian liquid film over an unsteady stretching sheetrdquoJournal of Non-Newtonian Fluid Mechanics vol 135 no 2-3 pp128ndash135 2006

[28] N F Noor O Abdulaziz and I Hashim ldquoMHD flow and heattransfer in a thin liquid film on an unsteady stretching sheetby the homotopy analysis methodrdquo International Journal forNumerical Methods in Fluids vol 63 no 3 pp 357ndash373 2010

[29] M S Abel N Mahesha and J Tawade ldquoHeat transfer in aliquid film over an unsteady stretching surface with viscousdissipation in presence of external magnetic fieldrdquo AppliedMathematical Modelling vol 33 no 8 pp 3430ndash3441 2009

[30] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[31] N F M Noor and I Hashim ldquoThermocapillarity and magneticfield effects in a thin liquid film on an unsteady stretchingsurfacerdquo International Journal of Heat and Mass Transfer vol53 no 9-10 pp 2044ndash2051 2010

[32] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[33] Y H Lin L C Zheng and X X Zhang ldquoRadiation effects onMarangoni convection flow and heat transfer in pseudo-plasticnon-Newtonian nanofluids with variable thermal conductivityrdquoInternational Journal of Heat and Mass Transfer vol 77 no 4pp 708ndash716 2014

[34] Y H Lin L C Zheng X X Zhang L X Ma and G ChenldquoMHD pseudo-plastic nanofluid unsteady flow and heat trans-fer in a finite thin film over stretching surface with internal heatgenerationrdquo International Journal of Heat and Mass Transfervol 84 pp 903ndash911 2015

[35] E Lim and Y M Hung ldquoThermophysical phenomena ofworking fluids of thermocapillary convection in evaporatingthin liquid filmsrdquo International Communications in Heat andMass Transfer vol 66 pp 203ndash211 2015

[36] D M Christopher and B-X Wang ldquoSimilarity simulationfor Marangoni convection around a vapor bubble duringnucleation and growthrdquo International Journal of Heat and MassTransfer vol 44 no 4 pp 799ndash810 2001

[37] S Maity Y Ghatani and B S Dandapat ldquoThermocapillary flowof a thin nanoliquid film over an unsteady stretching sheetrdquoJournal of Heat Transfer vol 138 no 4 Article ID 042401 2016

[38] E A Chinnov ldquoWave-thermocapillary effects in heated liquidfilms at high Reynolds numbersrdquo International Journal of Heatand Mass Transfer vol 71 no 1 pp 106ndash116 2014

[39] H Schichting Boundary Layer Theory McGraw-Hill NewYork NY USA 6th edition 1964

[40] K Arafune and A Hirata ldquoThermal and solutal marangoniconvection in In-Ga-Sb systemrdquo Journal of Crystal Growth vol197 no 4 pp 811ndash817 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of