Research Article Influence of Slip Condition on Unsteady ...downloads.hindawi.com/journals/aaa/2015/327975.pdf · Department of Mathematical Sciences, Fac ulty of Science, Universiti

Research ArticleInfluence of Slip Condition on Unsteady Free Convection Flowof Viscous Fluid with Ramped Wall Temperature

Sami Ul Haq1 Ilyas Khan2 Farhad Ali1 Arshad Khan3 and Tarek Nabil AhmedAbdelhameed24

1Department of Mathematics City University of Science and Information Technology Peshawar 25000 Pakistan2Department of Basic Sciences College of Engineering Majmaah University PO Box 66 Majmaah 11952 Saudi Arabia3Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia (UTM) 81310 Skudai Malaysia4Mathematics Department Faculty of Science Beni-Suef University Beni-Suef 62514 Egypt

Correspondence should be addressed to Sami Ul Haq samiulhaqmathsyahoocom

Received 27 June 2014 Revised 16 November 2014 Accepted 24 November 2014

Academic Editor Saeed Islam

Copyright copy 2015 Sami Ul Haq 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 objective of this study is to explore the influence of wall slip condition on a free convection flow of an incompressible viscousfluid with heat transfer and ramped wall temperature Exact solution of the problem is obtained by using Laplace transformtechnique Graphical results to see the effects of Prandtl number Pr time 119905 and slip parameter 120578 on velocity and skin frictionfor the case of ramped and constant temperature of the plate are provided and discussed

1 Introduction

Free convection flow occurs due to a buoyancy-inducedmotion resulting from the body forces acting on densitygradients and is particularly important in atmospheric andoceanic circulation in the problems of heat rejection andremoval in many devices in the design of spaceships andfiltration process Siegel [1] in his pioneering work studiedthe unsteady free convection flow past a semi-infinite verticalplate with uniform temperature However many practicalproblems usually require wall conditions which are nonuni-form or arbitrary In order to understand such problems it isimportant to investigate problems subject to a step change inthe wall temperature An early attempt was made by Schetz[2] by developing an approximate analytical model LaterHayday et al [3] used a numerical approach Of these worksMalhotra et al [4] mentioned that in the fabrication of thin-film photovoltaic devices ramped wall temperatures can beemployed to control the temperature uniformity of the sys-tem Periodic temperature step changes are also important inbuilding heat transfer applications for example in air condi-tioning where the conventional assumption of periodic out-door conditions may lead to considerable errors in the case

of a significant temporary deviation of the temperature fromperiodicity as discussed by Antonopoulos and Democritou[5] Due to the aforementioned significance of step changein the wall temperature Chandran et al [6] presented ananalytical solution to the unsteady natural convection flowof an incompressible viscous flow near a vertical plate withramped wall temperature Seth et al [7] elaborated theunsteady hydromagnetic natural convection flow of a viscousincompressible electrically conducting fluid with radiativeheat transfer near an impulsively moving vertical flat plateembedded in a porous medium with ramped wall tempera-ture Narahari et al [8] investigated mass transfer effects onfree convection flow past an infinite vertical plate subject todiscontinuous or nonuniform wall temperature conditions

The above achievements aremade with the assumption ofno-slip condition between the wall and the fluid The effectsof fluid slippage at the wall appear in many applications suchas in microchannels or nanochannels and in applicationswhere a thin film of light oil is attached to the moving platesor when the surface is coated with special coatings such asthickmonolayer of hydrophobic octadecyltrichlorosilane [9]The fluid problem in the slip flow regime is very importantin the era of modern science technology and vast ranging

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 327975 7 pageshttpdxdoiorg1011552015327975

2 Abstract and Applied Analysis

industrialization In view of such applications Makinde andOsausi [10] studied the combined effect of magnetic fieldand permeable wall slip velocity on the steady flow of anelectrically conducting fluid in a channel of uniform widthMakinde and Mhone [11] investigated the combined effectof a transverse magnetic field and radiative heat transfer tounsteady flow of a conducting optically thin fluid througha channel filled with saturated porous medium and nonuni-form walls temperature Mehmood and Ali [12] extended thework ofMakinde andMhone [11] by considering the fluid slipat the lower wall Few other attempts taking into account theslip boundary condition are [13ndash16] However the literaturelacks studies that take into consideration the combined effectof slippage and ramped temperature at the wall on theunsteady free convection flow of a viscous incompressiblefluid near a vertical flat plate This is the source of motivationto study the influence of slip condition on the unsteadyfree convection transient flow near a vertical flat plate withramped wall temperature

2 Formulation of the Problem and Solution

Let us consider the flow of an incompressible viscous fluidnear an infinite vertical plate The 1199091015840-axis is taken along thewall in the upward direction and 1199101015840-axis is taken perpendic-ular to it into the fluid At the initial moment 1199051015840 = 0 boththe plate and the fluid are at rest at a constant temperature1198791015840

infin At time 1199051015840 = 0

+ the temperature of the plate is raisedor lowered to 1198791015840

infin+ (1198791015840

119908minus 1198791015840

infin)(11990510158401199050) and then for 1199051015840 gt 119905

0

the temperature is maintained at the constant temperature1198791015840

119908 In view of the above assumptions as well as of the usual

Boussinesqrsquos approximation the governing equations reduceto those obtained by Chandran et al [6 Equations (1) and(2)]

1205971199061015840

1205971199051015840= ]

12059721199061015840

12059711991010158402+ 119892120573 (119879

1015840minus 1198791015840

infin)

1205971198791015840

1205971199051015840=

119896

120588119888119901

12059721198791015840

12059711991010158402

(1)

where 1199061015840 1198791015840 ] 120588 119892 120573 119896 and 119888119901are respectively the velocity

in the 1199091015840 direction temperature of the fluid kinematic

viscosity fluid density acceleration due to gravity volumetriccoefficient of thermal expansion thermal conductivity andspecific heat at constant pressure

The appropriate initial and boundary conditions are

1199061015840(1199101015840 0) = 0 119879

1015840(1199101015840 0) = 119879

1015840

infinfor 1199101015840 ge 0

1199061015840(0 1199051015840) minus 120582

1205971199061015840(0 1199051015840)

1205971199101015840

= 0 for 1199051015840 gt 0

1198791015840(0 1199051015840) =

1198791015840

infin+ (1198791015840

119908minus 1198791015840

infin)1199051015840

1199050

for 0 lt 1199051015840le 1199050

1198791015840

119908for 1199051015840 gt 119905

0

1199061015840(1199101015840 1199051015840) 997888rarr 0 119879

1015840(1199101015840 1199051015840) 997888rarr 119879

1015840

infin

as 1199101015840 997888rarr infin for 1199051015840 gt 0

(2)Introducing the following dimensionless variables

119906 = radic1199050

]1199061015840 119910 =

1

radic]1199050

1199101015840 119905 =

1199051015840

1199050

120579 =1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

Pr =120588]119888119901

119896 120578 =

120582

radic]1199050

(3)

and dropping out the prime notation from 119906 119910 and 119905 thegoverning equations (1) take the simplified forms

120597119906 (119910 119905)

120597119905=1205972119906 (119910 119905)

1205971199102

+ 120579 (119910 119905) 119910 119905 gt 0

Pr120597120579 (119910 119905)

120597119905=1205972120579 (119910 119905)

1205971199102

119910 119905 gt 0

(4)

where Pr = 120588]119888119901119896 is the Prandtl number According to the

above nondimensionalisation process the characteristic time1199050can be defined as

1199050= [

]11989221205732(1198791015840

119908minus 1198791015840

infin)2]

13

(5)

In dimensionless form the initial and boundary condi-tions (2) become

119906 (119910 0) = 0 120579 (119910 0) = 0 for 119910 ge 0

119906 (0 119905) minus 120578120597119906 (0 119905)

120597119910= 0 for 119905 gt 0

120579 (0 119905) = 119905 for 0 lt 119905 le 1

1 for 119905 gt 1

119906 (119910 119905) 997888rarr 0 120579 (119910 119905) 997888rarr 0


(6)

where 120578 is the dimensionless slip parameterEquations (4) are a coupled linear system of equations

which can be solved by the Laplace transform techniquesubject to the initial and boundary conditions (6) Thesolutions of energy and momentum equations are

120579 (119910 119905) = 119865 (119910 119905) minus 119865 (119910 119905 minus 1)119867 (119905 minus 1) (7)

119906 (119910 119905)

=

1

Prminus1[1199061(119910 119905) minus 119906

2(119910 119905) + radicPr119906

3(119910 119905)

minus 1199061(119910 119905 minus 1) minus 119906

2(119910 119905 minus 1)

+radicPr1199063(119910 119905 minus 1)119867 (119905 minus 1)]

for Pr = 1

1199063(119910 119905) + 119906

4(119910 119905)

minus 1199063(119910 119905 minus 1) + 119906

4(119910 119905 minus 1)119867 (119905 minus 1)

for Pr = 1

(8)

Abstract and Applied Analysis 3

where

119865 (119910 119905) = (Pr1199102

2+ 119905) erfc(

119910radicPr2radic119905

)

minus radicPr 119905120587119910 exp(

Pr1199102

4119905)

1199061(119910 119905) = 119905

2 exp(minus1198861199102

2) sinh(

119886119910

2)

+1199052

2exp (minus1198861199102) minus 119886

120587

times int

119905

0

int

infin

0

(1 minus 119890minus119909119904

1199092

)

times [119886 sin (119910radic119909)119909 + 1198862

+radic119909 cos (119910radic119909)

119909 + 1198862

]119889119909119889119904

1199062(119910 119905) =

1

2(Pr21199104

12+ Pr1199102119905 + 1199052) erfc(


)

minus radicPr 119905120587

119910

6exp(

Pr1199102

2+ 5119905) exp(minus

Pr1199102

4119905)

1199063(119910 119905) =

1

3119886radic120587

times [radic1205871199103+ radic119905 (4119905 minus 2119910

2) 119890minus11991024119905

minusradic1205871199103 erfc(

119910

2radic119905

)] minus (1

1198862+119910

119886)

times [(119905 +1199102

2) erfc(

119910

2radic119905

) minus radic119905

120587119910119890minus11991024119905]

+119890119886119910

1198862int

119905

0

1198901198862119904 erfc(

119910

2radic119904+ 119886radic119904)119889119904

1199064(119910 119905) = radic

119905

120587

119910

3(1199102+ 4119905) 119890

minus11991024119905

minus 1199102(1199102

6+ 119905) erfc(

119910

2radic119905

)

(9)

Here 119886 = 1120578 is a constant and 119867(119905 minus 1) is the unit stepfunction defined as

119867(119905 minus 119886) = 0 for 0 le 119905 lt 119886

1 for 119905 ge 119886(10)

It is important to note that (7) and (8) in the absence ofslip effect reduce to those obtained by Chandran et al [6Equations (1) and (2)]

21 Plate with Constant Temperature In order to showthe effect of the ramped temperature distribution of theboundary on the flow it is necessary to compare such a flowwith the one near a plate with constant temperature The

temperature and velocity variables for the flow near a platewith constant temperature can be expressed as

120579 (119910 119905) = erfc(119910

2radic119905

)

119906 (119910 119905) =

1

Prminus1[119886radicPr119906

5(119910 119905) + 119906

6(119910 119905)

minus 119865 (119910 119905)] for Pr = 1

1

2[1199101199067(119910 119905) minus 119886119906

5(119910 119905)] for Pr = 1

(11)

where

1199065(119910 119905) =

2

119886[radic

119905

120587119890minus11991024119905

minus (1

1198862+119910

119886) erfc(

119910

2radic119905

)

+1

1198862119890119886119910+1198862119905 erfc(

119910

2radic119905

+ 119886radic119905)]

1199066(119910 119905) =

1

120587int

infin

0

(119905

119909minus1 minus 119890minus119909119905

1199092

)

times [119886 sin (119910radic119909)119909 + 1198862

+radic119909 cos (119910radic119909)

119909 + 1198862

]119889119909

1199067(119910 119905) = 2radic

119905

120587minus 119910 erfc(

119910

2radic119905

)

(12)

The corresponding Nusselt number and skin frictionwhich are respectively the measures of the rate of heattransfer and shear stress at the plate can be determined byconsidering (7) into

Nu = Nu (119905) = minus120597120579(119910 119905)

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119905 gt 0

Nu (ramped) = 2radicPr120587[radic119905 minus radic119905 minus 1119867 (119905 minus 1)]

Nu (constant) = radicPr120587119905

(13)

and (8) into

120591 = 120591 (119905) =120597119906(119910 119905)

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119905 gt 0

120591 (ramped) = 1

Prminus1

times [1198681(119905) minus 119868

2(119905) + radicPr119868

3(119905)

minus 1198681(119905 minus 1) minus 119868

2(119905 minus 1) + radicPr119868

3(119905 minus 1)

times 119867 (119905 minus 1)]

120591 (constant) = 1

Prminus1[radicPr119868

4(119905) + 119868

5(119905) minus 119868

6(119905)]

(14)


Here

1198681(119905) =

1198861199052

2minus [

1198901198862119905

1198863erfc (119886radic119905) minus 1

1198863

minus119905

119886+411990532

3radic120587+2

1198862

radic119905

120587]

1198682(119905) =

1

3

radicPr12058711990532

1198683(119905) =

1

1198863[1198901198862119905 erfc (119886radic119905) + 2119886radic 119905

120587minus 1198862119905 minus 1]

1198684(119905) = 119890

1198862119905 erfc (119886radic119905) minus 1

1198685(119905) = 119905 minus [

1

1198862minus2

119886

radic119905

120587minus1198901198862119905

1198862erfc (119886radic119905)]

1198686(119905) = minus 2radic

Pr 119905120587

(15)

3 Results and Discussion

The problem of heat transfer to unsteady flow of a viscousincompressible fluid with ramped wall temperature and slipcondition at the wall is addressed in this study Numericalcalculations have been carried out for the dimensionlesstemperature 120579 velocity 119906 skin friction 120591 andNusselt numberNu for the case of ramped and constant temperature ofthe plate The effects of pertinent parameters such as slipparameter 120578 Prandtl number Pr and time 119905 on ramped andconstant profiles of temperature 120579 velocity 119906 skin friction120591 and Nusselt number Nu are shown graphically Figure 1depicts that the velocity in case of ramped temperatureplate decreases with increase of slip parameter for 119910 lt 1

and increases for 119910 gt 1 However in Figure 2 velocityin case of constant temperature plate is always increasingdue to increase in the values of 120578 In order to examine theeffect of ramped temperature against constant temperatureon the fluid velocity we have plotted Figures 3 and 4 Weobserve from Figure 3 that velocity in the case of rampedtemperature is always less than that of velocity in case ofconstant temperature This behaviour is in agreement withthe graphical results from [6 7] Furthermore Figures 3and 4 also elaborate the influence of Prandtl number Prand time 119905 on fluid velocity in case of ramped temperatureand constant temperature It is observed that velocity is adecreasing function of time However it is further noticedthat velocity near the plate is greater and is continuouslydecreasing with increasing distance from the plate and finallyapproaches to zero for large values of 119910 It is also observedthat the velocity of the fluid is greater for air (Pr = 071) thanthat of water (Pr = 70) Figure 5 is prepared for the constantvelocity profile for different values of time 119905 when 120578 = 002

and Pr = 071 This figure clearly shows that when time iszero the velocity satisfies initial condition given in (6)

4

3

2

1

0

u

0 05 1 15 2 25 3

y

120578 = 01

120578 = 02

120578 = 03

120578 = 04

Figure 1 Velocity profiles for different values of 120578 corresponding toramped temperature of the plate with 119905 = 09 (Pr = 071)

0 1 2 3 4y

0

02

04

06

08

1

12

u

120578 = 01

120578 = 02

120578 = 03

120578 = 04

Figure 2 Velocity profiles for different values of 120578 corresponding tostepped temperature of the plate with 119905 = 09 (Pr = 071)

0 1 2 3 4 5y

0

1

2

3

4

5

6

7

ut = 04 08 12

SteppedRamped

Figure 3 Velocity profiles for different values of 119905 with 120578 = 08 andPr = 071


0 1 2 3 4 5y

0

01

02

03

04

05

u

t = 04 08 12

SteppedRamped


0 05 1 15 2y

0

5

10

15

20

25

30

35

u

t = 01

t = 05

t = 07

t = 09

Figure 5 Velocity profiles for different values of 119905

Figures 6 and 7 illustrate the variations of temperatureprofiles for different values of Pr and 119905 Two different valuesof Pr such as Pr = 071 and 70 are chosen It is depicted fromFigure 6 that temperature profiles decrease with increasingvalues of Pr both in the case of ramped and constant temper-ature at the plate It is observed that the thermal boundarylayer thickness is maximum near the plate and decreaseswith increasing distance from the leading edge and finallyapproaches zero Furthermore ramped temperature profilesare found smaller than constant profiles of the temperatureIt is observed from Figure 7 that temperature increases withan increase in time in the case of ramped as well as constanttemperature at the plate From these graphs it is clearly seenthat for ramped wall temperature case the temperature takesthe values of time at the plate boundary whereas its values isone for all 119905 gt 1 Hence these graphs show that temperatureprofiles satisfy the imposed boundary conditions in (6)

The skin friction variations along time 119905 are shown inFigures 8 and 9 for different values of slip parameter 120578Figure 8 shows that skin friction is decreasingwith increasing

1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

Pr = 071Pr = 70

t = 20

17

13

08

06

02

Figure 6 Temperature profiles for two different values of Pr

1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

t = 20

17

13

08

06

02

SteppedRamped

Figure 7 Temperature profiles for different values of 119905

0 05 1 15 2t

0

10

20

30

40

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04

Figure 8 Variations of skin friction 120591 along 119905 for different values of120578 corresponding to ramped temperature of the plate with Pr = 071


0 05 1 15 2t

0

2

4

6

8

10

120578 = 01

120578 = 02

120578 = 03

120578 = 04

120591

Figure 9 Variations of skin friction 120591 along 119905 for different values of120578 corresponding to stepped temperature of the plate with Pr = 071

0 05 1 15 2t

0

05

1

15

2

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04


values of 120578 in case of ramped temperature of the plate whilein Figure 9 we observe that skin friction is increasing dueto increase in 120578 in case of constant temperature of the plateFrom the comparison of Figures 8 and 10 we deduce thatskin friction decreases due to the increase in Prandtl numberPr in case of ramped temperature of the plate Again fromthe comparison of Figures 9 and 11 we conclude that skinfriction is decreasing due to the increasing values of Prandtlnumber Pr in case of constant temperature of the plate Thevariation of Nusselt number is shown in Figures 12 and 13 fordifferent values of Pr Figures 12 and 13 depict that Nusseltnumber increases with increasing Pr both in case of rampedand constant temperatures of the plate It is further notedthat Nusselt number for water (Pr = 70) is greater thanelectrolytic solution (Pr = 10) air (Pr = 071) and mercury(Pr = 0015) Physically it is justified due to the fact thatlarge values of Prandtl are responsible to decrease the thermalconductivity and therefore heat diffusesmore slowly from theplate than for smaller values of Pr Hence the rate of heat

0 05 1 15 2t

120578 = 01

120578 = 02

120578 = 03

120578 = 04

minus02

0

02

04

06

120591


0 05 1 15 2t

0

05

1

15

2

25

3

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015

Figure 12 Variations of Nusselt number Nu along 119905 for differentvalues of Pr corresponding to ramped temperature of the plate

transfer is increased This behavior is in a proper agreementwith the observations made in [6 Figure 6]

4 Conclusions

The influence of slip condition on free convection flow ofan incompressible viscous fluid past a vertical plate withramped wall temperature is investigated Laplace transformprocedure is used for finding the exact solutions of theproblem The expressions for velocity and temperature areobtained in terms of the exponential and complementaryerror functions It is found that they satisfy all the imposedinitial and boundary conditions and as a special case canbe reduced to the similar solution existing in the literatureThe effects of different physical parameters such as slipparameter 120578 Prandtl number Pr and time 119905 on ramped andconstant profiles of temperature velocity skin friction andNusselt number are studied graphically It is observed that thevelocity increases with increasing slip parameter 120578 whereas


05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015

Figure 13 Variations of Nusselt number Nu along 119905 for differentvalues of Pr corresponding to stepped temperature of the plate

it decreases with increasing time Temperature in case of airis greater than water and decreases with increasing Pr and119905 Skin friction increases when the temperature is constantwhile it decreases for the ramped nature of temperatureNusselt number increases in both cases of constant andramped temperatures

Conflict of Interests

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

References

[1] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the American Society of Mechanical Engineersvol 80 pp 347ndash359 1958

[2] J A Schetz ldquoOn the approximate solution of viscous flowproblemsrdquoASME Journal of AppliedMechanics vol 30 pp 263ndash268 1963

[3] A A Hayday D A Bowlus and R A McGraw ldquoFree convec-tion from a vertical flat plate with step discontinuities in surfacetemperaturerdquo Journal of Heat Transfer vol 89 no 3 pp 244ndash249 1967

[4] C P Malhotra R L Mahajan W S Sampath K L Barth andRA Enzenroth ldquoControl of temperature uniformity during themanufacture of stable thin-film photovoltaic devicesrdquo Interna-tional Journal of Heat and Mass Transfer vol 49 no 17-18 pp2840ndash2850 2006

[5] K A Antonopoulos and F Democritou ldquoExperimental andnumerical study of unsteady non-periodic wall heat transferunder step ramp and cosine temperature perturbationsrdquo Inter-national Journal of Energy Research vol 18 no 6 pp 563ndash5791994

[6] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[7] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsively

moving plate with ramped wall temperaturerdquo Heat and MassTransfer vol 47 no 5 pp 551ndash561 2011

[8] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling of mass transfer and free convection current effectson unsteady viscous flowwith rampedwall temperaturerdquoWorldJournal of Mechanics vol 1 no 4 pp 176ndash184 2011

[9] D C Tretheway and C D Meinhart ldquoApparent fluid slip athydrophobic microchannel wallsrdquo Physics of Fluids vol 14 no3 pp L9ndashL12 2002

[10] O D Makinde and E Osausi ldquoMHD steady flow in a channelwith slip at the permeable boundariesrdquo Romanian Journal ofPhysics vol 51 pp 319ndash328 2006

[11] O D Makinde and P Y Mhone ldquoHeat transfer to MHDoscillatory flow in a channel Olled with porous mediumrdquoRomanian Journal of Physics vol 50 pp 931ndash938 2005

[12] A Mehmood and A Ali ldquoThe effect of slip condition onunsteady MHD oscillatory flow of a viscous fluid in a planerchannelrdquo Romanian Journal of Physics vol 52 pp 85ndash92 2007

[13] A M Rohni S Ahmad I Pop and J H Merkin ldquoUnsteadymixed convection boundary-layer flow with suction and tem-perature slip effects near the stagnation point on a verticalpermeable surface embedded in a porous mediumrdquo Transportin Porous Media vol 92 no 1 pp 1ndash14 2012

[14] T Hayat S A Shehzad and M Qasim ldquoSlip effects on theunsteady stagnation point flow with variable free streamrdquoWalailak Journal of Science and Technology vol 10 no 4 pp385ndash394 2013

[15] M Qasim Z H Khan W A Khan and I Ali Shah ldquoMHDboundary layer slip flow and heat transfer of ferrofluid alonga stretching cylinder with prescribed heat fluxrdquo PLoS ONE vol9 no 1 Article ID e83930 2014

[16] E A Ashmawy ldquoFully developed natural convectivemicropolarfluid flow in a vertical channel with sliprdquo Journal of the EgyptianMathematical Society 2014

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Stochastic AnalysisInternational Journal of


industrialization In view of such applications Makinde andOsausi [10] studied the combined effect of magnetic fieldand permeable wall slip velocity on the steady flow of anelectrically conducting fluid in a channel of uniform widthMakinde and Mhone [11] investigated the combined effectof a transverse magnetic field and radiative heat transfer tounsteady flow of a conducting optically thin fluid througha channel filled with saturated porous medium and nonuni-form walls temperature Mehmood and Ali [12] extended thework ofMakinde andMhone [11] by considering the fluid slipat the lower wall Few other attempts taking into account theslip boundary condition are [13ndash16] However the literaturelacks studies that take into consideration the combined effectof slippage and ramped temperature at the wall on theunsteady free convection flow of a viscous incompressiblefluid near a vertical flat plate This is the source of motivationto study the influence of slip condition on the unsteadyfree convection transient flow near a vertical flat plate withramped wall temperature

2 Formulation of the Problem and Solution

Let us consider the flow of an incompressible viscous fluidnear an infinite vertical plate The 1199091015840-axis is taken along thewall in the upward direction and 1199101015840-axis is taken perpendic-ular to it into the fluid At the initial moment 1199051015840 = 0 boththe plate and the fluid are at rest at a constant temperature1198791015840

infin At time 1199051015840 = 0

+ the temperature of the plate is raisedor lowered to 1198791015840

infin+ (1198791015840

119908minus 1198791015840

infin)(11990510158401199050) and then for 1199051015840 gt 119905

0

the temperature is maintained at the constant temperature1198791015840

119908 In view of the above assumptions as well as of the usual

Boussinesqrsquos approximation the governing equations reduceto those obtained by Chandran et al [6 Equations (1) and(2)]

1205971199061015840

1205971199051015840= ]

12059721199061015840

12059711991010158402+ 119892120573 (119879

1015840minus 1198791015840

infin)

1205971198791015840

1205971199051015840=

119896

120588119888119901

12059721198791015840

12059711991010158402

(1)

where 1199061015840 1198791015840 ] 120588 119892 120573 119896 and 119888119901are respectively the velocity

in the 1199091015840 direction temperature of the fluid kinematic

viscosity fluid density acceleration due to gravity volumetriccoefficient of thermal expansion thermal conductivity andspecific heat at constant pressure

The appropriate initial and boundary conditions are

1199061015840(1199101015840 0) = 0 119879

1015840(1199101015840 0) = 119879

1015840

infinfor 1199101015840 ge 0

1199061015840(0 1199051015840) minus 120582

1205971199061015840(0 1199051015840)

1205971199101015840

= 0 for 1199051015840 gt 0

1198791015840(0 1199051015840) =

1198791015840

infin+ (1198791015840

119908minus 1198791015840

infin)1199051015840

1199050

for 0 lt 1199051015840le 1199050

1198791015840

119908for 1199051015840 gt 119905

0

1199061015840(1199101015840 1199051015840) 997888rarr 0 119879

1015840(1199101015840 1199051015840) 997888rarr 119879

1015840

infin


(2)Introducing the following dimensionless variables

119906 = radic1199050

]1199061015840 119910 =

1

radic]1199050

1199101015840 119905 =

1199051015840

1199050

120579 =1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

Pr =120588]119888119901

119896 120578 =

120582

radic]1199050

(3)

and dropping out the prime notation from 119906 119910 and 119905 thegoverning equations (1) take the simplified forms

120597119906 (119910 119905)

120597119905=1205972119906 (119910 119905)

1205971199102

+ 120579 (119910 119905) 119910 119905 gt 0

Pr120597120579 (119910 119905)

120597119905=1205972120579 (119910 119905)

1205971199102

119910 119905 gt 0

(4)

where Pr = 120588]119888119901119896 is the Prandtl number According to the

above nondimensionalisation process the characteristic time1199050can be defined as

1199050= [

]11989221205732(1198791015840

119908minus 1198791015840

infin)2]

13

(5)

In dimensionless form the initial and boundary condi-tions (2) become

119906 (119910 0) = 0 120579 (119910 0) = 0 for 119910 ge 0

119906 (0 119905) minus 120578120597119906 (0 119905)

120597119910= 0 for 119905 gt 0

120579 (0 119905) = 119905 for 0 lt 119905 le 1

1 for 119905 gt 1

119906 (119910 119905) 997888rarr 0 120579 (119910 119905) 997888rarr 0


(6)

where 120578 is the dimensionless slip parameterEquations (4) are a coupled linear system of equations

which can be solved by the Laplace transform techniquesubject to the initial and boundary conditions (6) Thesolutions of energy and momentum equations are

120579 (119910 119905) = 119865 (119910 119905) minus 119865 (119910 119905 minus 1)119867 (119905 minus 1) (7)

119906 (119910 119905)

=

1

Prminus1[1199061(119910 119905) minus 119906

2(119910 119905) + radicPr119906

3(119910 119905)

minus 1199061(119910 119905 minus 1) minus 119906

2(119910 119905 minus 1)

+radicPr1199063(119910 119905 minus 1)119867 (119905 minus 1)]

for Pr = 1

1199063(119910 119905) + 119906

4(119910 119905)

minus 1199063(119910 119905 minus 1) + 119906

4(119910 119905 minus 1)119867 (119905 minus 1)

for Pr = 1

(8)


where

119865 (119910 119905) = (Pr1199102

2+ 119905) erfc(


)


Pr1199102

4119905)

1199061(119910 119905) = 119905

2 exp(minus1198861199102

2) sinh(

119886119910

2)

+1199052

2exp (minus1198861199102) minus 119886

120587

times int

119905

0

int

infin

0

(1 minus 119890minus119909119904

1199092

)

times [119886 sin (119910radic119909)119909 + 1198862

+radic119909 cos (119910radic119909)

119909 + 1198862

]119889119909119889119904

1199062(119910 119905) =

1

2(Pr21199104

12+ Pr1199102119905 + 1199052) erfc(


)


119910

6exp(

Pr1199102

2+ 5119905) exp(minus

Pr1199102

4119905)

1199063(119910 119905) =

1

3119886radic120587


2) 119890minus11991024119905


119910

2radic119905

)] minus (1

1198862+119910

119886)

times [(119905 +1199102

2) erfc(

119910

2radic119905

) minus radic119905

120587119910119890minus11991024119905]

+119890119886119910

1198862int

119905

0

1198901198862119904 erfc(

119910

2radic119904+ 119886radic119904)119889119904

1199064(119910 119905) = radic

119905

120587

119910

3(1199102+ 4119905) 119890

minus11991024119905

minus 1199102(1199102

6+ 119905) erfc(

119910

2radic119905

)

(9)


119867(119905 minus 119886) = 0 for 0 le 119905 lt 119886

1 for 119905 ge 119886(10)




120579 (119910 119905) = erfc(119910

2radic119905

)

119906 (119910 119905) =

1


5(119910 119905) + 119906

6(119910 119905)

minus 119865 (119910 119905)] for Pr = 1

1

2[1199101199067(119910 119905) minus 119886119906

5(119910 119905)] for Pr = 1

(11)

where

1199065(119910 119905) =

2

119886[radic

119905

120587119890minus11991024119905

minus (1

1198862+119910

119886) erfc(

119910

2radic119905

)

+1

1198862119890119886119910+1198862119905 erfc(

119910

2radic119905

+ 119886radic119905)]

1199066(119910 119905) =

1

120587int

infin

0

(119905


1199092

)

times [119886 sin (119910radic119909)119909 + 1198862

+radic119909 cos (119910radic119909)

119909 + 1198862

]119889119909

1199067(119910 119905) = 2radic

119905

120587minus 119910 erfc(

119910

2radic119905

)

(12)


Nu = Nu (119905) = minus120597120579(119910 119905)

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119905 gt 0



(13)

and (8) into

120591 = 120591 (119905) =120597119906(119910 119905)

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119905 gt 0

120591 (ramped) = 1

Prminus1

times [1198681(119905) minus 119868

2(119905) + radicPr119868

3(119905)


2(119905 minus 1) + radicPr119868

3(119905 minus 1)

times 119867 (119905 minus 1)]

120591 (constant) = 1


4(119905) + 119868

5(119905) minus 119868

6(119905)]

(14)


Here

1198681(119905) =

1198861199052

2minus [

1198901198862119905

1198863erfc (119886radic119905) minus 1

1198863

minus119905

119886+411990532

3radic120587+2

1198862

radic119905

120587]

1198682(119905) =

1

3

radicPr12058711990532

1198683(119905) =

1

1198863[1198901198862119905 erfc (119886radic119905) + 2119886radic 119905

120587minus 1198862119905 minus 1]

1198684(119905) = 119890

1198862119905 erfc (119886radic119905) minus 1

1198685(119905) = 119905 minus [

1

1198862minus2

119886

radic119905

120587minus1198901198862119905

1198862erfc (119886radic119905)]

1198686(119905) = minus 2radic

Pr 119905120587

(15)





4

3

2

1

0

u

0 05 1 15 2 25 3

y

120578 = 01

120578 = 02

120578 = 03

120578 = 04


0 1 2 3 4y

0

02

04

06

08

1

12

u

120578 = 01

120578 = 02

120578 = 03

120578 = 04


0 1 2 3 4 5y

0

1

2

3

4

5

6

7

ut = 04 08 12

SteppedRamped



0 1 2 3 4 5y

0

01

02

03

04

05

u

t = 04 08 12

SteppedRamped


0 05 1 15 2y

0

5

10

15

20

25

30

35

u

t = 01

t = 05

t = 07

t = 09




1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

Pr = 071Pr = 70

t = 20

17

13

08

06

02


1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

t = 20

17

13

08

06

02

SteppedRamped


0 05 1 15 2t

0

10

20

30

40

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

0

2

4

6

8

10

120578 = 01

120578 = 02

120578 = 03

120578 = 04

120591


0 05 1 15 2t

0

05

1

15

2

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

120578 = 01

120578 = 02

120578 = 03

120578 = 04

minus02

0

02

04

06

120591


0 05 1 15 2t

0

05

1

15

2

25

3

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015



4 Conclusions



05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119865 (119910 119905) = (Pr1199102

2+ 119905) erfc(


)


Pr1199102

4119905)

1199061(119910 119905) = 119905

2 exp(minus1198861199102

2) sinh(

119886119910

2)

+1199052

2exp (minus1198861199102) minus 119886

120587

times int

119905

0

int

infin

0

(1 minus 119890minus119909119904

1199092

)

times [119886 sin (119910radic119909)119909 + 1198862

+radic119909 cos (119910radic119909)

119909 + 1198862

]119889119909119889119904

1199062(119910 119905) =

1

2(Pr21199104

12+ Pr1199102119905 + 1199052) erfc(


)


119910

6exp(

Pr1199102

2+ 5119905) exp(minus

Pr1199102

4119905)

1199063(119910 119905) =

1

3119886radic120587


2) 119890minus11991024119905


119910

2radic119905

)] minus (1

1198862+119910

119886)

times [(119905 +1199102

2) erfc(

119910

2radic119905

) minus radic119905

120587119910119890minus11991024119905]

+119890119886119910

1198862int

119905

0

1198901198862119904 erfc(

119910

2radic119904+ 119886radic119904)119889119904

1199064(119910 119905) = radic

119905

120587

119910

3(1199102+ 4119905) 119890

minus11991024119905

minus 1199102(1199102

6+ 119905) erfc(

119910

2radic119905

)

(9)


119867(119905 minus 119886) = 0 for 0 le 119905 lt 119886

1 for 119905 ge 119886(10)




120579 (119910 119905) = erfc(119910

2radic119905

)

119906 (119910 119905) =

1


5(119910 119905) + 119906

6(119910 119905)

minus 119865 (119910 119905)] for Pr = 1

1

2[1199101199067(119910 119905) minus 119886119906

5(119910 119905)] for Pr = 1

(11)

where

1199065(119910 119905) =

2

119886[radic

119905

120587119890minus11991024119905

minus (1

1198862+119910

119886) erfc(

119910

2radic119905

)

+1

1198862119890119886119910+1198862119905 erfc(

119910

2radic119905

+ 119886radic119905)]

1199066(119910 119905) =

1

120587int

infin

0

(119905


1199092

)

times [119886 sin (119910radic119909)119909 + 1198862

+radic119909 cos (119910radic119909)

119909 + 1198862

]119889119909

1199067(119910 119905) = 2radic

119905

120587minus 119910 erfc(

119910

2radic119905

)

(12)


Nu = Nu (119905) = minus120597120579(119910 119905)

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119905 gt 0



(13)

and (8) into

120591 = 120591 (119905) =120597119906(119910 119905)

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119905 gt 0

120591 (ramped) = 1

Prminus1

times [1198681(119905) minus 119868

2(119905) + radicPr119868

3(119905)


2(119905 minus 1) + radicPr119868

3(119905 minus 1)

times 119867 (119905 minus 1)]

120591 (constant) = 1


4(119905) + 119868

5(119905) minus 119868

6(119905)]

(14)


Here

1198681(119905) =

1198861199052

2minus [

1198901198862119905

1198863erfc (119886radic119905) minus 1

1198863

minus119905

119886+411990532

3radic120587+2

1198862

radic119905

120587]

1198682(119905) =

1

3

radicPr12058711990532

1198683(119905) =

1

1198863[1198901198862119905 erfc (119886radic119905) + 2119886radic 119905

120587minus 1198862119905 minus 1]

1198684(119905) = 119890

1198862119905 erfc (119886radic119905) minus 1

1198685(119905) = 119905 minus [

1

1198862minus2

119886

radic119905

120587minus1198901198862119905

1198862erfc (119886radic119905)]

1198686(119905) = minus 2radic

Pr 119905120587

(15)





4

3

2

1

0

u

0 05 1 15 2 25 3

y

120578 = 01

120578 = 02

120578 = 03

120578 = 04


0 1 2 3 4y

0

02

04

06

08

1

12

u

120578 = 01

120578 = 02

120578 = 03

120578 = 04


0 1 2 3 4 5y

0

1

2

3

4

5

6

7

ut = 04 08 12

SteppedRamped



0 1 2 3 4 5y

0

01

02

03

04

05

u

t = 04 08 12

SteppedRamped


0 05 1 15 2y

0

5

10

15

20

25

30

35

u

t = 01

t = 05

t = 07

t = 09




1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

Pr = 071Pr = 70

t = 20

17

13

08

06

02


1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

t = 20

17

13

08

06

02

SteppedRamped


0 05 1 15 2t

0

10

20

30

40

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

0

2

4

6

8

10

120578 = 01

120578 = 02

120578 = 03

120578 = 04

120591


0 05 1 15 2t

0

05

1

15

2

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

120578 = 01

120578 = 02

120578 = 03

120578 = 04

minus02

0

02

04

06

120591


0 05 1 15 2t

0

05

1

15

2

25

3

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015



4 Conclusions



05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Here

1198681(119905) =

1198861199052

2minus [

1198901198862119905

1198863erfc (119886radic119905) minus 1

1198863

minus119905

119886+411990532

3radic120587+2

1198862

radic119905

120587]

1198682(119905) =

1

3

radicPr12058711990532

1198683(119905) =

1

1198863[1198901198862119905 erfc (119886radic119905) + 2119886radic 119905

120587minus 1198862119905 minus 1]

1198684(119905) = 119890

1198862119905 erfc (119886radic119905) minus 1

1198685(119905) = 119905 minus [

1

1198862minus2

119886

radic119905

120587minus1198901198862119905

1198862erfc (119886radic119905)]

1198686(119905) = minus 2radic

Pr 119905120587

(15)





4

3

2

1

0

u

0 05 1 15 2 25 3

y

120578 = 01

120578 = 02

120578 = 03

120578 = 04


0 1 2 3 4y

0

02

04

06

08

1

12

u

120578 = 01

120578 = 02

120578 = 03

120578 = 04


0 1 2 3 4 5y

0

1

2

3

4

5

6

7

ut = 04 08 12

SteppedRamped



0 1 2 3 4 5y

0

01

02

03

04

05

u

t = 04 08 12

SteppedRamped


0 05 1 15 2y

0

5

10

15

20

25

30

35

u

t = 01

t = 05

t = 07

t = 09




1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

Pr = 071Pr = 70

t = 20

17

13

08

06

02


1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

t = 20

17

13

08

06

02

SteppedRamped


0 05 1 15 2t

0

10

20

30

40

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

0

2

4

6

8

10

120578 = 01

120578 = 02

120578 = 03

120578 = 04

120591


0 05 1 15 2t

0

05

1

15

2

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

120578 = 01

120578 = 02

120578 = 03

120578 = 04

minus02

0

02

04

06

120591


0 05 1 15 2t

0

05

1

15

2

25

3

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015



4 Conclusions



05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5y

0

01

02

03

04

05

u

t = 04 08 12

SteppedRamped


0 05 1 15 2y

0

5

10

15

20

25

30

35

u

t = 01

t = 05

t = 07

t = 09




1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

Pr = 071Pr = 70

t = 20

17

13

08

06

02


1

08

06

04

02

0

120579

0 05 1 15 2 25 3

y

t = 20

17

13

08

06

02

SteppedRamped


0 05 1 15 2t

0

10

20

30

40

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

0

2

4

6

8

10

120578 = 01

120578 = 02

120578 = 03

120578 = 04

120591


0 05 1 15 2t

0

05

1

15

2

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

120578 = 01

120578 = 02

120578 = 03

120578 = 04

minus02

0

02

04

06

120591


0 05 1 15 2t

0

05

1

15

2

25

3

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015



4 Conclusions



05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2t

0

2

4

6

8

10

120578 = 01

120578 = 02

120578 = 03

120578 = 04

120591


0 05 1 15 2t

0

05

1

15

2

120591

120578 = 01

120578 = 02

120578 = 03

120578 = 04



0 05 1 15 2t

120578 = 01

120578 = 02

120578 = 03

120578 = 04

minus02

0

02

04

06

120591


0 05 1 15 2t

0

05

1

15

2

25

3

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015



4 Conclusions



05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05 1 15 2t

0

1

2

3

4

5

6

Nu

Pr = 70Pr = 10

Pr = 071Pr = 0015





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Influence of Slip Condition on Unsteady ...downloads.hindawi.com/journals/aaa/2015/327975.pdf · Department of Mathematical Sciences, Fac ulty of Science, Universiti