Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 583615 10 pageshttpdxdoiorg1011552013583615

Research ArticleEffect of Internal Heat GenerationAbsorption onDusty Fluid Flow over an Exponentially Stretching Sheet withViscous Dissipation

G M Pavithra and B J Gireesha

Department of Studies and Research in Mathematics Kuvempu University Shankaraghatta Shimoga Karnataka 577 451 India

Correspondence should be addressed to B J Gireesha bjgireesurediffmailcom

Received 8 January 2013 Accepted 2 June 2013

Academic Editor Pierpaolo DrsquoUrso

Copyright copy 2013 G M Pavithra and B J Gireesha This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A numerical analysis has been carried out to describe the boundary layer flow and heat transfer of a dusty fluid over an exponentiallystretching surface in the presence of viscous dissipation and internal heat generationabsorptionThe governing partial differentialequations are reduced to nonlinear ordinary differential equations by a similarity transformation before being solved numericallyby Runge-Kutta-Fehlberg 45methodThe heat transfer analysis has been carried out for both PEST and PEHF casesThe numericalresults are compared with the earlier study and found to be in excellent agreement Some important features of the flow and heattransfer in terms of velocities and temperature distributions for different values of the governing parameters like fluid-particleinteraction parameter Prandtl number Eckert number Number density heat sourcesink parameter and suction parameter whichare of physical and engineering interests are analyzed discussed and presented through tables and graphs

1 Introduction

An investigation on boundary layer flow and heat transfer ofviscous fluids over a moving continuous stretching surfacehas considerable practical applications in industries andengineering since the study of heat transfer has becomeimportant industrially for determining the quality of finalproduct which greatly depends on the rate of cooling Manyresearchers inspired by Sakiadis [1 2] who initiated theboundary layer behavior studied the stretching flow problemin various aspects Extension to that an exact solution wasgiven by Crane [3] for a boundary layer flow caused bystretching surface A new dimension to the boundary layerflow was given by Magyari and Keller [4] by considering thenonstandard stretching flow known as exponentially stretch-ing surfaceThey described themass and heat transfer charac-teristics of the boundary layer After that Elbashbeshy [5] wasthe first who considered the heat transfer problem over anexponentially stretching sheet with suction parameter Theeffect of viscous dissipation on the boundary layer flow alongvertical exponential stretching sheet was explained by Parthaet al [6]

Sanjayanand andKhan [7] discussed heat andmass trans-fer in a viscoelastic boundary layer flow over an exponentiallystretching sheet The numerical solution for the boundarylayer flow with thermal radiation over an exponentiallystretching sheet was given by Bidin andNazar [8] Samad andMohebujjaman [9] worked on both heat and mass transferof free convective flow along a vertical stretching sheetin presence of magnetic field with heat generation AgainElbashbeshy andAldawody [10] investigated the heat transferover an unsteady stretching surface embedded in a porousmediumwith variable heat flux in the presence of heat sourceor sink Ishak [11] and Bala and Bhaskar [12] discussed theboundary layer flow over an exponentially stretching sheetwithmagnetic field and thermal radiation effect On the otherhand Sahoo and Poncet [13] carried out an exponentiallystretching flow problem by considering partial slip boundarycondition for third grade fluid The flow and heat transferof a Jeffrey fluid past an exponentially stretching sheet withthermal radiation is analyzed by Nadeem et al [14] Kumar[15] discussed the heat transfer over a stretching porous sheetsubjected to power law heat flux in presence of heat source

2 Journal of Mathematics

Very recently Bhattacharyya [16] analyzed the effects ofradiation and heat sourcesink on unsteady MHD boundarylayer flow and heat transfer over a shrinking sheet withsuctioninjection

All the works as mentioned above are restricted only forfluids induced by stretching sheet In fact a fluid flow withdust particles is a significant type of flow due to its wide rangeof its applications in the boundary layer that include the use ofdust in gas cooling systems centrifugal separation of matterfromfluid polymer technology andfluid droplets sprays thatis powder technology and paint spraying The fluid-particlesystem was initially described by Saffman [17] and derivedthe motion of gas equations carrying the dust particlesFurther Datta and Mishra [18] have investigated dusty fluidin boundary layer flow over a semi-infinite flat plate Gireeshaet al [19] have obtained the boundary layer and heat transferof a dusty fluid flow over a stretching sheet with nonuniformheat sourcesink and concluded that the temperature depen-dent heat sinks are better suited for cooling purposes AlsoGireesha et al [20 21] have discussed the boundary layerflow and heat transfer of a dusty fluid over a stretching sheetby considering the viscous dissipation for both steady andunsteady flows

Onbasis of previous observations we have considered theflow over an exponentially stretching sheet The main aim ofthe present investigation is to study the effect of viscous dis-sipation and internal heat generationabsorption on flow andheat transfer of a dusty fluid over an exponentially stretchingsheet with suction parameter by taking PEST and PEHFcases The study of heat generation or absorption in movingfluids is important in problems dealing with chemical reac-tions and these concerned with dissociating fluids Flows inporousmedia have several applications in geothermal oil res-ervoir engineering and astrophysics The governing bound-ary layer equations have been simplified using suitable simi-larity transformations and then have been solved numericallyusing Runge-Kutta-Fehlberg 45 method with the help ofMaple

2 Mathematical Formulation andSolution of the Problem

A steady two-dimensional laminar boundary layer flow andheat transfer of an incompressible viscous dusty fluid near apermeable plane wall stretching with velocity 119880

119908= 1198800119890(119909119871)

is considered (Figure 1) The 119909-axis is chosen along the sheetand 119910-axis normal to it Two equal and opposite forces areapplied along the sheet so that the wall is stretched exponen-tially

Under these assumptions the two-dimensional boundarylayer equations can be written as

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= 120592

1205972

119906

1205971199102+119870119873

120588(119906119901minus 119906) (2)

Slit

Permeablestretching sheet

Momentum boundary layer

Thermal boundary layer

y

x

S

UT minus Tinfin

U = U0exL T = Tinfin + T0e

x2L

Figure 1 Schematic representation of boundary layer flow

120597119906119901

120597119909+

120597V119901

120597119910= 0 (3)

119906119901

120597119906119901

120597119909+ V119901

120597119906119901

120597119910=119870

119898(119906 minus 119906

119901) (4)

where 119909 and 119910 represent coordinate axes along the contin-uous surface in the direction of motion and perpendicularto it respectively (119906 V) and (119906

119901 V119901) denote the velocity

components of the fluid and particle phase along the 119909 and119910 directions respectively 120592 is the coefficient of viscosityof fluid 120588 is the density of the fluid phase 119870 is Stokersquosresistance119873 is the number density of dust particles119898 is themass concentration of dust particles and 120591V = 119898119870 is therelaxation time of particle phase

In order to solve the governing boundary layer equationsconsider the following appropriate boundary conditions onvelocity

119906 = 119880119908(119909) V = 119881

119908(119909) at 119910 = 0

119906 997888rarr 0 119906119901997888rarr 0 V

119901997888rarr V as 119910 997888rarr infin

(5)

where 119880119908(119909) = 119880

0119890(119909119871) is the sheet velocity 119881

119908(119909) =

minus119878radic11988001205922119871119890

(1199092119871) is the suction velocity 1198800is reference

velocity and 119871 is the reference lengthEquations (1)ndash(4) that are subject to boundary condition

(5) admit self-similar solutions in terms of the similarity func-tion 119891 and the similarity variable 120578 as

119906 = 1198800119890119909119871

1198911015840

(120578)

V = minusradic1199060120592

21198711198901199092119871

[119891 (120578) + 1205781198911015840

(120578)]

119906119901= 1198800119890119909119871

1198651015840

(120578)

V119901= minusradic

1199060120592

21198711198901199092119871

[119865 (120578) + 1205781198651015840

(120578)]

120578 = radic1199060

21205921198711198901199092119871

119910

(6)

Journal of Mathematics 3

Table 1 Comparison of the results for the dimensionless wall temperature gradient 1205791015840(0) (PEST case) by varying the Pr and Ec with 120573 = 119873 =

120582 = 119878 = 0

Pr Ec = 0 Ec = 02 Ec = 09Nadeem et al

[14]Bidin andNazar [8]

Presentstudy 1205791015840(0)

Nadeem et al[14]

Bidin andNazar [8]

Presentstudy 1205791015840(0)

Nadeem et al[14]

Bidin andNazar [8]

Presentstudy 1205791015840(0)

1 minus0955 minus0955 minus0955 minus0863 minus0862 minus0863 minus0539 minus0539 minus05392 minus1471 minus1471 minus1471 minus1306 minus1306 minus1305 minus0725 minus0725 minus07243 minus1870 minus1869 minus1869 minus1639 minus1688 minus1638 minus0830 minus0830 minus0830

Table 2 Values of wall temperature gradient 1205791015840(0) (for PEST case)and wall temperature 120579(0) (for PEHF case)

120573 Pr Ec 120582 119873 1205791015840

(0) 120579(0)

020610

072 05 05 05minus175024minus166464minus164077

063348067408068511

060721015

05 05 05minus166464minus201014minus253037

067408059634052418

06 07200510

05 05minus203930minus166464minus128998

049036067408085780

06 072 05051015

05minus166464minus139884minus105000

067408077824096894

06 072 05 05051015

minus166464minus239962minus294768

067408051431044797

These equations identically satisfy the governing equa-tions (1) and (3) Substitute (6) into (2)ndash(4) and on equatingthe coefficient of (119909 119871)0 on both sides and then one can get

119891101584010158401015840

(120578) + 119891 (120578) 11989110158401015840

(120578) minus 21198911015840

(120578)2

+ 2119897120573 [1198651015840

(120578) minus 1198911015840

(120578)] = 0

119865 (120578) 11986510158401015840

(120578) minus 21198651015840

(120578)2

+ 2120573 [1198911015840

(120578) minus 1198651015840

(120578)] = 0

(7)

where prime denotes the differentiation with respect to 120578 119897 =119898119873120588 is the mass concentration and 120573 = 119871120591V1198800 is the fluid-particle interaction parameter for velocity

Similarity boundary conditions (5) will become

1198911015840

(120578) = 1 119891 (120578) = 119878 at 120578 = 0

1198911015840

(120578) = 0 1198651015840

(120578) = 0

119865 (120578) = 119891 (120578) + 1205781198911015840

(120578) minus 1205781198651015840

(120578) as 120578 997888rarr infin

(8)

where 119878 gt 0 is a suction parameterThe important physical parameter for the boundary layer

flow is the skin-friction coefficient which is defined as

119862119891=

120591119908

1205881198802119908

(9)

where the skin friction 120591119908is given by

120591119908= 120583(

120597119906

120597119910)

119910=0

(10)

Table 3 Values of skin friction coefficient 11989110158401015840(0)

120573 119878 11989110158401015840

(0)

020610

05minus155365minus156928minus157897

06051015

minus156928minus188171minus223651

Using the nondimensional variables one obtains

radic2Re119862119891= 11989110158401015840

(0) (11)

where Re = 1198800119871120592 is the Reynolds number

3 Heat Transfer Analysis

The governing steady boundary layer heat transport equa-tions for both fluid and dust phases with viscous dissipationand heat generationabsorption are given by

120588119888119901[119906

120597119879

120597119909+ V

120597119879

120597119910] = 119896

1205972

119879

1205971199102+

119873119888119901

120591119879

(119879119901minus 119879) +

119873

120591V

(119906119901minus 119906)2

+ 120583(120597119906

120597119910)

2

+ 119876 (119879 minus 119879infin)

119873119888119898[119906119901

120597119879119901

120597119909+ V119901

120597119879119901

120597119910] = minus

119873119888119901

120591119879

(119879119901minus 119879)

(12)

where 119879 and 119879119901are the temperatures of the fluid and dust

particles inside the boundary layer 119888119901and 119888119898are the specific

heat of fluid and dust particles 120591119879is the thermal equilibrium

time that is it is time required by a dust cloud to adjust itstemperature to the fluid 119896 is the thermal conductivity 120591V isthe relaxation time of the of dust particle that is the timerequired by a dust particle to adjust its velocity relative to thefluid and 119876 represents the heat source when 119876 gt 0 and thesink when 119876 lt 0

We solved the heat transfer phenomenon for two types ofheating process namely

(1) prescribed exponential order surface temperature(PEST)

(2) prescribed exponential order heat flux (PEHF)

4 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120573 = 02 06 1

f998400 (120578)F998400 (120578)

120578

Figure 2 Effect of 120573 on velocity profiles with119873 = 120582 = Ec = 119878 = 05and Pr = 072

Case 1 (prescribed exponential order surface temperature)For this heating process we employ the following boundaryconditions

119879 = 119879119908(119909) at 119910 = 0

119879 997888rarr 119879infin 119879119901997888rarr 119879infin

as 119910 997888rarr infin(13)

where 119879119908= 119879infin+ 119879011989011988811199092119871 is the temperature distribution in

the stretching surface 1198790is a reference temperature and 119888

1is

constantIntroduce the dimensionless variables for the tempera-

tures 120579(120578) and 120579119901(120578) as follows

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120579119901(120578) =

119879119901minus 119879infin

119879119908minus 119879infin

(14)

where 119879 minus 119879infin= 119879011989011988811199092119871

120579(120578)Using the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(15)

where Pr = 120583119888119901119896 is the Prandtl number Ec = 119880

2

0119888119901119879 is the

Eckert number 120573 = 119871120591V1198800 and 120573120591 = 1198711205911198791198800are the fluid-

particle interaction parameter for velocity and temperatureand 120574 = 119888

119901119888119898is the ratio of specific heat

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

f998400 (120578)F998400 (120578)

120578

S = 2 3 4

Figure 3 Effect of 119878 on velocity profiles with 119873 = 120582 = Ec = 05Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

120579 (120578) = 1 at 120578 = 0

120579 (120578) 997888rarr 0 120579119901(120578) 997888rarr 0 as 120578 997888rarr infin

(16)

Case 2 (prescribed exponential order heat flux) For thisheating process consider the boundary conditions as follows

120597119879

120597119910= minus

119902119908(119909)

119896at 119910 = 0

119879 997888rarr 119879infin 119879

119901997888rarr 119879infin

as 120578 997888rarr infin

(17)

where 119902119908(119909) = 119879

1119890(1198882+1)1199092119871 119879

1is reference temperature and

1198882is constantUsing the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(18)

where Ec = 119896119880201198881199011198791radic11988002120592119871 is the Eckert number

Journal of Mathematics 5

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(b)

Figure 4 Effect of 120573 on temperature profiles for both PEST and PEHF cases with119873 = 120582 = Ec = 119878 = 05 and Pr = 072

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(b)

Figure 5 Effect of 120582 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 119878 = 05 Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

1205791015840

(120578) = minus1 at 120578 = 0120579 (120578) 997888rarr 0 120579

119901(120578) 997888rarr 0 as 120578 997888rarr infin

(19)

4 Numerical Solution

Two-dimensional boundary layer flow and heat transfer ofa dusty fluid over an exponential stretching sheet is consid-ered The system of highly nonlinear ordinary differential

equations (7) (15) for PEST case and (18) for PEHF case aresolved numerically using RKF-45 method with the help of analgebraic software Maple In this method we choose suitablefinite values of 120578 rarr infin as 120578 = 5

Here we have given the comparison of our results ofminus1205791015840

(0) with Nadeem et al [14] and Bidin and Nazar [8] asin Table 1 for various values of Pr and Ec From this tableone can notice that there is a close agreement with theseapproaches and thus verifies the accuracy of themethod usedThe thermal characteristics at the wall that are examinedfor the values of temperature gradient 1205791015840(0) in PEST case

6 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEST case120579(120578)120579 p(120578)

(a)

06

045

03

015

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEHF case

120579(120578)120579 p(120578)

(b)

Figure 6 Effect of 119878 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

N = 05 1 2

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

N = 05 1 2

1

08

06

04

02

00

(b)

Figure 7 Effect of119873 on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

and the temperature 120579(0) in PEHF case are tabulated inTable 2 The computed values of skin friction coefficient11989110158401015840

(0) is tabulated in Table 3 for different values of fluid-particle interaction parameter (120573) and suction parameter (119878)Further we studied the effects of viscous dissipation and heatgenerationabsorption on velocity and temperature profilesthat are depicted graphically for different values of fluid-particle interaction parameter (120573) Suction parameter (119878)heat sourcesink parameter (120582) Number density (119873) Prandtlnumber (Pr) and Eckert number (Ec)

5 Results and Discussion

Numerical calculations are performed for velocity and tem-perature profiles for various values of physical parameterssuch as fluid-particle interaction parameter (120573) Suctionparameter (119878) heat sourcesink parameter (120582) Number den-sity (119873) Prandtl number (Pr) and Eckert number (Ec) andare depicted graphically (from Figure 2 to Figure 11) Com-parison values of wall-temperature gradient are tabulated inTable 1 From Table 3 we observed that the skin friction

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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

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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

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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

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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 Journal of Mathematics

Very recently Bhattacharyya [16] analyzed the effects ofradiation and heat sourcesink on unsteady MHD boundarylayer flow and heat transfer over a shrinking sheet withsuctioninjection

All the works as mentioned above are restricted only forfluids induced by stretching sheet In fact a fluid flow withdust particles is a significant type of flow due to its wide rangeof its applications in the boundary layer that include the use ofdust in gas cooling systems centrifugal separation of matterfromfluid polymer technology andfluid droplets sprays thatis powder technology and paint spraying The fluid-particlesystem was initially described by Saffman [17] and derivedthe motion of gas equations carrying the dust particlesFurther Datta and Mishra [18] have investigated dusty fluidin boundary layer flow over a semi-infinite flat plate Gireeshaet al [19] have obtained the boundary layer and heat transferof a dusty fluid flow over a stretching sheet with nonuniformheat sourcesink and concluded that the temperature depen-dent heat sinks are better suited for cooling purposes AlsoGireesha et al [20 21] have discussed the boundary layerflow and heat transfer of a dusty fluid over a stretching sheetby considering the viscous dissipation for both steady andunsteady flows

Onbasis of previous observations we have considered theflow over an exponentially stretching sheet The main aim ofthe present investigation is to study the effect of viscous dis-sipation and internal heat generationabsorption on flow andheat transfer of a dusty fluid over an exponentially stretchingsheet with suction parameter by taking PEST and PEHFcases The study of heat generation or absorption in movingfluids is important in problems dealing with chemical reac-tions and these concerned with dissociating fluids Flows inporousmedia have several applications in geothermal oil res-ervoir engineering and astrophysics The governing bound-ary layer equations have been simplified using suitable simi-larity transformations and then have been solved numericallyusing Runge-Kutta-Fehlberg 45 method with the help ofMaple

2 Mathematical Formulation andSolution of the Problem

A steady two-dimensional laminar boundary layer flow andheat transfer of an incompressible viscous dusty fluid near apermeable plane wall stretching with velocity 119880

119908= 1198800119890(119909119871)

is considered (Figure 1) The 119909-axis is chosen along the sheetand 119910-axis normal to it Two equal and opposite forces areapplied along the sheet so that the wall is stretched exponen-tially

Under these assumptions the two-dimensional boundarylayer equations can be written as

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= 120592

1205972

119906

1205971199102+119870119873

120588(119906119901minus 119906) (2)

Slit

Permeablestretching sheet

Momentum boundary layer

Thermal boundary layer

y

x

S

UT minus Tinfin

U = U0exL T = Tinfin + T0e

x2L

Figure 1 Schematic representation of boundary layer flow

120597119906119901

120597119909+

120597V119901

120597119910= 0 (3)

119906119901

120597119906119901

120597119909+ V119901

120597119906119901

120597119910=119870

119898(119906 minus 119906

119901) (4)

where 119909 and 119910 represent coordinate axes along the contin-uous surface in the direction of motion and perpendicularto it respectively (119906 V) and (119906

119901 V119901) denote the velocity

components of the fluid and particle phase along the 119909 and119910 directions respectively 120592 is the coefficient of viscosityof fluid 120588 is the density of the fluid phase 119870 is Stokersquosresistance119873 is the number density of dust particles119898 is themass concentration of dust particles and 120591V = 119898119870 is therelaxation time of particle phase

In order to solve the governing boundary layer equationsconsider the following appropriate boundary conditions onvelocity

119906 = 119880119908(119909) V = 119881

119908(119909) at 119910 = 0

119906 997888rarr 0 119906119901997888rarr 0 V

119901997888rarr V as 119910 997888rarr infin

(5)

where 119880119908(119909) = 119880

0119890(119909119871) is the sheet velocity 119881

119908(119909) =

minus119878radic11988001205922119871119890

(1199092119871) is the suction velocity 1198800is reference

velocity and 119871 is the reference lengthEquations (1)ndash(4) that are subject to boundary condition

(5) admit self-similar solutions in terms of the similarity func-tion 119891 and the similarity variable 120578 as

119906 = 1198800119890119909119871

1198911015840

(120578)

V = minusradic1199060120592

21198711198901199092119871

[119891 (120578) + 1205781198911015840

(120578)]

119906119901= 1198800119890119909119871

1198651015840

(120578)

V119901= minusradic

1199060120592

21198711198901199092119871

[119865 (120578) + 1205781198651015840

(120578)]

120578 = radic1199060

21205921198711198901199092119871

119910

(6)

Journal of Mathematics 3

Table 1 Comparison of the results for the dimensionless wall temperature gradient 1205791015840(0) (PEST case) by varying the Pr and Ec with 120573 = 119873 =

120582 = 119878 = 0

Pr Ec = 0 Ec = 02 Ec = 09Nadeem et al

[14]Bidin andNazar [8]

Presentstudy 1205791015840(0)

Nadeem et al[14]

Bidin andNazar [8]

Presentstudy 1205791015840(0)

Nadeem et al[14]

Bidin andNazar [8]

Presentstudy 1205791015840(0)

1 minus0955 minus0955 minus0955 minus0863 minus0862 minus0863 minus0539 minus0539 minus05392 minus1471 minus1471 minus1471 minus1306 minus1306 minus1305 minus0725 minus0725 minus07243 minus1870 minus1869 minus1869 minus1639 minus1688 minus1638 minus0830 minus0830 minus0830

Table 2 Values of wall temperature gradient 1205791015840(0) (for PEST case)and wall temperature 120579(0) (for PEHF case)

120573 Pr Ec 120582 119873 1205791015840

(0) 120579(0)

020610

072 05 05 05minus175024minus166464minus164077

063348067408068511

060721015

05 05 05minus166464minus201014minus253037

067408059634052418

06 07200510

05 05minus203930minus166464minus128998

049036067408085780

06 072 05051015

05minus166464minus139884minus105000

067408077824096894

06 072 05 05051015

minus166464minus239962minus294768

067408051431044797

These equations identically satisfy the governing equa-tions (1) and (3) Substitute (6) into (2)ndash(4) and on equatingthe coefficient of (119909 119871)0 on both sides and then one can get

119891101584010158401015840

(120578) + 119891 (120578) 11989110158401015840

(120578) minus 21198911015840

(120578)2

+ 2119897120573 [1198651015840

(120578) minus 1198911015840

(120578)] = 0

119865 (120578) 11986510158401015840

(120578) minus 21198651015840

(120578)2

+ 2120573 [1198911015840

(120578) minus 1198651015840

(120578)] = 0

(7)

where prime denotes the differentiation with respect to 120578 119897 =119898119873120588 is the mass concentration and 120573 = 119871120591V1198800 is the fluid-particle interaction parameter for velocity

Similarity boundary conditions (5) will become

1198911015840

(120578) = 1 119891 (120578) = 119878 at 120578 = 0

1198911015840

(120578) = 0 1198651015840

(120578) = 0

119865 (120578) = 119891 (120578) + 1205781198911015840

(120578) minus 1205781198651015840

(120578) as 120578 997888rarr infin

(8)

where 119878 gt 0 is a suction parameterThe important physical parameter for the boundary layer

flow is the skin-friction coefficient which is defined as

119862119891=

120591119908

1205881198802119908

(9)

where the skin friction 120591119908is given by

120591119908= 120583(

120597119906

120597119910)

119910=0

(10)

Table 3 Values of skin friction coefficient 11989110158401015840(0)

120573 119878 11989110158401015840

(0)

020610

05minus155365minus156928minus157897

06051015

minus156928minus188171minus223651

Using the nondimensional variables one obtains

radic2Re119862119891= 11989110158401015840

(0) (11)

where Re = 1198800119871120592 is the Reynolds number

3 Heat Transfer Analysis

The governing steady boundary layer heat transport equa-tions for both fluid and dust phases with viscous dissipationand heat generationabsorption are given by

120588119888119901[119906

120597119879

120597119909+ V

120597119879

120597119910] = 119896

1205972

119879

1205971199102+

119873119888119901

120591119879

(119879119901minus 119879) +

119873

120591V

(119906119901minus 119906)2

+ 120583(120597119906

120597119910)

2

+ 119876 (119879 minus 119879infin)

119873119888119898[119906119901

120597119879119901

120597119909+ V119901

120597119879119901

120597119910] = minus

119873119888119901

120591119879

(119879119901minus 119879)

(12)

where 119879 and 119879119901are the temperatures of the fluid and dust

particles inside the boundary layer 119888119901and 119888119898are the specific

heat of fluid and dust particles 120591119879is the thermal equilibrium

time that is it is time required by a dust cloud to adjust itstemperature to the fluid 119896 is the thermal conductivity 120591V isthe relaxation time of the of dust particle that is the timerequired by a dust particle to adjust its velocity relative to thefluid and 119876 represents the heat source when 119876 gt 0 and thesink when 119876 lt 0

We solved the heat transfer phenomenon for two types ofheating process namely

(1) prescribed exponential order surface temperature(PEST)

(2) prescribed exponential order heat flux (PEHF)

4 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120573 = 02 06 1

f998400 (120578)F998400 (120578)

120578

Figure 2 Effect of 120573 on velocity profiles with119873 = 120582 = Ec = 119878 = 05and Pr = 072

Case 1 (prescribed exponential order surface temperature)For this heating process we employ the following boundaryconditions

119879 = 119879119908(119909) at 119910 = 0

119879 997888rarr 119879infin 119879119901997888rarr 119879infin

as 119910 997888rarr infin(13)

where 119879119908= 119879infin+ 119879011989011988811199092119871 is the temperature distribution in

the stretching surface 1198790is a reference temperature and 119888

1is

constantIntroduce the dimensionless variables for the tempera-

tures 120579(120578) and 120579119901(120578) as follows

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120579119901(120578) =

119879119901minus 119879infin

119879119908minus 119879infin

(14)

where 119879 minus 119879infin= 119879011989011988811199092119871

120579(120578)Using the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(15)

where Pr = 120583119888119901119896 is the Prandtl number Ec = 119880

2

0119888119901119879 is the

Eckert number 120573 = 119871120591V1198800 and 120573120591 = 1198711205911198791198800are the fluid-

particle interaction parameter for velocity and temperatureand 120574 = 119888

119901119888119898is the ratio of specific heat

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

f998400 (120578)F998400 (120578)

120578

S = 2 3 4

Figure 3 Effect of 119878 on velocity profiles with 119873 = 120582 = Ec = 05Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

120579 (120578) = 1 at 120578 = 0

120579 (120578) 997888rarr 0 120579119901(120578) 997888rarr 0 as 120578 997888rarr infin

(16)

Case 2 (prescribed exponential order heat flux) For thisheating process consider the boundary conditions as follows

120597119879

120597119910= minus

119902119908(119909)

119896at 119910 = 0

119879 997888rarr 119879infin 119879

119901997888rarr 119879infin

as 120578 997888rarr infin

(17)

where 119902119908(119909) = 119879

1119890(1198882+1)1199092119871 119879

1is reference temperature and

1198882is constantUsing the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(18)

where Ec = 119896119880201198881199011198791radic11988002120592119871 is the Eckert number

Journal of Mathematics 5

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(b)

Figure 4 Effect of 120573 on temperature profiles for both PEST and PEHF cases with119873 = 120582 = Ec = 119878 = 05 and Pr = 072

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(b)

Figure 5 Effect of 120582 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 119878 = 05 Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

1205791015840

(120578) = minus1 at 120578 = 0120579 (120578) 997888rarr 0 120579

119901(120578) 997888rarr 0 as 120578 997888rarr infin

(19)

4 Numerical Solution

Two-dimensional boundary layer flow and heat transfer ofa dusty fluid over an exponential stretching sheet is consid-ered The system of highly nonlinear ordinary differential

equations (7) (15) for PEST case and (18) for PEHF case aresolved numerically using RKF-45 method with the help of analgebraic software Maple In this method we choose suitablefinite values of 120578 rarr infin as 120578 = 5

Here we have given the comparison of our results ofminus1205791015840

(0) with Nadeem et al [14] and Bidin and Nazar [8] asin Table 1 for various values of Pr and Ec From this tableone can notice that there is a close agreement with theseapproaches and thus verifies the accuracy of themethod usedThe thermal characteristics at the wall that are examinedfor the values of temperature gradient 1205791015840(0) in PEST case

6 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEST case120579(120578)120579 p(120578)

(a)

06

045

03

015

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEHF case

120579(120578)120579 p(120578)

(b)

Figure 6 Effect of 119878 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

N = 05 1 2

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

N = 05 1 2

1

08

06

04

02

00

(b)

Figure 7 Effect of119873 on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

and the temperature 120579(0) in PEHF case are tabulated inTable 2 The computed values of skin friction coefficient11989110158401015840

(0) is tabulated in Table 3 for different values of fluid-particle interaction parameter (120573) and suction parameter (119878)Further we studied the effects of viscous dissipation and heatgenerationabsorption on velocity and temperature profilesthat are depicted graphically for different values of fluid-particle interaction parameter (120573) Suction parameter (119878)heat sourcesink parameter (120582) Number density (119873) Prandtlnumber (Pr) and Eckert number (Ec)

5 Results and Discussion

Numerical calculations are performed for velocity and tem-perature profiles for various values of physical parameterssuch as fluid-particle interaction parameter (120573) Suctionparameter (119878) heat sourcesink parameter (120582) Number den-sity (119873) Prandtl number (Pr) and Eckert number (Ec) andare depicted graphically (from Figure 2 to Figure 11) Com-parison values of wall-temperature gradient are tabulated inTable 1 From Table 3 we observed that the skin friction

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

Submit your manuscripts athttpwwwhindawicom

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

Table 1 Comparison of the results for the dimensionless wall temperature gradient 1205791015840(0) (PEST case) by varying the Pr and Ec with 120573 = 119873 =

120582 = 119878 = 0

Pr Ec = 0 Ec = 02 Ec = 09Nadeem et al

[14]Bidin andNazar [8]

Presentstudy 1205791015840(0)

Nadeem et al[14]

Bidin andNazar [8]

Presentstudy 1205791015840(0)

Nadeem et al[14]

Bidin andNazar [8]

Presentstudy 1205791015840(0)

1 minus0955 minus0955 minus0955 minus0863 minus0862 minus0863 minus0539 minus0539 minus05392 minus1471 minus1471 minus1471 minus1306 minus1306 minus1305 minus0725 minus0725 minus07243 minus1870 minus1869 minus1869 minus1639 minus1688 minus1638 minus0830 minus0830 minus0830

Table 2 Values of wall temperature gradient 1205791015840(0) (for PEST case)and wall temperature 120579(0) (for PEHF case)

120573 Pr Ec 120582 119873 1205791015840

(0) 120579(0)

020610

072 05 05 05minus175024minus166464minus164077

063348067408068511

060721015

05 05 05minus166464minus201014minus253037

067408059634052418

06 07200510

05 05minus203930minus166464minus128998

049036067408085780

06 072 05051015

05minus166464minus139884minus105000

067408077824096894

06 072 05 05051015

minus166464minus239962minus294768

067408051431044797

These equations identically satisfy the governing equa-tions (1) and (3) Substitute (6) into (2)ndash(4) and on equatingthe coefficient of (119909 119871)0 on both sides and then one can get

119891101584010158401015840

(120578) + 119891 (120578) 11989110158401015840

(120578) minus 21198911015840

(120578)2

+ 2119897120573 [1198651015840

(120578) minus 1198911015840

(120578)] = 0

119865 (120578) 11986510158401015840

(120578) minus 21198651015840

(120578)2

+ 2120573 [1198911015840

(120578) minus 1198651015840

(120578)] = 0

(7)

where prime denotes the differentiation with respect to 120578 119897 =119898119873120588 is the mass concentration and 120573 = 119871120591V1198800 is the fluid-particle interaction parameter for velocity

Similarity boundary conditions (5) will become

1198911015840

(120578) = 1 119891 (120578) = 119878 at 120578 = 0

1198911015840

(120578) = 0 1198651015840

(120578) = 0

119865 (120578) = 119891 (120578) + 1205781198911015840

(120578) minus 1205781198651015840

(120578) as 120578 997888rarr infin

(8)

where 119878 gt 0 is a suction parameterThe important physical parameter for the boundary layer

flow is the skin-friction coefficient which is defined as

119862119891=

120591119908

1205881198802119908

(9)

where the skin friction 120591119908is given by

120591119908= 120583(

120597119906

120597119910)

119910=0

(10)

Table 3 Values of skin friction coefficient 11989110158401015840(0)

120573 119878 11989110158401015840

(0)

020610

05minus155365minus156928minus157897

06051015

minus156928minus188171minus223651

Using the nondimensional variables one obtains

radic2Re119862119891= 11989110158401015840

(0) (11)

where Re = 1198800119871120592 is the Reynolds number

3 Heat Transfer Analysis

The governing steady boundary layer heat transport equa-tions for both fluid and dust phases with viscous dissipationand heat generationabsorption are given by

120588119888119901[119906

120597119879

120597119909+ V

120597119879

120597119910] = 119896

1205972

119879

1205971199102+

119873119888119901

120591119879

(119879119901minus 119879) +

119873

120591V

(119906119901minus 119906)2

+ 120583(120597119906

120597119910)

2

+ 119876 (119879 minus 119879infin)

119873119888119898[119906119901

120597119879119901

120597119909+ V119901

120597119879119901

120597119910] = minus

119873119888119901

120591119879

(119879119901minus 119879)

(12)

where 119879 and 119879119901are the temperatures of the fluid and dust

particles inside the boundary layer 119888119901and 119888119898are the specific

heat of fluid and dust particles 120591119879is the thermal equilibrium

time that is it is time required by a dust cloud to adjust itstemperature to the fluid 119896 is the thermal conductivity 120591V isthe relaxation time of the of dust particle that is the timerequired by a dust particle to adjust its velocity relative to thefluid and 119876 represents the heat source when 119876 gt 0 and thesink when 119876 lt 0

We solved the heat transfer phenomenon for two types ofheating process namely

(1) prescribed exponential order surface temperature(PEST)

(2) prescribed exponential order heat flux (PEHF)

4 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120573 = 02 06 1

f998400 (120578)F998400 (120578)

120578

Figure 2 Effect of 120573 on velocity profiles with119873 = 120582 = Ec = 119878 = 05and Pr = 072

Case 1 (prescribed exponential order surface temperature)For this heating process we employ the following boundaryconditions

119879 = 119879119908(119909) at 119910 = 0

119879 997888rarr 119879infin 119879119901997888rarr 119879infin

as 119910 997888rarr infin(13)

where 119879119908= 119879infin+ 119879011989011988811199092119871 is the temperature distribution in

the stretching surface 1198790is a reference temperature and 119888

1is

constantIntroduce the dimensionless variables for the tempera-

tures 120579(120578) and 120579119901(120578) as follows

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120579119901(120578) =

119879119901minus 119879infin

119879119908minus 119879infin

(14)

where 119879 minus 119879infin= 119879011989011988811199092119871

120579(120578)Using the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(15)

where Pr = 120583119888119901119896 is the Prandtl number Ec = 119880

2

0119888119901119879 is the

Eckert number 120573 = 119871120591V1198800 and 120573120591 = 1198711205911198791198800are the fluid-

particle interaction parameter for velocity and temperatureand 120574 = 119888

119901119888119898is the ratio of specific heat

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

f998400 (120578)F998400 (120578)

120578

S = 2 3 4

Figure 3 Effect of 119878 on velocity profiles with 119873 = 120582 = Ec = 05Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

120579 (120578) = 1 at 120578 = 0

120579 (120578) 997888rarr 0 120579119901(120578) 997888rarr 0 as 120578 997888rarr infin

(16)

Case 2 (prescribed exponential order heat flux) For thisheating process consider the boundary conditions as follows

120597119879

120597119910= minus

119902119908(119909)

119896at 119910 = 0

119879 997888rarr 119879infin 119879

119901997888rarr 119879infin

as 120578 997888rarr infin

(17)

where 119902119908(119909) = 119879

1119890(1198882+1)1199092119871 119879

1is reference temperature and

1198882is constantUsing the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(18)

where Ec = 119896119880201198881199011198791radic11988002120592119871 is the Eckert number

Journal of Mathematics 5

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(b)

Figure 4 Effect of 120573 on temperature profiles for both PEST and PEHF cases with119873 = 120582 = Ec = 119878 = 05 and Pr = 072

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(b)

Figure 5 Effect of 120582 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 119878 = 05 Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

1205791015840

(120578) = minus1 at 120578 = 0120579 (120578) 997888rarr 0 120579

119901(120578) 997888rarr 0 as 120578 997888rarr infin

(19)

4 Numerical Solution

Two-dimensional boundary layer flow and heat transfer ofa dusty fluid over an exponential stretching sheet is consid-ered The system of highly nonlinear ordinary differential

equations (7) (15) for PEST case and (18) for PEHF case aresolved numerically using RKF-45 method with the help of analgebraic software Maple In this method we choose suitablefinite values of 120578 rarr infin as 120578 = 5

Here we have given the comparison of our results ofminus1205791015840

(0) with Nadeem et al [14] and Bidin and Nazar [8] asin Table 1 for various values of Pr and Ec From this tableone can notice that there is a close agreement with theseapproaches and thus verifies the accuracy of themethod usedThe thermal characteristics at the wall that are examinedfor the values of temperature gradient 1205791015840(0) in PEST case

6 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEST case120579(120578)120579 p(120578)

(a)

06

045

03

015

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEHF case

120579(120578)120579 p(120578)

(b)

Figure 6 Effect of 119878 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

N = 05 1 2

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

N = 05 1 2

1

08

06

04

02

00

(b)

Figure 7 Effect of119873 on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

and the temperature 120579(0) in PEHF case are tabulated inTable 2 The computed values of skin friction coefficient11989110158401015840

(0) is tabulated in Table 3 for different values of fluid-particle interaction parameter (120573) and suction parameter (119878)Further we studied the effects of viscous dissipation and heatgenerationabsorption on velocity and temperature profilesthat are depicted graphically for different values of fluid-particle interaction parameter (120573) Suction parameter (119878)heat sourcesink parameter (120582) Number density (119873) Prandtlnumber (Pr) and Eckert number (Ec)

5 Results and Discussion

Numerical calculations are performed for velocity and tem-perature profiles for various values of physical parameterssuch as fluid-particle interaction parameter (120573) Suctionparameter (119878) heat sourcesink parameter (120582) Number den-sity (119873) Prandtl number (Pr) and Eckert number (Ec) andare depicted graphically (from Figure 2 to Figure 11) Com-parison values of wall-temperature gradient are tabulated inTable 1 From Table 3 we observed that the skin friction

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120573 = 02 06 1

f998400 (120578)F998400 (120578)

120578

Figure 2 Effect of 120573 on velocity profiles with119873 = 120582 = Ec = 119878 = 05and Pr = 072

Case 1 (prescribed exponential order surface temperature)For this heating process we employ the following boundaryconditions

119879 = 119879119908(119909) at 119910 = 0

119879 997888rarr 119879infin 119879119901997888rarr 119879infin

as 119910 997888rarr infin(13)

where 119879119908= 119879infin+ 119879011989011988811199092119871 is the temperature distribution in

the stretching surface 1198790is a reference temperature and 119888

1is

constantIntroduce the dimensionless variables for the tempera-

tures 120579(120578) and 120579119901(120578) as follows

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120579119901(120578) =

119879119901minus 119879infin

119879119908minus 119879infin

(14)

where 119879 minus 119879infin= 119879011989011988811199092119871

120579(120578)Using the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(15)

where Pr = 120583119888119901119896 is the Prandtl number Ec = 119880

2

0119888119901119879 is the

Eckert number 120573 = 119871120591V1198800 and 120573120591 = 1198711205911198791198800are the fluid-

particle interaction parameter for velocity and temperatureand 120574 = 119888

119901119888119898is the ratio of specific heat

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

f998400 (120578)F998400 (120578)

120578

S = 2 3 4

Figure 3 Effect of 119878 on velocity profiles with 119873 = 120582 = Ec = 05Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

120579 (120578) = 1 at 120578 = 0

120579 (120578) 997888rarr 0 120579119901(120578) 997888rarr 0 as 120578 997888rarr infin

(16)

Case 2 (prescribed exponential order heat flux) For thisheating process consider the boundary conditions as follows

120597119879

120597119910= minus

119902119908(119909)

119896at 119910 = 0

119879 997888rarr 119879infin 119879

119901997888rarr 119879infin

as 120578 997888rarr infin

(17)

where 119902119908(119909) = 119879

1119890(1198882+1)1199092119871 119879

1is reference temperature and

1198882is constantUsing the similarity variable 120578 and (14) into (12) and on

equating the coefficient of (119909 119871)0 on both sides one canarrive at the following system of equations

12057910158401015840

(120578) + Pr [119891 (120578) 1205791015840 (120578) minus 11988811198911015840

(120578) 120579 (120578)]

+2119873

120588120573119879Pr [120579119901(120578) minus 120579 (120578)]

+2119873

120588120573Pr Ec[1198651015840 (120578) minus 1198911015840 (120578)]

2

+ Pr Ec[11989110158401015840 (120578)]2

+ 2Pr120582120579 (120578) = 0

11988811198651015840

(120578) 120579119901(120578) minus 119865 (120578) 120579

1015840

119901(120578) + 2120573

119879120574 [120579119901(120578) minus 120579 (120578)] = 0

(18)

where Ec = 119896119880201198881199011198791radic11988002120592119871 is the Eckert number

Journal of Mathematics 5

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(b)

Figure 4 Effect of 120573 on temperature profiles for both PEST and PEHF cases with119873 = 120582 = Ec = 119878 = 05 and Pr = 072

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(b)

Figure 5 Effect of 120582 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 119878 = 05 Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

1205791015840

(120578) = minus1 at 120578 = 0120579 (120578) 997888rarr 0 120579

119901(120578) 997888rarr 0 as 120578 997888rarr infin

(19)

4 Numerical Solution

Two-dimensional boundary layer flow and heat transfer ofa dusty fluid over an exponential stretching sheet is consid-ered The system of highly nonlinear ordinary differential

equations (7) (15) for PEST case and (18) for PEHF case aresolved numerically using RKF-45 method with the help of analgebraic software Maple In this method we choose suitablefinite values of 120578 rarr infin as 120578 = 5

Here we have given the comparison of our results ofminus1205791015840

(0) with Nadeem et al [14] and Bidin and Nazar [8] asin Table 1 for various values of Pr and Ec From this tableone can notice that there is a close agreement with theseapproaches and thus verifies the accuracy of themethod usedThe thermal characteristics at the wall that are examinedfor the values of temperature gradient 1205791015840(0) in PEST case

6 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEST case120579(120578)120579 p(120578)

(a)

06

045

03

015

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEHF case

120579(120578)120579 p(120578)

(b)

Figure 6 Effect of 119878 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

N = 05 1 2

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

N = 05 1 2

1

08

06

04

02

00

(b)

Figure 7 Effect of119873 on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

and the temperature 120579(0) in PEHF case are tabulated inTable 2 The computed values of skin friction coefficient11989110158401015840

(0) is tabulated in Table 3 for different values of fluid-particle interaction parameter (120573) and suction parameter (119878)Further we studied the effects of viscous dissipation and heatgenerationabsorption on velocity and temperature profilesthat are depicted graphically for different values of fluid-particle interaction parameter (120573) Suction parameter (119878)heat sourcesink parameter (120582) Number density (119873) Prandtlnumber (Pr) and Eckert number (Ec)

5 Results and Discussion

Numerical calculations are performed for velocity and tem-perature profiles for various values of physical parameterssuch as fluid-particle interaction parameter (120573) Suctionparameter (119878) heat sourcesink parameter (120582) Number den-sity (119873) Prandtl number (Pr) and Eckert number (Ec) andare depicted graphically (from Figure 2 to Figure 11) Com-parison values of wall-temperature gradient are tabulated inTable 1 From Table 3 we observed that the skin friction

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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

Journal of Mathematics 5

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120573 = 02 06 1

120578

120579(120578)120579 p(120578)

(b)

Figure 4 Effect of 120573 on temperature profiles for both PEST and PEHF cases with119873 = 120582 = Ec = 119878 = 05 and Pr = 072

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEST case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(a)

1

08

06

04

02

00 1 2 3 4 5

Fluid phase

PEHF case

Dust phase

120578

120579(120578)120579 p(120578)

120582 = minus1 minus05 0 05 1

(b)

Figure 5 Effect of 120582 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 119878 = 05 Pr = 072 and 120573 = 06

Corresponding thermal boundary conditions becomes

1205791015840

(120578) = minus1 at 120578 = 0120579 (120578) 997888rarr 0 120579

119901(120578) 997888rarr 0 as 120578 997888rarr infin

(19)

4 Numerical Solution

Two-dimensional boundary layer flow and heat transfer ofa dusty fluid over an exponential stretching sheet is consid-ered The system of highly nonlinear ordinary differential

equations (7) (15) for PEST case and (18) for PEHF case aresolved numerically using RKF-45 method with the help of analgebraic software Maple In this method we choose suitablefinite values of 120578 rarr infin as 120578 = 5

Here we have given the comparison of our results ofminus1205791015840

(0) with Nadeem et al [14] and Bidin and Nazar [8] asin Table 1 for various values of Pr and Ec From this tableone can notice that there is a close agreement with theseapproaches and thus verifies the accuracy of themethod usedThe thermal characteristics at the wall that are examinedfor the values of temperature gradient 1205791015840(0) in PEST case

6 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEST case120579(120578)120579 p(120578)

(a)

06

045

03

015

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEHF case

120579(120578)120579 p(120578)

(b)

Figure 6 Effect of 119878 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

N = 05 1 2

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

N = 05 1 2

1

08

06

04

02

00

(b)

Figure 7 Effect of119873 on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

and the temperature 120579(0) in PEHF case are tabulated inTable 2 The computed values of skin friction coefficient11989110158401015840

(0) is tabulated in Table 3 for different values of fluid-particle interaction parameter (120573) and suction parameter (119878)Further we studied the effects of viscous dissipation and heatgenerationabsorption on velocity and temperature profilesthat are depicted graphically for different values of fluid-particle interaction parameter (120573) Suction parameter (119878)heat sourcesink parameter (120582) Number density (119873) Prandtlnumber (Pr) and Eckert number (Ec)

5 Results and Discussion

Numerical calculations are performed for velocity and tem-perature profiles for various values of physical parameterssuch as fluid-particle interaction parameter (120573) Suctionparameter (119878) heat sourcesink parameter (120582) Number den-sity (119873) Prandtl number (Pr) and Eckert number (Ec) andare depicted graphically (from Figure 2 to Figure 11) Com-parison values of wall-temperature gradient are tabulated inTable 1 From Table 3 we observed that the skin friction

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEST case120579(120578)120579 p(120578)

(a)

06

045

03

015

00 1 2 3 4 5

Fluid phaseDust phase

120578

S = 2 3 4

PEHF case

120579(120578)120579 p(120578)

(b)

Figure 6 Effect of 119878 on temperature profiles for both PEST and PEHF cases with119873 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

N = 05 1 2

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

N = 05 1 2

1

08

06

04

02

00

(b)

Figure 7 Effect of119873 on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 120582 = 05 Pr = 072 and 120573 = 06

and the temperature 120579(0) in PEHF case are tabulated inTable 2 The computed values of skin friction coefficient11989110158401015840

(0) is tabulated in Table 3 for different values of fluid-particle interaction parameter (120573) and suction parameter (119878)Further we studied the effects of viscous dissipation and heatgenerationabsorption on velocity and temperature profilesthat are depicted graphically for different values of fluid-particle interaction parameter (120573) Suction parameter (119878)heat sourcesink parameter (120582) Number density (119873) Prandtlnumber (Pr) and Eckert number (Ec)

5 Results and Discussion

Numerical calculations are performed for velocity and tem-perature profiles for various values of physical parameterssuch as fluid-particle interaction parameter (120573) Suctionparameter (119878) heat sourcesink parameter (120582) Number den-sity (119873) Prandtl number (Pr) and Eckert number (Ec) andare depicted graphically (from Figure 2 to Figure 11) Com-parison values of wall-temperature gradient are tabulated inTable 1 From Table 3 we observed that the skin friction

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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

Journal of Mathematics 7

1

08

06

04

02

00 1 2 3 4 5

Pr = 072 1 15

Fluid phaseDust phase

120578

PEST case120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Pr = 072 1 15

(b)

Figure 8 Effect of Pr on temperature profiles for both PEST and PEHF cases with 119878 = Ec = 119873 = 120582 = 05 and 120573 = 06

1

08

06

04

02

00 1 2 3 4 5

Ec = 0 05 1

Fluid phaseDust phase

120578

PEST case

120579(120578)120579 p(120578)

(a)

1 2 3 4 5

Fluid phaseDust phase

120578

PEHF case

120579(120578)120579 p(120578)

1

08

06

04

02

00

Ec = 0 05 1

(b)

Figure 9 Effect of Ec on temperature profiles for both PEST and PEHF cases with 119878 = 119873 = 120582 = 05 Pr = 072 and 120573 = 06

coefficient decreases with increasing values of fluid-particleinteraction parameter (120573) as well as for the suction parameter(119878) Physically negative values of 11989110158401015840(0)mean that the surfaceexerts a drag force on the fluid so that the stretching surfacewill induce the flow

From Figure 2 the observation shows that an increasein fluid-particle interaction parameter 120573 decreases the fluidvelocity 1198911015840(120578) and increases the particle velocity 1198651015840(120578) andalso as 120573 increases the velocity of fluid will be equal to thevelocity of the dust particle Figure 3 represents the velocityprofiles for different values of suction parameter 119878 It shows

that the velocity of fluid and dust decreases when suctionparameter increases So that the momentum boundary layerthickness becomes thinner

Figure 4 depicts the temperature profiles 120579(120578) and 120579119901(120578)

versus 120578 for different values of fluid-particle interactionparameter 120573 We infer from this figure that the temperatureincrease with increase in the fluid-particle interaction param-eter 120573 and it indicates that the fluid-particle temperature isparallel to that of dust phase Also one can observ that fluidphase temperature is higher than that of dust phase FromFigure 5 the observation shows the effect of heat sourcesink

8 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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 Journal of Mathematics

1

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

04

03

02

01

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(a)

08

06

04

02

00 1 2 3 4 5

For fluid phase

120578

120579(120578)

120582 = minus05 0 05

S = 0

S = 2

02

03

015

025

01

005

00 1 2 3 4 5

For dust phase

120578

120582 = minus05 0 05

S = 0

S = 2

120579 p(120578)

(b)

Figure 10 (a) Effect of 120582 on temperature profiles for PEST case (b) effect of 120582 on temperature profiles for PEHF case

parameter 120582 on temperature profiles As 120582 increases temper-ature profiles for both fluid and dust phases increase in bothPEST and PEHF cases It is clear that the temperature in thecase of heat source is higher than in the case of sink Thisis very much significant in which the heat transfer is givenprime importance

The effects of suction parameter 119878on the temperature pro-files are depicted as in Figure 6 This figure explains that thetemperature will decrease as 119878 increasesThis is due to the factthat the fluid at ambient conditions is brought closer to thesurface which results in thinning of thermal boundary layerthickness This causes an increase in the rate of heat transfer

So suction can be used as a means for cooling the surfaceFigure 7 shows the temperature distributions 120579(120578) and 120579

119901(120578)

versus 120578 for different values of number density 119873 We inferfrom this figure that the temperature decreases with increasesin119873 for both cases

The temperature field for various values of the Prandtlnumber (Pr) is represent in Figure 8 The relative thickeningof momentum and thermal boundary layers is controlled byPrandtl number (Pr) Since small values of Pr will possesshigher thermal conductivities so that heat can diffuse fromthe sheet very quickly compared to the velocity From thisfigure it reveals that the temperature decreases with increase

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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

Journal of Mathematics 9

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus15

minus2

minus25

minus3

minus35

minus4

minus1 minus05 0 05 1120582

PEST caseEC = 05 S = 2

minus120579998400 (120578)

(a)

Pr = 072

Pr = 1

Pr = 15

Pr = 2

minus1 minus05 0 05 1120582

PEHF caseEC = 05 S = 2

120579(0)

072

064

056

048

04

(b)

Figure 11 Heat transfer characteristics for different values of Pr and 120582 for both PEST and PEHF cases

in the value of Pr Hence Prandtl number can be used toincrease the rate of cooling Figure 9 explains the effect of vis-cous dissipation on temperature profiles Viscous dissipationchanges the temperature distribution by playing a role like anenergy source which leads to affect heat transfer rates Herethe temperature increaseswith increase in the value of Ec dueto the heat energy that is stored in the liquid and frictionalheating and this is true in both cases

Figures 10(a) and 10(b) show the effect of internal heatsourcesink parameter in the presence and in the absence ofsuction parameter 119878 This shows that there is an increase oftemperature as 120582 increases which results in the reduction ofthe thermal boundary layer thickness It is also noticed thatthe temperature is less in presence of suction parameter thanin absence of suction parameter

The rate of heat transfer from the sheet that is evaluatedby the variation of wall temperature gradient 1205791015840(0) at sheet ispresent in Figure 11 for various values of Pr and 120582 It is seenfrom this figure that the rate of heat transfer decreases withincreases in Pr It is also evident that 1205791015840(0) which is negativemeans heat transfer and 120579(0) which is positive means heatabsorption and also it is clear from Table 2 We have usedthe values of 120573

119879= 06 119888

1= 1 119897 = 01 and 120588 = 1 throughout

our thermal analysis

6 Conclusions

Thepresent work deals with the boundary layer flow and heattransfer of a steady dusty fluid over an exponential stretchingsheet with viscous dissipation and heat generation or absorp-tion for both PEST andPEHF casesThe set of nonlinear ordi-nary differential equations (7) (15) for PEST case or (18) forPEHF case is solved numerically by applying RKF-45 ordermethod using the software Maple The results of the thermalcharacteristics at the wall are examined for the values of tem-perature gradient function 1205791015840(0) in PEST case and the tem-perature function 120579(0) in PEHF case which are tabulated in

Table 2 Also the results of skin friction coefficient 11989110158401015840(0) aretabulated in Table 3 for various values of fluid particle inter-action parameter (120573) and suction parameter (119878) The velocityand temperature profiles are obtained for various values ofphysical parameters like fluid-particle interaction parameter(120573) Suction parameter (119878) heat sourcesink parameter (120582)Number density (119873) Prandtl number (Pr) and Eckert num-ber (Ec) The numerical results obtained are agrees with pre-viously reported cases available in the literature [8 14]

The major findings from the present study can be sum-marized as follows

(i) Suction parameter reduces the velocity and tempera-ture profiles for both PEST and PEHF cases

(ii) Heat sourcesink effect is less in permeable than inimpermeable stretching sheet

(iii) The PEHF boundary condition is better suited foreffective cooling of the stretching sheet

(iv) The effect of increasing the values of 120573 Ec and 120582 is toincrease the wall temperature gradient function 1205791015840(0)and wall temperature function 120579(0) and decrease theincreasing values of Pr and 119873 both PEST and PEHFcases

(v) The effect of heat sourcesink on temperature is quiteopposite to that of suction parameter

(vi) If 120573 rarr 0119873 rarr 0 and 119878 rarr 0 then our results coin-cide with the results of Nadeem et al [14] and Bidinand Nazar [8] for different values of Prandtl and Eck-ert numbers

(vii) The effect of Ec increases while Pr decreases the ther-mal boundary layer thickness

(viii) Fluid phase temperature is higher than that of dustphase

10 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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 Journal of Mathematics

Acknowledgment

Theauthors would like to acknowledge the reviewers for theirsuggestions which lead to the present form of themanuscriptFurther one of the co-author G M Pavithra gratefullyacknowledges the financial support of Rajiv Gandhi NationalFellowship (RGNF) UGCNewDelhi India for pursuing thiswork

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface i boundary-layer equations for two dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurface ii boundary layer behavior on continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash235 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift Fur Ange-wandte Mathematik und Physik ZAMP vol 21 no 4 pp 645ndash647 1970

[4] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D vol 32 no 5 pp 577ndash585 1999

[5] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Me-chanics vol 53 no 6 pp 643ndash651 2001

[6] M K Partha P V S N Murthy and G P Rajasekhar ldquoEffect ofviscous dissipation on the mixed convection heat transfer froman exponentially stretching surfacerdquo Heat and Mass Transfervol 41 no 4 pp 360ndash366 2005

[7] E Sanjayanand and S K Khan ldquoOn heat and mass transferin a viscoelastic boundary layer flow over an exponentiallystretching sheetrdquo International Journal of Thermal Sciences vol45 no 8 pp 819ndash828 2006

[8] B Bidin and R Nazar ldquoNumerical solution of the boundarylayer flow over an exponentially stretching sheet with thermalradiationrdquo European Journal of Scientific Research vol 33 no 4pp 710ndash717 2009

[9] M A Samad and M Mohebujjaman ldquoMHD heat and masstransfer free convection flow along a vertical stretching sheet inpresence of magnetic field with heat generationrdquo Research Jour-nal of Applied Sciences vol 1 no 3 pp 98ndash106 2009

[10] E M A Elbashbeshy and D A Aldawody ldquoHeat transfer overan unsteady stretching surface with variable heat flux in thepresence of a heat source or sinkrdquo Computers and Mathematicswith Applications vol 60 no 10 pp 2806ndash2811 2010

[11] A Ishak ldquoMHD boundary layer flow due to an exponentiallystretching sheet with radiation effectrdquo SainsMalaysiana vol 40no 4 pp 391ndash395 2011

[12] A R P Bala and R N Bhaskar ldquoThermal radiation effects onhydro-magnetic flow due to an exponentially stretching sheetrdquoInternational Journal of Applied Mathematics vol 3 no 4 pp300ndash306 2011

[13] B Sahoo and S Poncet ldquoFlow and heat transfer of a third gradefluid past an exponentially stretching sheet with partial slipboundary conditionrdquo International Journal of Heat and MassTransfer vol 54 no 23-24 pp 5010ndash5019 2011

[14] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[15] H Kumar ldquoRadiative heat transfer with hydromagnetic flowand viscous dissipation over a stretching surface in the presenceof variable heat fluxrdquoThermal Science vol 13 no 2 pp 163ndash1692009

[16] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[17] P G Saffman ldquoOn the stability of laminar flow of a dusty gasrdquoJournal of Fluid Mechanics vol 13 pp 120ndash128 1962

[18] N Datta and S K Mishra ldquoBoundary layer flow of a dusty fluidover a semi-infinite flat platerdquo Acta Mechanica vol 42 no 1-2pp 71ndash83 1982

[19] B J Gireesha G K Ramesh S Abel and C S BagewadildquoBoundary layer flow and heat transfer of a dusty fluid flow overa stretching sheet with non-uniform heat sourcesinkrdquo Interna-tional Journal of Multiphase Flow vol 37 no 8 pp 977ndash9822011

[20] B J Gireesha G K Ramesh and C S Bagewadi ldquoHeat transferinMHDflowof a dusty fluid over a stretching sheet with viscousdissipationrdquo Journal of Applied Sciences Research vol 3 no 4pp 2392ndash2401 2012

[21] B J Gireesha G S Roopa andC S Bagewadi ldquoEffect of viscousdissipation and heat source on flow and heat transfer of dustyfluid over unsteady stretching sheetrdquo Applied Mathematics andMechanics vol 33 no 8 pp 1001ndash1014 2012

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

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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