Research Article Viscous Flows Driven by Uniform Shear over a … · 2019. 7. 31. · a stretching sheet in presence of uniform shear-free stream. Xu [ ] obtained the analytic solution

Research ArticleViscous Flows Driven by Uniform Shear over a PorousStretching Sheet in the Presence of SuctionBlowing

Samir Kumar Nandy1 and Swati Mukhopadhyay2

1 Department of Mathematics AKPC Mahavidyalaya Bengai Hooghly 712 611 India2Department of Mathematics The University of Burdwan West Bengal 713104 India

Correspondence should be addressed to Samir Kumar Nandy nandysamiryahoocom

Received 21 February 2014 Revised 10 May 2014 Accepted 12 May 2014 Published 25 May 2014

Academic Editor Boming Yu

Copyright copy 2014 S K Nandy and S Mukhopadhyay 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

An analysis is carried out to study the steady two-dimensional flow of an incompressible viscous fluid past a porous deformablesheet which is stretched in its own plane with a velocity proportional to the distance from the fixed point subject to uniform suctionor blowing A uniform shear flow of strain rate 120573 is considered over the stretching sheet The analysis of the result obtained showsthat the magnitude of the wall shear stress increases with the increase of suction velocity and decreases with the increase of blowingvelocity and this effect is more pronounced for suction than blowing It is seen that the horizontal velocity component (at a fixedstreamwise position along the plate) increases with the increase in the ratio of shear rate 120573 and stretching rate (119888) (ie 120573119888) andthere is an indication of flow reversal It is also observed that this flow reversal region increases with the increase in 120573119888

1 Introduction

Suction or injection (blowing) of a fluid through the bound-ing surface can significantly change the flow field andconsequently affects the heat transfer rate from the surfaceInjection or withdrawal of fluid through a porous boundingheatedcooled wall is of great general interest in practicalproblems such as cooling of films cooling of wires and coat-ing of polymer fiber The process of suction or blowing hasits importance in many engineering and industrial activitiessuch as in the design of thrust bearing and radial diffusersand thermal oil recovery Suction is also applied to chemicalprocesses to remove reactants whereas blowing is used to addreactants prevent corrosion or scaling and reduce the drag

During the last decades the study of flowover a stretchingsurface has attracted much more interest of the researchersdue to its various industrial applications such as extrusion ofpolymer sheets continuous stretchingmanufacturing plasticfilms and artificial fibers In a melt-spinning process theextrudate from the die is generally drawn and simultaneouslystretched into a sheet which is then solidified throughquenching or gradual cooling by direct contact with the

waterThemechanical properties of the final product dependcrucially on the rate of coolingheating along the surfacewhile being stretched Sakiadas [1] was the first to studyboundary layer flow due to a rigid flat continuous surfacemoving in its own plane Erickson et al [2] analyzed theboundary layer flow due to the motion of a porous flatplate when the transverse velocity at the surface is nonzeroA detailed analysis of the boundary layer flow due to astretching sheet has been carried out by Danberg and Fansler[3]

Later Crane [4] gave an exact similarity solution inclosed analytical form for steady boundary layer flow of anincompressible viscous fluid caused solely by the stretchingof an elastic flat sheet which moves in its own plane witha velocity varying linearly with distance from a fixed pointWang [5] investigated a uniform shear flows over a quiescentliquid and he showed that an interfacial boundary layerdevelops both in the air and the liquid P S Gupta andA S Gupta [6] investigated the heat and mass transfercorresponding to the similarity solution for the boundarylayer flow over a stretching sheet subject to uniform suctionor blowing Rajagopal et al [7] studied the boundary layer

Hindawi Publishing CorporationJournal of FluidsVolume 2014 Article ID 206846 7 pageshttpdxdoiorg1011552014206846

2 Journal of Fluids

flow over a stretching surface for second-order fluid andobtained similarity solutions of the boundary layer equationsDandapat and Gupta [8] extended the same problem withheat transfer and found exact analytical solutions

Shear driven flows namely wall driven Couette flowwind driven Ekman flow and so forth are the classical topicsof fluid mechanics Due to their wide range of applicationsshear driven flows attract the attention of the researchersThe study of boundary layer flow driven by uniform shear isseen to have fewer contributors in fluidmechanics Rajagopalet al [9] studied the nonsimilar boundary layer flow of asecond-order fluid over a stretching sheet in the presence of auniform shear flow Weidman et al [10] reported a similaritysolution of the boundary layer flow over a flat impermeableplate with free shear flows driven by rotational velocitiesThe extension of the same problem to a permeable flat platewas analyzed by Magyari et al [11] Cossali [12] gave thesimilarity solutions of the energy and momentum boundarylayer equations for a power-law shear driven flow over asemi-infinite flat plate The thermal boundary layer beneathan external uniform shear flow was considered by Magyariand Weidman [13] Weidman et al [14] also investigatedthe effects of plate extension and transpiration on uniformshear flow over a semi-infinite flat plate by considering theboundary layer approximations Due to the presence of shearin the free stream the free stream is no longer rotation-freeAs a result the flow behaviours are quite different from therotation-free flow In most of the works the flat surface waskept stationary But the interaction of shear flow and wallstretching affects the flow significantly Fang [15] analyzedthe heat transfer characteristics for boundary layer flow pasta stretching sheet in presence of uniform shear-free streamXu [16] obtained the analytic solution in case of boundarylayer flow driven by power-law shear Very recently the heattransfer characteristics of a viscous incompressible fluid overa stretchingshrinking sheet in a uniform shear flow witha convective surface boundary condition were analyzed byAman et al [17]

Motivated by the above investigations in this paperthe steady two-dimensional incompressible viscous fluidflow past a porous stretching sheet in presence of uniformsuctionblowing has been investigated A uniform shear flowof strain rate 120573 is considered over the stretching surface Thebehaviours of the horizontal component of the flow velocityare explained and it is seen that there is a flow reversal in caseof suction or blowing at the sheet

2 Flow Analysis

Consider the steady two-dimensional flow of a viscous fluidtowards a stretching surface coinciding with the plane 119910 = 0the flow being confined to the region 119910 gt 0 We choose thecoordinate system such that the 119909-axis is along the sheet 119910-axis is normal to the sheet and the origin of the coordinatesystem is located at a finite position on the sheet Two equaland opposing forces are applied on the stretching surfacealong the 119909-axis so that the surface is stretched keeping theorigin fixed Here we consider a uniform shear flow of strainrate 120573 over the stretching surface The free stream velocity is

XX998400

u(x y)

V0V0 V0 V0 V0

Y

O

Uniform shear flow

Free stream

Figure 1 A sketch of the physical problem and the coordinatesystem involved

uniform shear and the surface is subject to uniform suctionor blowing It is to be noted that due to the presence ofblowing or suction a streamwise pressure gradient is requiredto maintain the flow The flow configuration is shown inFigure 1

The inviscid version of the present steady flow is given interms of the stream function 120595

0as

1205950= 120573119910(

119910

2minus 1205751) + 1205752119909 (1)

where1205751and1205752are two constantsHere120575

1is the displacement

thickness arising out of the boundary layer on the stretchingsurface and 120575

2is the parameter which controls the horizon-

tal pressure gradient that produces the shear flow So theparameters 120575

1and 1205752have some effects on the flow field (see

Drazin and Riley [18]) The velocity components along 119909 and119910 directions respectively for the flow described by (1) are

1198801= 120573 (119910 minus 120575

1) 119881

1= minus1205752 (2)

The boundary conditions at the stretching surface are

119906 = 119888119909 V = V119908

at 119910 = 0 (3)

where 119888 gt 0 is a constantThe boundary conditions at infinityare

119906 997888rarr 1198801(119909 119910) V 997888rarr 119881

1(119909 119910) as 119910 997888rarr infin (4)

where1198801and119881

1are given by (2) Near the stretching surface

we take the stream function in the form120595

]= 120585119865 (120578) +119882(120578) (5)

Journal of Fluids 3

where

120585 = (119888

])12

119909 120578 = (119888

])12

119910 (6)

and ] is the kinematic viscosity The velocity components 119906and V along 119909 and 119910 directions are given by

119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (7)

Hence the dimensionless velocity components 119880 and 119881 areobtained from (5) and (6) as

119880 = 1205851198651015840 (120578) + 1198821015840 (120578)

119881 = minus 119865 (120578) (8)

where 119880 = 119906(119888])12 and 119881 = V(119888])12 Substituting (8) intothe Navier-Stokes equations we get

minus1

120588

120597119901

120597119909= ]2(

119888

])32

times [120585 (11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840) + (11986510158401198821015840 minus 11986511988210158401015840 minus119882101584010158401015840)]

minus1

120588

120597119901

120597119910= ]2(

119888

])32

[1198651198651015840 + 11986510158401015840]

(9)

where a prime denotes differentiation with respect to 120578Eliminating 119901 between (9) and equating the coefficients of 1205850and 1205851 we get upon integration

11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840 = 1198621

11986510158401198821015840 minus 11986511988210158401015840 minus119882101584010158401015840 = 1198622

(10)

where 1198621and 119862

2are the constants of integration

The boundary conditions (3) and (4) become

119865 (0) = minus1198810 1198651015840 (0) = 1 1198651015840 (infin) = 0 (11)

119882(0) = 0 1198821015840 (0) = 0

1198821015840 (120578) =120573

119888(120578 minus 119889

1) as 120578 997888rarr infin

(12)

where 1198810= V119908(119888])12 and 119889

1= (119888])12120575

1 It is to be noted

that the boundary condition 119882(0) = 0 is taken due to thesteam function Let us assume that 119865(120578) rarr 119889

2as 120578 rarr infin

where 1198892= 1205752(119888])12 Here 119889

2is the dimensionless shear rate

parameter linked to the shear flow and1198891is the dimensionless

displacement thickness parameter The constants 1198621and 119862

2

are now obtained from (10) using (11) and (12) as 1198621= 0 and

1198622= minus(120573119888)119889

2 Hence (10) becomes

11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840 = 0 (13)

11986510158401198821015840 minus 11986511988210158401015840 minus119882101584010158401015840 = minus120573

1198881198892 (14)

Here the dimensionless shear rate parameter 1198892

isobtained by integrating (13) numerically by using the bound-ary condition (11)When (13) and (14) are substituted into (9)we get the dimensionless pressure distribution 119875(120585 120578) afterintegration as

minus119875 =1

21198652 + 1198651015840 minus

120573

1198881198892120585 + constant (15)

where 119875 = 119901(119909 119910)(120588119888]) The dimensionless wall shear stress119879 is given as

119879 = 12058511986510158401015840 (0) + 11988210158401015840(0) (16)

The dimensionless stream function for the flow can beobtained as

120595 (120585 120578) = 120585119865 (120578) + int120578

0

1198821015840 (120578) 119889120578 (17)

where 120585 is the dimensionless distance along the surface

3 Method of Numerical Solution

In the absence of an analytic solution of a problem a numeri-cal solution is indeed an obvious and natural choiceThus thegoverning equations (13) and (14) along with the boundaryconditions (11)-(12) are solved numerically by fourth-orderRunge-Kutta method with shooting technique To do this wefirst transform the nonlinear differential equation (13) to asystem of three first-order differential equations as

11991010158401= 1199102 1199101015840

2= 1199103 1199101015840

3= 11991022minus 11991011199103 (18)

where1199101= 119865(120578)119910

2= 1198651015840(120578)119910

3= 11986510158401015840(120578) and a prime denotes

differentiationwith respect to the independent variable 120578Theboundary conditions (11) become

1199101= minus1198810 119910

2= 1 at 120578 = 0

1199102997888rarr 0 as 120578 997888rarr infin

(19)

For a given value of 1198810 the values of 119910

1and 119910

2are known at

the starting point 120578 = 0 Now the value of 1199102as 120578 rarr infin is

replaced by 1199102at a finite value 120578 = 120578

infinto be determined later

The value of 1199102at 120578 = 0 is guessed in order to initiate the

integration scheme Starting from the given values of 1199101and

1199102at 120578 = 0 and the guessed value of 119910

3at 120578 = 0 we integrate

the system of first-order equations (18) by using a fourth-order Runge-Kutta method up to the end-point 120578 = 120578

infin The

computed value of 1199102at 120578 = 120578

infinis then compared with 119910

2

at 120578 = 120578infin The absolute difference between these two values

should be as small as possible To this end we use a Newton-Raphson iteration procedure to assure quadratic convergenceof the iterations The value of 120578

infinis then increased till 119910

2

attains the value zero asymptoticallyUsing the numerical values of 119865(120578) obtained from the

solutions of (11) and (13) (14) along with the boundaryconditions (12) is solved numerically using the same methodas described above to obtain119882(120578)

4 Journal of Fluids

120573c

120585F998400998400(0)+W

998400998400(0)

minus2

minus1

minus3

minus4

minus5

minus6

minus7

minus8

00 02 04 06 08 10 12 14 16 18 20

V0 = 00

V0 = minus10

V0 = minus20V0 = minus30

Figure 2 Variation of wall shear stress with 120573119888 for several valuesof 1198810(le 0) (ie suction at the plate) when 120585 = 10 and 119889

1= 10

120573c

00 02 04 06 08 10 12 14 16 18 20

V0 = 00V0 = 10

V0 = 20V0 = 30

120585F998400998400(0)+W

998400998400(0)

minus02

minus04

minus06

minus08

minus10

minus12

minus14

minus16

minus18

minus20

Figure 3 Variation of wall shear stress with 120573119888 for several valuesof 1198810(ge 0) (ie blowing at the plate) when 120585 = 10 and 119889

1= 10

4 Results and Discussion

Figures 2 and 3 show the variation of dimensionlesswall shearstress with 120573 for several values of119881

0keeping the values of the

other parameters fixed Note that 1198810= 0 corresponds to the

case when there is no suction or blowing at the sheet 1198810lt 0

corresponds to suction and 1198810gt 0 corresponds to blowing

at the sheet Figure 2 reveals that as the suction velocityincreases themagnitude of thewall shear stress increases Onthe other hand Figure 3 indicates that with the increase ofblowing velocity at the plate the magnitude of the wall shearstress decreases From these figures it is also noticed that theeffect of wall shear stress ismore pronounced for suction thanblowing at the stretching sheet

Figure 4 shows the variation of 119880(120585 120578) the horizontalcomponent of velocity with 120578 for several values of 120573119888 whenthere is no suction or blowing velocity at the sheet Figure 5shows the same behaviour for suction (119881

0= minus20) at the

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)

minus3 minus2 minus1 0 1 2 3

14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 00

Figure 4 Variation of 119880(120585 120578) with 120578 for several values of 120573119888 with120585 = 10 119889

1= 30 and119881

0= 0 (ie no suction or blowing at the plate)

120573c = 10

120573c = 20120573c = 30

U(120585 120578)

minus3minus4minus5 minus2 minus1 0 1 2 3 4 5 6 7 8 9 10

V0 = minus20

12057814

12

10

8

6

4

2

0


1= 30 and 119881

0= minus20 (ie for suction at the plate)

stretching sheet and Figure 6 shows for blowing (1198810= 20)

at the stretching sheet keeping other parameters fixed Fromthese figures it is observed that 119880(120585 120578) increases with theincrease of 120573119888 except in a small region near the sheet Thefigures indicate that there is a flow reversal very near thesheetThis region of flow reversal gradually increases with theincrease of 120573119888

Figures 7 and 8 show the variation of 119880(120585 120578) with 120578for several values of the suction and blowing at the platerespectively Figure 7 reveals that as |119881

0| increases (ie

suction velocity increases) the horizontal velocity at a pointdecreases except in a small region near the sheet Figure 8shows that as the blowing velocity increases velocity at apoint increases Notice that there is an indication of flowreversal From these two figures it can also be concluded that

Journal of Fluids 5

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)

minus2 minus1 0 1 2 3

14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 20


1= 30 and 119881

0= 20 (ie for blowing at the plate)

120578

U(120585 120578)

0 1 2 3 4 50

1

2

3

4

5

V0 = 00

V0 = minus10V0 = minus20V0 = minus30

Figure 7 Variation of 119880(120585 120578) with 120578 for several values of 1198810(le 0)

(ie suction at the plate) when 120573119888 = 20 120585 = 10 and 1198891= 10

with the increase of 1198810(magnitude) the flow reversal region

increasesFigure 9 shows the variation of 119880(120585 120578) with 120578 for several

values of the shear rate parameter 1198891in the absence of suction

or blowing at the sheet Figure 10 shows the same for suctionand Figure 11 stands for blowing at the sheet for the sameset of parameters From these figures it is seen that with theincrease of shear rate parameter 119889

1 the region of flow reversal

increases The tendency of flow reversal is more prominentin case of suction than blowing It is also observed that asthe shear rate parameter 119889

1increases velocity at a point also

increases Figure 12 depicts the variation of the horizontalcomponent of velocity 119880(120585 120578) with 120578 for several values of 120585(negative zero and positive) in the presence of blowing at thesheetThe figure reveals that as 120585 increases velocity at a pointdecreases

V0 = 00V0 = 10

V0 = 20V0 = 30

U(120585 120578)

minus2 minus1 0 1 2 3 4 5 6

120578

14

12

10

8

6

4

2

0

Figure 8 Variation of 119880(120585 120578) with 120578 for several values of 1198810(ge 0)

(ie blowing at the plate) when 120573119888 = 20 120585 = 10 and 1198891= 10

V0 = 00120578

14

12

10

8

6

4

2

0

U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

d1 = 10d1 = 20

d1 = 30

Figure 9 Variation of 119880(120585 120578) with 120578 for several values of 1198891with

120585 = 10 120573119888 = 20 and 1198810= 0 (ie no suction or blowing at the

plate)

Presentation of full stability analysis is beyond the scopeof the present work since a stability analysis requires anunsteady flow whereas our problem is a steady one But aswe know that in boundary layer flow the reverse profilessuggest temporal instability so one can expect that in the caseof full Navier-Stokes solution the same feature holds goodThis can be verified using the stability analysis by adoptingthe techniques of Merkin [19] and Mahapatra et al [20]

5 Concluding Remark

We have investigated the steady two-dimensional flow ofincompressible viscous fluid past a porous deformable sheetwhich is stretched in its own plane with velocity 119888119909 119909being the distance along the sheet from the fixed pointwith uniform suction or blowing A uniform shear flow of

6 Journal of Fluids

U(120585 120578)

minus3minus4 minus2 minus1 0 1 2 3 4 5 6

120578

10

8

7

9

6

5

4

3

2

1

0

V0 = minus20

d1 = 10d1 = 20

d1 = 30


120585 = 10 120573119888 = 20 and 1198810= minus20 (ie for suction at the plate)

U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

120578

10

8

7

9

6

5

4

3

2

1

0

d1 = 10d1 = 20

d1 = 30

V0 = 20


120585 = 10 120573119888 = 20 and 1198810= 20 (ie for blowing at the plate)

strain rate 120573 is considered here over the stretching surfaceIt is seen that the magnitude of wall skin friction increaseswith the increase of suction velocity (magnitude) but itdecreases with the increase of blowing velocity and this effectis more pronounced in suction than blowing The horizontalcomponent of velocity at a fixed point increases with theincrease of 120573119888 and there is a flow reversal This region offlow reversal increaseswith the increase of120573119888Thebehaviourof the horizontal velocity component is shown for differentparameters

Conflict of Interests

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

U(120585 120578)minus1 0 1 2 3 4 5

120578

4

3

2

1

0

120585 = minus10120585 = 00

120585 = 10

V0 = minus05

Figure 12 Variation of 119880(120585 120578) with 120578 for several values of 120585 with1198891= 10 120573119888 = 20 and 119881


Acknowledgments

The author Samir Kumar Nandy is grateful to the Univer-sity Grants Commission New Delhi India for providingfinancial support under a minor research project Grant noF PSW-00213-14 to carry out the work Authors would liketo thank referees for the valuable comments which enhancedthe quality of the paper

References

[1] B C Sakiadas ldquoBoundary layer behaviour on continuous solidsurfacerdquo AIChE Journal vol 7 pp 26ndash28 1961

[2] L E Erickson L T Fan and V G Fox ldquoHeat and mass transferon a moving continuous flat plate with suction or injectionrdquoIndustrial and Engineering Chemistry Fundamentals vol 5 no1 pp 19ndash25 1966

[3] J E Danberg and K S Fansler ldquoA non-similar moving-wallboundary-layer problemrdquo Quarterly of Applied Mathematicsvol 34 pp 305ndash309 1976

[4] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furAngewandte Mathematik und Physik vol 21 no 4 pp 645ndash6471970

[5] C Y Wang ldquoThe boundary layers due to shear flow over a stillfluidrdquo Physics of Fluids A vol 4 no 6 pp 1304ndash1306 1992

[6] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquoTheCanadian Journalof Chemical Engineering vol 55 pp 744ndash746 1977

[7] K R Rajagopal T Y Na andA S Gupta ldquoFlow of a viscoelasticfluid over a stretching sheetrdquo Rheologica Acta vol 24 no 2 pp213ndash215 1984

[8] B S Dandapat and A S Gupta ldquoFlow and heat transfer in aviscoelastic fluid over a stretching sheetrdquo International Journalof Non-Linear Mechanics vol 24 no 3 pp 215ndash219 1989

Journal of Fluids 7

[9] K R Rajagopal T Y Na and A S Gupta ldquoA non-similarboundary layer on a stretching sheet in a non-Newtonian fluidwith uniform free streamrdquo Journal ofMathematical and PhysicalSciences vol 21 pp 189ndash200 1987

[10] P D Weidman D G Kubitschek and S N Brown ldquoBoundarylayer similarity flow driven by power-law shearrdquo Acta Mechan-ica vol 120 pp 199ndash215 1997

[11] E Magyari B Keller and I Pop ldquoBoundary-layer similarityflows driven by a power-law shear over a permeable planesurfacerdquo Acta Mechanica vol 163 no 3-4 pp 139ndash146 2003

[12] G E Cossali ldquoSimilarity solutions of energy and momentumboundary layer equations for a power-law shear driven flowover a semi-infinite flat platerdquo European Journal of MechanicsmdashBFluids vol 25 no 1 pp 18ndash32 2006

[13] E Magyari and P D Weidman ldquoHeat transfer on a platebeneath an external uniform shear flowrdquo International Journalof Thermal Sciences vol 45 no 2 pp 110ndash115 2006

[14] P D Weidman A M J Davis and D G Kubitschek ldquoCroccovariable formulation for uniform shear flow over a stretchingsurface with transpiration multiple solutions and stabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 58 no2 pp 313ndash332 2008

[15] T Fang ldquoFlow and heat transfer characteristics of the boundarylayers over a stretching surface with a uniform-shear freestreamrdquo International Journal of Heat andMass Transfer vol 51no 9-10 pp 2199ndash2213 2008

[16] H Xu ldquoHomotopy analysis of a self-similar boundary-flowdriven by a power-law shearrdquo Archive of Applied Mechanics vol78 no 4 pp 311ndash320 2008

[17] F Aman A Ishak and I Pop ldquoHeat transfer at a stretch-ingshrinking surface beneath an external uniform shear flowwith a convective boundary conditionrdquo Sains Malaysiana vol40 no 12 pp 1369ndash1374 2011

[18] P G Drazin and N RileyThe Navier-Stokes Equations A Clas-sification of Flows and Exact Solutions Cambridge UniversityPress Cambridge UK 2006

[19] J H Merkin ldquoMixed convection boundary layer flow on avertical surface in a saturated porous mediumrdquo Journal ofEngineering Mathematics vol 14 no 4 pp 301ndash313 1980

[20] T R Mahapatra S K Nandy and A S Gupta ldquoDual solutionof MHD stagnation point flow towards a stretching surfacerdquoEngineering vol 2 pp 299ndash305 2010

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2 Journal of Fluids

flow over a stretching surface for second-order fluid andobtained similarity solutions of the boundary layer equationsDandapat and Gupta [8] extended the same problem withheat transfer and found exact analytical solutions

Shear driven flows namely wall driven Couette flowwind driven Ekman flow and so forth are the classical topicsof fluid mechanics Due to their wide range of applicationsshear driven flows attract the attention of the researchersThe study of boundary layer flow driven by uniform shear isseen to have fewer contributors in fluidmechanics Rajagopalet al [9] studied the nonsimilar boundary layer flow of asecond-order fluid over a stretching sheet in the presence of auniform shear flow Weidman et al [10] reported a similaritysolution of the boundary layer flow over a flat impermeableplate with free shear flows driven by rotational velocitiesThe extension of the same problem to a permeable flat platewas analyzed by Magyari et al [11] Cossali [12] gave thesimilarity solutions of the energy and momentum boundarylayer equations for a power-law shear driven flow over asemi-infinite flat plate The thermal boundary layer beneathan external uniform shear flow was considered by Magyariand Weidman [13] Weidman et al [14] also investigatedthe effects of plate extension and transpiration on uniformshear flow over a semi-infinite flat plate by considering theboundary layer approximations Due to the presence of shearin the free stream the free stream is no longer rotation-freeAs a result the flow behaviours are quite different from therotation-free flow In most of the works the flat surface waskept stationary But the interaction of shear flow and wallstretching affects the flow significantly Fang [15] analyzedthe heat transfer characteristics for boundary layer flow pasta stretching sheet in presence of uniform shear-free streamXu [16] obtained the analytic solution in case of boundarylayer flow driven by power-law shear Very recently the heattransfer characteristics of a viscous incompressible fluid overa stretchingshrinking sheet in a uniform shear flow witha convective surface boundary condition were analyzed byAman et al [17]

Motivated by the above investigations in this paperthe steady two-dimensional incompressible viscous fluidflow past a porous stretching sheet in presence of uniformsuctionblowing has been investigated A uniform shear flowof strain rate 120573 is considered over the stretching surface Thebehaviours of the horizontal component of the flow velocityare explained and it is seen that there is a flow reversal in caseof suction or blowing at the sheet

2 Flow Analysis

Consider the steady two-dimensional flow of a viscous fluidtowards a stretching surface coinciding with the plane 119910 = 0the flow being confined to the region 119910 gt 0 We choose thecoordinate system such that the 119909-axis is along the sheet 119910-axis is normal to the sheet and the origin of the coordinatesystem is located at a finite position on the sheet Two equaland opposing forces are applied on the stretching surfacealong the 119909-axis so that the surface is stretched keeping theorigin fixed Here we consider a uniform shear flow of strainrate 120573 over the stretching surface The free stream velocity is

XX998400

u(x y)

V0V0 V0 V0 V0

Y

O

Uniform shear flow

Free stream

Figure 1 A sketch of the physical problem and the coordinatesystem involved

uniform shear and the surface is subject to uniform suctionor blowing It is to be noted that due to the presence ofblowing or suction a streamwise pressure gradient is requiredto maintain the flow The flow configuration is shown inFigure 1

The inviscid version of the present steady flow is given interms of the stream function 120595

0as

1205950= 120573119910(

119910

2minus 1205751) + 1205752119909 (1)

where1205751and1205752are two constantsHere120575

1is the displacement

thickness arising out of the boundary layer on the stretchingsurface and 120575

2is the parameter which controls the horizon-

tal pressure gradient that produces the shear flow So theparameters 120575

1and 1205752have some effects on the flow field (see

Drazin and Riley [18]) The velocity components along 119909 and119910 directions respectively for the flow described by (1) are

1198801= 120573 (119910 minus 120575

1) 119881

1= minus1205752 (2)

The boundary conditions at the stretching surface are

119906 = 119888119909 V = V119908

at 119910 = 0 (3)

where 119888 gt 0 is a constantThe boundary conditions at infinityare

119906 997888rarr 1198801(119909 119910) V 997888rarr 119881

1(119909 119910) as 119910 997888rarr infin (4)

where1198801and119881

1are given by (2) Near the stretching surface

we take the stream function in the form120595

]= 120585119865 (120578) +119882(120578) (5)

Journal of Fluids 3

where

120585 = (119888

])12

119909 120578 = (119888

])12

119910 (6)


119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (7)


119880 = 1205851198651015840 (120578) + 1198821015840 (120578)

119881 = minus 119865 (120578) (8)


minus1

120588

120597119901

120597119909= ]2(

119888

])32


minus1

120588

120597119901

120597119910= ]2(

119888

])32

[1198651198651015840 + 11986510158401015840]

(9)


11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840 = 1198621

11986510158401198821015840 minus 11986511988210158401015840 minus119882101584010158401015840 = 1198622

(10)

where 1198621and 119862



119865 (0) = minus1198810 1198651015840 (0) = 1 1198651015840 (infin) = 0 (11)

119882(0) = 0 1198821015840 (0) = 0

1198821015840 (120578) =120573

119888(120578 minus 119889

1) as 120578 997888rarr infin

(12)

where 1198810= V119908(119888])12 and 119889

1= (119888])12120575

1 It is to be noted



where 1198892= 1205752(119888])12 Here 119889




2


1198622= minus(120573119888)119889


11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840 = 0 (13)


1198881198892 (14)



minus119875 =1

21198652 + 1198651015840 minus

120573

1198881198892120585 + constant (15)


119879 = 12058511986510158401015840 (0) + 11988210158401015840(0) (16)


120595 (120585 120578) = 120585119865 (120578) + int120578

0

1198821015840 (120578) 119889120578 (17)




11991010158401= 1199102 1199101015840

2= 1199103 1199101015840

3= 11991022minus 11991011199103 (18)

where1199101= 119865(120578)119910

2= 1198651015840(120578)119910



1199101= minus1198810 119910

2= 1 at 120578 = 0


(19)


1and 119910

2are known at









infin The



2




2



4 Journal of Fluids

120573c

120585F998400998400(0)+W

998400998400(0)

minus2

minus1

minus3

minus4

minus5

minus6

minus7

minus8

00 02 04 06 08 10 12 14 16 18 20

V0 = 00

V0 = minus10



1= 10

120573c

00 02 04 06 08 10 12 14 16 18 20

V0 = 00V0 = 10

V0 = 20V0 = 30

120585F998400998400(0)+W

998400998400(0)

minus02

minus04

minus06

minus08

minus10

minus12

minus14

minus16

minus18

minus20


1= 10









0= minus20) at the

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 00


1= 30 and119881


120573c = 10

120573c = 20120573c = 30

U(120585 120578)


V0 = minus20

12057814

12

10

8

6

4

2

0


1= 30 and 119881





0| increases (ie


Journal of Fluids 5

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 20


1= 30 and 119881


120578

U(120585 120578)

0 1 2 3 4 50

1

2

3

4

5

V0 = 00












V0 = 00V0 = 10

V0 = 20V0 = 30

U(120585 120578)

minus2 minus1 0 1 2 3 4 5 6

120578

14

12

10

8

6

4

2

0



V0 = 00120578

14

12

10

8

6

4

2

0

U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

d1 = 10d1 = 20

d1 = 30



plate)


5 Concluding Remark


6 Journal of Fluids

U(120585 120578)


120578

10

8

7

9

6

5

4

3

2

1

0

V0 = minus20

d1 = 10d1 = 20

d1 = 30



U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

120578

10

8

7

9

6

5

4

3

2

1

0

d1 = 10d1 = 20

d1 = 30

V0 = 20






U(120585 120578)minus1 0 1 2 3 4 5

120578

4

3

2

1

0

120585 = minus10120585 = 00

120585 = 10

V0 = minus05



Acknowledgments


References









Journal of Fluids 7


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 3

where

120585 = (119888

])12

119909 120578 = (119888

])12

119910 (6)


119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (7)


119880 = 1205851198651015840 (120578) + 1198821015840 (120578)

119881 = minus 119865 (120578) (8)


minus1

120588

120597119901

120597119909= ]2(

119888

])32


minus1

120588

120597119901

120597119910= ]2(

119888

])32

[1198651198651015840 + 11986510158401015840]

(9)


11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840 = 1198621

11986510158401198821015840 minus 11986511988210158401015840 minus119882101584010158401015840 = 1198622

(10)

where 1198621and 119862



119865 (0) = minus1198810 1198651015840 (0) = 1 1198651015840 (infin) = 0 (11)

119882(0) = 0 1198821015840 (0) = 0

1198821015840 (120578) =120573

119888(120578 minus 119889

1) as 120578 997888rarr infin

(12)

where 1198810= V119908(119888])12 and 119889

1= (119888])12120575

1 It is to be noted



where 1198892= 1205752(119888])12 Here 119889




2


1198622= minus(120573119888)119889


11986510158402 minus 11986511986510158401015840 minus 119865101584010158401015840 = 0 (13)


1198881198892 (14)



minus119875 =1

21198652 + 1198651015840 minus

120573

1198881198892120585 + constant (15)


119879 = 12058511986510158401015840 (0) + 11988210158401015840(0) (16)


120595 (120585 120578) = 120585119865 (120578) + int120578

0

1198821015840 (120578) 119889120578 (17)




11991010158401= 1199102 1199101015840

2= 1199103 1199101015840

3= 11991022minus 11991011199103 (18)

where1199101= 119865(120578)119910

2= 1198651015840(120578)119910



1199101= minus1198810 119910

2= 1 at 120578 = 0


(19)


1and 119910

2are known at









infin The



2




2



4 Journal of Fluids

120573c

120585F998400998400(0)+W

998400998400(0)

minus2

minus1

minus3

minus4

minus5

minus6

minus7

minus8

00 02 04 06 08 10 12 14 16 18 20

V0 = 00

V0 = minus10



1= 10

120573c

00 02 04 06 08 10 12 14 16 18 20

V0 = 00V0 = 10

V0 = 20V0 = 30

120585F998400998400(0)+W

998400998400(0)

minus02

minus04

minus06

minus08

minus10

minus12

minus14

minus16

minus18

minus20


1= 10









0= minus20) at the

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 00


1= 30 and119881


120573c = 10

120573c = 20120573c = 30

U(120585 120578)


V0 = minus20

12057814

12

10

8

6

4

2

0


1= 30 and 119881





0| increases (ie


Journal of Fluids 5

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 20


1= 30 and 119881


120578

U(120585 120578)

0 1 2 3 4 50

1

2

3

4

5

V0 = 00












V0 = 00V0 = 10

V0 = 20V0 = 30

U(120585 120578)

minus2 minus1 0 1 2 3 4 5 6

120578

14

12

10

8

6

4

2

0



V0 = 00120578

14

12

10

8

6

4

2

0

U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

d1 = 10d1 = 20

d1 = 30



plate)


5 Concluding Remark


6 Journal of Fluids

U(120585 120578)


120578

10

8

7

9

6

5

4

3

2

1

0

V0 = minus20

d1 = 10d1 = 20

d1 = 30



U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

120578

10

8

7

9

6

5

4

3

2

1

0

d1 = 10d1 = 20

d1 = 30

V0 = 20






U(120585 120578)minus1 0 1 2 3 4 5

120578

4

3

2

1

0

120585 = minus10120585 = 00

120585 = 10

V0 = minus05



Acknowledgments


References









Journal of Fluids 7


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



4 Journal of Fluids

120573c

120585F998400998400(0)+W

998400998400(0)

minus2

minus1

minus3

minus4

minus5

minus6

minus7

minus8

00 02 04 06 08 10 12 14 16 18 20

V0 = 00

V0 = minus10



1= 10

120573c

00 02 04 06 08 10 12 14 16 18 20

V0 = 00V0 = 10

V0 = 20V0 = 30

120585F998400998400(0)+W

998400998400(0)

minus02

minus04

minus06

minus08

minus10

minus12

minus14

minus16

minus18

minus20


1= 10









0= minus20) at the

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 00


1= 30 and119881


120573c = 10

120573c = 20120573c = 30

U(120585 120578)


V0 = minus20

12057814

12

10

8

6

4

2

0


1= 30 and 119881





0| increases (ie


Journal of Fluids 5

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 20


1= 30 and 119881


120578

U(120585 120578)

0 1 2 3 4 50

1

2

3

4

5

V0 = 00












V0 = 00V0 = 10

V0 = 20V0 = 30

U(120585 120578)

minus2 minus1 0 1 2 3 4 5 6

120578

14

12

10

8

6

4

2

0



V0 = 00120578

14

12

10

8

6

4

2

0

U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

d1 = 10d1 = 20

d1 = 30



plate)


5 Concluding Remark


6 Journal of Fluids

U(120585 120578)


120578

10

8

7

9

6

5

4

3

2

1

0

V0 = minus20

d1 = 10d1 = 20

d1 = 30



U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

120578

10

8

7

9

6

5

4

3

2

1

0

d1 = 10d1 = 20

d1 = 30

V0 = 20






U(120585 120578)minus1 0 1 2 3 4 5

120578

4

3

2

1

0

120585 = minus10120585 = 00

120585 = 10

V0 = minus05



Acknowledgments


References









Journal of Fluids 7


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 5

120573c = 10

120573c = 20120573c = 30

120578

U(120585 120578)


14

12

10

8

6

4

4 5 6 7 8

2

0

V0 = 20


1= 30 and 119881


120578

U(120585 120578)

0 1 2 3 4 50

1

2

3

4

5

V0 = 00












V0 = 00V0 = 10

V0 = 20V0 = 30

U(120585 120578)

minus2 minus1 0 1 2 3 4 5 6

120578

14

12

10

8

6

4

2

0



V0 = 00120578

14

12

10

8

6

4

2

0

U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

d1 = 10d1 = 20

d1 = 30



plate)


5 Concluding Remark


6 Journal of Fluids

U(120585 120578)


120578

10

8

7

9

6

5

4

3

2

1

0

V0 = minus20

d1 = 10d1 = 20

d1 = 30



U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

120578

10

8

7

9

6

5

4

3

2

1

0

d1 = 10d1 = 20

d1 = 30

V0 = 20






U(120585 120578)minus1 0 1 2 3 4 5

120578

4

3

2

1

0

120585 = minus10120585 = 00

120585 = 10

V0 = minus05



Acknowledgments


References









Journal of Fluids 7


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



6 Journal of Fluids

U(120585 120578)


120578

10

8

7

9

6

5

4

3

2

1

0

V0 = minus20

d1 = 10d1 = 20

d1 = 30



U(120585 120578)

minus1 0 1 2 3 4 5 6 7 8 9

120578

10

8

7

9

6

5

4

3

2

1

0

d1 = 10d1 = 20

d1 = 30

V0 = 20






U(120585 120578)minus1 0 1 2 3 4 5

120578

4

3

2

1

0

120585 = minus10120585 = 00

120585 = 10

V0 = minus05



Acknowledgments


References









Journal of Fluids 7


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 7


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Viscous Flows Driven by Uniform Shear over a … · 2019. 7. 31. · a stretching sheet in presence of uniform shear-free stream. Xu [ ] obtained the analytic solution