SLIP EFFECT ON DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A VERTICAL...

Slip Effect on Diffusion of Chemically ReactiveSpecies in Boundary Layer Flow over a Vertical

Stretching Sheet with Suction or Blowing

KRISHNENDU BHATTACHARYYA ANDG. C. LAYEK

Department of Mathematics, The University of Burdwan, Burdwan,West Bengal, India

In the present investigation, we study the effects of slip boundary condition on thediffusion of chemically reactive species in steady boundary layer flow of viscousincompressible fluid over a vertical stretching sheet with suction or blowing. Thefirst-order chemical reaction is considered and wall concentration varies linearlyalong the sheet. The self-similar equations are obtained using similarity transforma-tions and are solved numerically using shooting method. Our study reveals that dueto the increase of diffusion parameter and blowing, the velocity increases, and itdecreases with suction, Schmidt number, and reaction rate parameter. Importantly,for increase of slip parameter, the boundary layer thickness increases. In contrast,the concentration at a point increases only for increasing slip and blowing, whileit decreases for increase of all other parameters.

Keywords Boundary layer flow; Chemical reaction; Diffusion; Slip effect;Stretching sheet; Suction=blowing

Introduction

The boundary layer flow of a viscous incompressible fluid over a stretching sheet hasvarious engineering applications, such as extrusion of a polymer sheet from a dye,paper production, drawing of plastic film, and many others. Boundary layer flowdue to a linearly stretching sheet was first investigated by Crane (1970) under theassumption of steady flow condition and obtained a similarity solution in closed ana-lytical form. The pioneering work of Crane was extended by many researchers suchas Gupta and Gupta (1977), Carragher and Crane (1982), Elbashbeshy and Bazid(2004), and Abel et al. (2010) under different physical conditions.

The effects of chemically reactive species on fluid flow due to a stretchingsheet also bear equal importance in engineering research. The chemical reactioneffects were studied by many researchers on several physical aspects. The diffusionof a chemically reactive species in laminar boundary layer flow over a flatplate was demonstrated by Chambre and Young (1958). The effect of transfer of

Address correspondence to Krishnendu Bhattacharyya, Department of Mathematics,The University of Burdwan, Burdwan – 713104, West Bengal, India. E-mail: [email protected]; [email protected]

Chem. Eng. Comm., 198:1354–1365, 2011Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print=1563-5201 onlineDOI: 10.1080/00986445.2011.560515

1354

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chemically reactive species in the laminar flow over a stretching sheet was explainedby Andersson et al. (1994). Afify (2004) discussed the magnetohydrodynamic(MHD) free convective flow of viscous incompressible fluid and mass transfer overa stretching sheet with chemical reaction. Liu (2005) studied the momentum, heat,and mass transfer of a hydromagnetic flow past a stretching sheet in the presenceof a uniform transverse magnetic field. Akyildiz et al. (2006) obtained a solutionfor diffusion of chemically reactive species in flow of a non-Newtonian fluid overa stretching sheet immersed in a porous medium. Cortell (2007) investigated themotion and mass transfer for two classes of viscoelastic fluid over a porous stretch-ing sheet with chemically reactive species. Chamkha et al. (2010) demonstratedthe effects of chemical reaction on unsteady free convective heat and mass transferon a porous stretching surface in a porous medium. El-Aziz (2010) explainedunsteady flow due to a stretching sheet with mass and heat transfer and chemicalreaction. Bhattacharyya and Layek (2010) discussed the behavior of chemicallyreactive solute distribution in MHD boundary layer flow over a permeablestretching sheet.

In all previously mentioned studies, the flow obeys the conventional no-slipboundary condition. But, no-slip assumption is not consistent with all physical char-acteristics, i.e., some practical flow situations appear where it becomes necessary toreplace the no-slip boundary condition by partial slip boundary condition. Beaversand Joseph (1967) investigated the fluid flow over a permeable wall using the slipboundary condition. The effects of slip condition at the boundary on the flow of vis-cous incompressible fluid past a stretching sheet were studied by Andersson (2002)and Wang (2002). Ariel et al. (2006) considered the flow of a viscoelastic fluid overa stretching sheet with partial slip condition. Fang et al. (2009) obtained the exactanalytic solution of MHD flow under slip condition over a permeable stretchingsheet. Recently, Bhattacharyya et al. (2011) showed the slip effects on the boundarylayer flow over a shrinking sheet near a stagnation point.

In the present investigation, we study the effect of diffusion of chemicallyreactive species on a steady boundary layer flow due to a vertical stretching sheetwith partial slip condition at the boundary and subject to externally applied suctionor blowing. The first-order reaction and linearly variable concentration distributionalong the sheet is considered. Introducing similarity transformation, the governingpartial differential equations are transformed into a set of self-similar nonlinear ordi-nary differential equations. Then, the transformed coupled self-similar equations aresolved numerically by standard and efficient shooting method. The numerical resultsare presented through some figures, and the characteristics of the velocity and con-centration distributions are discussed in detail.

Mathematical Analysis of the Problem

We consider the steady flow of viscous incompressible fluid over a linearly stretchingsurface and a concentration distribution undergoing a first-order chemical reaction.Using boundary layer approximation, the equations for steady two-dimensional flowand the reactive concentration can be written in usual notation as

@u

@xþ @v

@y¼ 0 ð1Þ

Diffusion over Vertical Stretching Sheet 1355


u@u

@xþ v

@u

@y¼ t

@2u

@y2� gb�ðC � C1Þ ð2Þ

and

u@C

@xþ v

@C

@y¼ D

@2C

@y2� k1ðC � C1Þ ð3Þ

where u and v are velocity components in x- and y-directions respectively, t¼ l=q isthe kinematic fluid viscosity, q is the fluid density, l is the coefficient of fluid vis-cosity, b� is the volumetric coefficient of concentration expansion, g is the acceler-ation due to gravity, C is the concentration, D is the diffusion coefficient, and k1is the reaction rate of the solute. The last term on the right-hand side of Equation(2) represents the influence of the diffusion of chemically reactive species on the flowfield, with the ‘‘þ’’ sign corresponding to the flow assisting region and the ‘‘�’’ signthe flow opposing region. In Figure 1, the physical model of this type of flow in avertical stretching sheet is presented.

The appropriate boundary conditions for the velocity components and theconcentration are given by

u ¼ axþ Lð@u@yÞ; v ¼ vw at y ¼ 0; u ! 0 as y ! 1 ð4Þ

and

C ¼ Cw ¼ C1 þ C0x at y ¼ 0; C ! C1 as y ! 1 ð5Þ

where a(>0) is stretching constant, L denotes the slip length, Cw¼C1þC0x is vari-able concentration distribution along the sheet, C1 is the free stream concentrationassumed to be constant, and C0 is a constant that measures the rate of increase ofconcentration along the sheet. Here vw is prescribed suction (vw< 0) or blowing(vw> 0).

Figure 1. A sketch of the physical problem.

1356 K. Bhattacharyya and G. C. Layek


We now introduce the stream function w(x, y) as

u ¼ @w@y

and v ¼ � @w@x

ð6Þ

Now for relation (6), the continuity equation (1) is satisfied automatically and themomentum equation (2) and the concentration equation (3) take the followingforms:

@w@y

@2w@x@y

� @w@x

@2w@y2

¼ t@3w@y3

� gb�ðC � C1Þ ð7Þ

and

@w@y

@C

@x� @w

@x

@C

@y¼ D

@2C

@y2� k1ðC � C1Þ ð8Þ

The boundary conditions (4) of the flow reduce to

@w@y

¼ axþ L@2w@y2

;@w@x

¼ �vw at y ¼ 0;@w@y

! 0 as y ! 1 ð9Þ

Next, we introduce the dimensionless variables for w and C as given below:

w ¼ffiffiffiffiffi

atp

xf ðgÞ and C ¼ C1 þ ðCw � C1Þ/ðgÞ ð10Þ

where the similarity variable g is defined as g¼ y(a=t)1=2.Using (10) we obtain the following self-similar equations:

f 000 þ ff 00 � f 02 þ k/ ¼ 0 ð11Þ

and

/00 þ Sc f/0 � Scðf 0 þ bÞ/ ¼ 0 ð12Þ

where k¼�Grcx=Re2x (the significance of ‘‘�’’ is same as above) is the diffusion

parameter, k> 0 and k< 0 correspond to the flow assisting and flow opposingregions respectively, Grcx¼ gb�(Cw�C1)x3=t2 is the local modified Grashof num-ber, Rex¼ ax2=t is the local Reynolds number, Sc¼ t=D is the Schmidt number,and b¼ k1=a is the reaction rate parameter.

The boundary conditions (9) and (5) reduce to the following forms:

f ðgÞ ¼ S; f 0ðgÞ ¼ 1þ df 0ðgÞ at g ¼ 0; f 0ðgÞ ! 0 as g ! 1 ð13Þ

and

/ðgÞ ¼ 1 at g ¼ 0; /ðgÞ ! 0 as g ! 1 ð14Þ



where S¼�vw=(at)1=2 is the suction or blowing parameter S> 0 (i.e., vw< 0) corre-

sponds to suction and S< 0 (i.e., vw> 0) corresponds to blowing, and d¼L(a=t)1=2 isthe slip parameter.

Numerical Solution and Discussions

The nonlinear coupled self-similar equations (11) and (12) along with the boundaryconditions (13) and (14) form a boundary value problem (BVP), which is solved byshooting technique using the fourth-order Runge-Kutta method after converting itinto an initial value problem (IVP). First of all, we have to choose a suitable finitevalue of g!1, say g1. We set following first-order system:

f 0 ¼ p; p0 ¼ q; q0 ¼ p2 � fq� k/ ð15Þ

and

/0 ¼ r; r0 ¼ �Sc frþ Scðpþ bÞ/ ð16Þ

with the boundary conditions

f ð0Þ ¼ S; pð0Þ ¼ 1þ dqð0Þ; /ð0Þ ¼ 1 ð17Þ

To solve Equations (15) and (16) with (17) as an IVP we need values for q(0), i.e.,f 00(0), and r(0), i.e., /0(0), but no such values are given in the problem. The initialguess values for f 00(0) and /0(0) are chosen, and applying the Runge-Kutta methodthe solution is obtained. We compare the calculated values of f 0(g) and /(g) atg1(¼8) with the given boundary conditions f 0(g1)¼ 0 and /(g1)¼ 0 and adjustvalues f 00(0) and /0(0) using the secant method to find better approximation forthe solution. The step size is taken as Dg¼ 0.01. The process is repeated until weget results correct up to the desired accuracy of 10�5 level.

Figure 2. Velocity profiles f 0(g) for various values of k.



Numerical computations have been carried out for various values of the physicalparameters: the diffusion parameter k, the suction or blowing parameter S, the slipparameter d, the Schmidt number Sc, and the reaction rate parameter b. The com-puted values are represented in some figures and clarifications in the physical pointof view are given for all cases.

First, we discuss the variation of velocity field and reactive concentration distri-bution for several values of the diffusion parameter k. In Figures 2 and 3, the dimen-sionless velocity and concentration profiles are depicted for several values of k. Withincreasing values of k the velocity at a point increases, but the reverse effect is

Figure 3. Concentration profiles /(g) for various values of k.

Figure 4. Velocity profiles f 0(g) for various values of S.



observed for concentration profiles, i.e., /(g) at a fixed g decreases with k. This effectof diffusion of reactive species on the flow field is very significant in physical andpractical points of view.

The effects of suction or blowing parameter S on velocity and concentrationprofiles are plotted in Figures 4 and 5. With the increase of suction, the values ofthe velocity and the concentration profiles at a fixed point decrease. On the otherhand, the opposite nature is observed for blowing, i.e., with increasing blowing,the dimensionless velocity and the concentration increase.

Figures 6 and 7 demonstrate the effect of partial slip condition at the boundaryon the velocity and concentration profiles. In the velocity profiles in Figure 6 we

Figure 5. Concentration profiles /(g) for various values of S.

Figure 6. Velocity profiles f 0(g) for various values of d.



have found a special point g� 2.05, before and after the point at which the behaviorof the velocity profiles changes completely. The velocity decreases with the increasingvalues of slip parameter d up to that point, and it increases to some extent after this.Consequently, with increase of slip the thicknesses of the boundary layer increasesexcept for very large values of d, because if we increase d to a value tending to infinitythen the boundary layer structure will disappear. From Figure 7 it is seen that thevalue of the concentration profile at a point increases with d. In Table I we compareour results, the skin-friction coefficient f 00(0) for various values of slip parameter dwith the available results of Andersson (2002) with k¼ 0 and S¼ 0 and have foundin excellent agreement.

Next, we concentrate on the effects of the Schmidt number Sc on velocityfield as well as the concentration distribution. In Figures 8 and 9, the velocity and

Figure 7. Concentration profiles /(g) for various values of d.

Table I. Skin-friction coefficient f 00(0) for various values of d with k¼ 0 and S¼ 0

d Andersson (2002) Present study

0.0 �1.0000 �1.0000000.1 �0.8721 �0.8720830.2 �0.7764 �0.7763770.5 �0.5912 �0.5911951.0 �0.4302 �0.4301602.0 �0.2840 �0.2839805.0 �0.1448 �0.14484010.0 �0.0812 �0.08124220.0 �0.0438 �0.04378950.0 �0.0186 �0.018597100.0 �0.0095 �0.009551



concentration profiles are presented for various values of Sc. Velocity at a fixedpoint decreases for increase in Sc and the boundary layer thickness also decreases.Similarly, the value of the concentration profile at a particular point decreases sig-nificantly with increasing Sc, and consequently the concentration boundary layerthickness becomes thinner.

The reaction rate parameter b affects the flow velocity in addition to the concen-tration distribution, and this can be verified from Figures 10 and 11. From the fig-ures it can be noted that the values of the velocity and also the concentration profilesdecrease with increasing b, i.e., the chemical reaction opposes the flow and thediffusion of concentration distribution.

Figure 8. Velocity profiles f 0(g) for various values of Sc.

Figure 9. Concentration profiles /(g) for various values of Sc.



The values of skin-friction coefficient f 00(0) and the mass transfer coefficient�/0(0) for various values of parameters k, S, d, Sc, and b are presented inTable II. It is very clear that with increasing k, f 00(0) decreases, whereas �/0(0)increases. The values of both f 00(0) and �/0(0) increase with suction anddecrease with blowing. The increase in slip at the boundary causes the reductionof both skin-friction and mass transfer coefficients. Finally, for an increase ofSchmidt number Sc and reaction rate parameter b the values of f 00(0) and �/0(0)increase.

Figure 10. Velocity profiles f 0(g) for various values of b.

Figure 11. Concentration profiles /(g) for various values of b.



Conclusions

Our objective is to analyze the slip effects on diffusion of chemically reactive speciesin steady boundary layer flow of a viscous fluid due to a vertical stretching sheet withsuction or blowing. The whole analysis is performed by considering the first orderchemical reaction and linearly varying wall concentration. The transformed self-similar equations are solved numerically using shooting method. From our studyit can be concluded that all parameters involved in the governing equations—the dif-fusion parameter, the suction or blowing parameter, the slip parameter, the Schmidtnumber, and the reaction rate parameter—significantly affect the velocity field aswell as concentration distribution. The velocity increases with the diffusion para-meter and blowing, whereas it decreases with suction, Schmidt number, and reactionrate parameter. For increase of slip parameter, the opposite behavior is shown beforeand after a certain point, but ultimately boundary layer thickness increases with slip.On the other hand, the value of the concentration increases only for increasing blow-ing and slip, while it decreases for increase of any other parameter. Also, both theskin-friction and mass transfer coefficients decrease due to increasing slip at theboundary.

Acknowledgments

The authors are indebted to the honorable reviewer for his=her valuable suggestions.One of the authors (K. Bhattacharyya) gratefully acknowledges the financial supportof National Board for Higher Mathematics (NBHM), DAE, Mumbai, India forpursuing this work.

References

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Table II. Values of f 00((0) and �/0(0) for different values of k, S, d, Sc, and b

k S d Sc b f 00(0) �/0(0)

�0.05 0.5 0.1 0.7 0.5 �1.127252 1.8183060.0 0.5 0.1 0.7 0.5 �1.092404 1.8503950.05 0.5 0.1 0.7 0.5 �1.059916 1.8783910.05 0.0 0.1 0.7 0.5 �0.841970 1.3590570.05 �0.5 0.1 0.7 0.5 �0.668112 1.0357720.05 0.5 0.0 0.7 0.5 �1.244549 1.9702950.05 0.5 0.2 0.7 0.5 �0.927152 1.8093230.05 0.5 0.1 0.3 0.5 �1.042770 0.7841260.05 0.5 0.1 1.0 0.5 �1.067182 2.9641540.05 0.5 0.1 0.7 0.0 �1.055790 1.4343360.05 0.5 0.1 0.7 1.0 �1.062286 2.248456



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SLIP EFFECT ON DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A VERTICAL...

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