# Viscoelastic Effects on Free Convective Three-Dimensional Flow with Heat and Mass Transfer

Authors: Rita Choudhury; Utpal Jyoti Das
Journal: ISRN Computational Mathematics
 (2012)

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## Abstract

A theoretical study of free convective three-dimensional heat and mass transfer flow of a viscoelastic fluid along a steadily moving porous vertical plate in presence of transverse sinusoidal suction velocity distribution, and uniform free stream velocity has been considered. The flow becomes three dimensional due to this suction velocity. The governing equations of the flow field are solved by using series expansion method, and the expressions for velocity field, temperature field, skin friction, heat flux in terms of Nusselt number, and mass flux in terms of Sherwood number are obtained. The effects of the viscoelastic parameter on velocity profiles and shear stress with the combination of the other flow parameters are discussed graphically.


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## Body

## 1. Introduction

The study of combined heat and mass transfer problems with chemical reaction is of great practical importance to engineers and scientists because of its almost universal occurrence in many branches of science and engineering. Such phenomenon is observed in buoyancy-induced motions in the atmosphere, in bodies of water, quasisolid bodies such as earth, and so on. In nature and industrial applications, many transport processes exist where the heat and mass transfer takes place simultaneously as a result of combined effects of thermal diffusion and diffusion of chemical species. In addition, the phenomenon of heat and mass transfer is also encountered in chemical processes industries such as food processing and polymer production. Soundalgekar and Warve [1] have analyzed two-dimensional unsteady free convection flow, past an infinite vertical plate with oscillating wall temperature and constant suction. Lin and Wu [2] have analyzed the problem of simultaneous heat and mass transfer with entire range of buoyancy ratio for most practical chemical species in dilute and aqueous solutions. Muthucumaraswamy et al. [3] studied the heat and mass transfer effects on flow past an impulsively started infinite vertical plate.Many research workers are doing investigation of the problem of laminar flow control due to its importance in the field of aeronautical engineering, in view of its applications to reduce drag and enhance the vehicle power requirement by a substantial amount. Initially this subject has been developed by Lachmann [4]. Theoretical and experimental investigations have shown that the transition from laminar to the turbulent flow, which causes the drag coefficient to increase, may be prevented by suction of the fluid and heat and mass transfer from boundary to the wall.Singh et al. [5] studied the effect on wall shear stress and heat transfer of the flow caused by the periodic suction velocity perpendicular to the flow direction when the difference between the wall temperature and the free stream temperature gives rise to buoyancy force in the direction of the free stream. The effect of the porous medium on the three-dimensional Couette flow and heat transfer was presented by Singh and Sharma [6]. Chaudhary and Sharma [7] studied the three-dimensional convection flow through a porous medium and estimated the effect in heat and mass transfer. Ahmed [8] has studied the effects of heat and mass transfer on the steady three-dimensional flow of a viscous incompressible fluid along a steadily moving porous vertical plate subjected to a transverse sinusoidal velocity.The present paper is concerned with the free convective three-dimensional heat and mass transfer flow of visco-elastic incompressible fluid characterized by second-order fluid along a steadily moving porous vertical plate in presence of transverse sinusoidal suction velocity distribution and uniform free stream velocity.The constitutive equation for the incompressible second-order fluid is(1)σ=-pI+μ1A1+μ2A2+μ3(A1)2,
where σ is the stress tensor, pis hydrostatic pressure, Iis unit tensor, An(n=1,2)are the kinematic Rivlin-Ericksen tensors, and μ1,μ2,andμ3are the material coefficients describing the viscosity, elasticity, and cross-viscosity, respectively. The material coefficients μ1,μ2,andμ3 are taken constants with μ1 and μ3 as positive and μ2 as negative (Markovitz and Coleman [9]). Equation (1) was derived by Coleman and Noll [10] from that of simple fluids by assuming that the stress is more sensitive to the recent deformation than to the deformation that occurred in the distant past.The expression forA(1)ijand A(2)ijis given by(2)A(1)ij=vi,j+vj,i,A(2)ij=ai,j+aj,i+2vm,ivm,j,
where viand aiare the ith component of the velocity and acceleration vectors, respectively, and a comma denotes covariant differentiation with respect to the symbol following it.


## 2. Basic Equations

A rectangular Cartesian co-ordinate system is introduced such that the plate lies inxz-plane, x-axis being vertically upwards in the direction of buoyancy force which arises out of a difference in temperature of the plate and the free stream. The y-axis is taken perpendicular to the plate and directed into the fluid which is flowing laminarly with free stream velocityU. The transverse sinusoidal suction velocity distribution is assumed to be of the form(3)v̅w(z̅)=-v0(1+εcos⁡πz̅L),
where ε≪1 andLis the wave length of the periodic suction. All physical quantities are independent ofx̅ for this problem of fully developed laminar flow.The boundary conditions relevant to the problem are(4)y̅=0:u̅=V̅,v̅=v̅w,w̅=0,T̅=T̅w,C̅=C̅w,y̅⟶∞:u̅⟶U̅,v̅⟶-v0,w̅⟶0,T̅⟶T̅∞,p̅⟶p̅∞,C̅⟶C̅∞.
We introduce the following nondimensional quantities:(5)y=y̅L,z=z̅L,u=u̅v0,v=v̅v0,w=w̅v0,U=U̅v0,θ=T̅-T̅∞Tw-T∞,C=C̅-C̅∞C̅w-C̅∞,V=V̅v0,Pr=υ1α,S=υ1D,p=p̅ρ(υ1/L)2,p∞=p̅∞ρ(υ1/L)2,Gr=Lgβ(T̅w-T̅∞)v02,Gm=Lgβ̅(T̅w-T̅∞)v02,Re=v0Lυ1.
Here, (u̅,v̅,w̅)are the velocity components along the (x̅,y̅,z̅)directions, respectively, gthe acceleration due to gravity, υi=(μi/ρ)(i=1,2,3),ρ is the density, β is the coefficient of volume expansion for heat transfer, β̅ is the co-efficient of volume expansion for mass transfer, p̅ is the pressure, T̅ is the fluid temperature, C̅ is the species concentration, α is the thermal diffusivity, Dis the chemical molecular diffusivity, v0 is the dimensionless suction velocity, Pr is the Prandtl number, S is the Schmidt number, Gris the Grashof number for heat transfer, Gm is the Grashof number for mass transfer, Reis the Reynolds number, T̅w is the fluid temperature on the wall, and T̅∞ is the fluid temperature in the free stream.In view of the above nondimensional quantities, the governing equations for heat and mass transfer flow are(6)∂v∂y+∂w∂z=0,(7)v∂u∂y+w∂u∂z=Grθ+GmC+1Re(∂2u∂y2+∂2u∂z2)+α1(2∂v∂y∂2u∂y2+v∂3u∂y3+∂2v∂y2∂u∂y+∂2w∂y2∂u∂z+2∂w∂y∂2u∂y∂z+w∂3u∂y2∂z+∂2v∂z2∂u∂y+2∂v∂z∂2u∂y∂z+v∂3u∂y∂z2+∂2w∂z2∂u∂z+2∂w∂z∂2u∂z2+w∂3u∂z3)+α2(∂2v∂y2∂u∂y+2∂v∂y∂2u∂y2+2∂2u∂y∂z∂w∂y+∂u∂z∂2w∂y2+2∂2u∂y∂z∂v∂z+∂u∂z∂2w∂z2+∂u∂y∂2v∂z2+2∂2u∂z2∂w∂z),(8)v∂v∂y+w∂v∂z=-1Re2∂p∂y+1Re(∂2v∂y2+∂2v∂z2)+α1(v∂3v∂y3+w∂3v∂y2∂z+v∂3v∂y∂z2+w∂3v∂z3+2∂w∂y∂2v∂y∂z+3∂v∂z∂2v∂y∂z+2∂2u∂y∂z∂u∂z+13∂v∂y∂2v∂y2+3∂2w∂y2∂v∂z+4∂u∂y∂2u∂y2+4∂w∂y∂2w∂y2+2∂u∂y∂2u∂z2+∂v∂y∂2v∂z2)+α2(2∂u∂y∂2u∂y2+8∂v∂y∂2v∂y2+2∂2v∂y∂z∂w∂y+2∂2w∂y2∂v∂z+2∂v∂z∂2v∂y∂z+2∂w∂y∂2w∂y2+∂2u∂y∂z∂u∂z+∂u∂y∂2u∂z2),(9)v∂w∂y+w∂w∂z=-1Re2∂p∂z+1Re(∂2w∂y2+∂2w∂z2)+α1(w∂3v∂y3+v∂3w∂y∂z2+w∂3w∂z3+v∂3w∂y3+2∂v∂z∂2w∂y∂z+3∂w∂y∂2w∂y∂z+2∂u∂y∂2u∂y∂z+13∂w∂z∂2w∂z2+3∂2v∂z2∂w∂y+4∂u∂z∂2u∂z2+4∂v∂z∂2v∂z2+2∂u∂z∂2u∂y2+∂w∂z∂2w∂y2)+α2(2∂u∂z∂2u∂z2+8∂w∂z∂2w∂z2+2∂v∂z∂2w∂y∂z+2∂w∂y∂2v∂z2+2∂w∂y∂2w∂y∂z+2∂v∂z∂2v∂z2+∂2u∂y∂z∂u∂y+∂u∂z∂2u∂y2),(10)v∂θ∂y+w∂θ∂z=1RePr(∂2θ∂y2+∂2θ∂z2),(11)v∂C∂y+w∂C∂z=1ReS(∂2C∂y2+∂2C∂z2),
where α1=υ2/L2and α2=υ3/L2.The corresponding boundary conditions are(12)y=0:u=V,v(z)=-(1+εcos⁡πz),w=0,θ=1,C=1,y⟶∞:u⟶U,v⟶-1,w⟶0,θ⟶0,C⟶0,p⟶p∞.


## 3. Method of Solution

When the amplitudeε(≪1) of the suction velocity is small, we assume the solutions of the nonlinear partial differential equations (7) to (11) of the form(13)u(y,z)=u0(y)+εu1(y,z)+o(ε2),v(y,z)=v0(y)+εv1(y,z)+o(ε2),w(y,z)=w0(y)+εw1(y,z)+o(ε2),p(y,z)=p0(y)+εp1(y,z)+o(ε2),θ(y,z)=θ0(y)+εθ1(y,z)+o(ε2),C(y,z)=C0(y)+εC1(y,z)+o(ε2).
When ε=0, the problem reduces to the two-dimensional case with w=w0=0and p=p0=p∞. Substituting (13) into (6) to (11), the terms free from ε of both sides are(14)v0′=0,(15)α1v0u0′′+1Reu0′′-v0u0′+Grθ0+GmC0=0,(16)-1Re2∂p0∂y+(4α1+2α2)∂u0∂y∂2u0∂y2=0,(17)-∂θ0∂y=1RePr∂2θ0∂y2,(18)-∂C0∂y=1ReS∂2C0∂y2,
where prime denotes differentiation with respect to y.The corresponding boundary conditions are(19)y=0:u0=V,v0=-1,w0=0,θ0=1,C0=1,y⟶∞:u0⟶U,v0⟶-1,w0⟶0,θ0⟶0,C0⟶0,p0⟶p∞.
Solving (14), (17), and (18) under boundary conditions (19), we get(20)v0=-1,θ0=exp⁡(-RePry)C0=exp⁡(-ReSy).,
As |α1|≪1 (due to small shear rate) therefore substituting
(21)u0(y)=u00(y)+α1u01(y)+o(α12),
into (15) and boundary conditions (19) up to the first order of α1, and equating the coefficients of like powers of α1, we obtain the following sets of ordinary differential equations and corresponding boundary conditions(22)u00′′+Reu00′+Re{Grexp(-RePry)+Gmexp⁡(-ReSy)}=0,(23)Reu00′′′-u01′′-Reu01′=0,(24)y=0:u00=V,u01=0,y⟶∞:u00⟶U,u01⟶0.
Solving (22) and (23) under boundary conditions (24) and then substituting into (21) (neglecting α12), we get(25)u0=U+(V-U)exp⁡(-Rey)+GrRePr(Pr-1){exp⁡(-Rey)-exp⁡(-RePry)}+GmReS(S-1){exp⁡(-Rey)-exp⁡(-ReSy)}-α1Re[A2exp⁡(-RePry)Pr(Pr-1)-A3exp⁡(-ReSy)S(S-1){A2Pr(Pr-1)+A3S(S-1)}exp⁡(-Rey)+A1Reyexp⁡(-Rey)-A2exp⁡(-RePry)Pr(Pr-1)-A3exp⁡(-ReSy)S(S-1)],
where A1=Re3{U-V-(Gr/PrRe(Pr-1))-(Gm/ReS(S-1))},A2=(GrRe2Pr2/(Pr-1)),A3=GmRe2S2/(S-1)withPr≠1,S≠1.Equating the coefficients ofε from both sides after substitution of (13) in (6) to (11), neglecting those of ε2, using (16), and assuming ∂p0/∂y=0, we get(26)∂v1∂u+∂w1∂z=0,(27)v1∂u0∂y-∂u1∂y=Grθ1+GmC1+1Re(∂2u1∂y2+∂2u1∂z2)+α1(-2∂v1∂y∂2u0∂y2-∂3u1∂y3+v1∂3u0∂y3-∂2v1∂y2∂u0∂y-∂u0∂y∂2v1∂z2-∂3u1∂y∂z2),(28)-∂v1∂y=-1Re2∂p1∂y+1Re(∂2v1∂y2+∂2v1∂z2)-α1(∂3v1∂y3+∂3v1∂y∂z2),(29)-∂w1∂y=-1Re2∂p1∂z+1Re(∂2w1∂y2+∂2w1∂z2)-α1(∂3w1∂y∂z2+∂3w1∂y3),(30)-∂θ1∂y+v1∂θ0∂y=1RePr(∂2θ1∂y2+∂2θ1∂z2),(31)-∂C1∂y+v1∂C0∂y=1ReS(∂2C1∂y2+∂2C1∂z2),
with relevant boundary conditions(32)y=0:u1=0,v1=-cos⁡πz,w1=0,θ1=0,p1=0,C1=0,y⟶∞:u1⟶0,v1⟶0,w1⟶0,θ1⟶0,p1⟶0,C1⟶0.
Equations (26), (28), and (29) govern the cross-flow, and (27), (30), and (31) govern the main flow, the temperature and the species concentration, respectively.


## 4. Cross-Flow Solution

In order to solve (26), (28), and (29), being independent of the main flow component u1 and the temperature field θ1,we assume that(33)v1(y,z)=-πv11(y)cos⁡πz,(34)w1(y,z)=v11′(y)sin⁡πz,(35)p1(y,z)=Re2p11(y)cos⁡πz.
The prime in v11′ denotes differentiation with respect to y. Equations (33) and (34) have been chosen so that the continuity equation (26) is satisfied. Substituting these equations into (28) and (29), two ordinary differential equations for v11 and p11are obtained:(36)v11′′+Rev11′-π2v11+α1Re(π2v11′-v11′′′)=-p11′πRe,(37)v11′′′+Rev11′′-π2v11′+α1Re(π2v11′′-v11IV)=-πp11Re,
with the boundary conditions(38)y=0:v11=1π,v11′=0,y⟶0:v11⟶0,v11′⟶0.
On eliminating the pressure p11′from (36) and (37), we get the following differential equation in v11 as(39)v11IV+Rev11′′′-2π2v11′-π2Rev11′+π4v11+α1Re(2π2v11′′′-v11V-π4v11′)=0.
To solve (39), we note that α1<1 for small shear, and so we can assume that(40)v11(y)=v110(y)+α1v111(y)+o(α12).
Substituting from (40) into (39) then equating the like powers of α1 and neglecting the higher powers of α1,we get(41)v110IV+Rev110′′′-2π2v110′′-π2Rev110′+π4v110=0,(42)v111IV+Rev111′′′-2π2v111′′-π2Rev111′+π4v111+2Reπ2v110′′′-Rev110V-π4Rev110′=0.The corresponding boundary conditions are:(43)y=0:v110=1π,v111=0,v110′=0,v111′=0,y⟶∞:v110⟶0,v111⟶0,v110′⟶0,v111′⟶0.
Solving (41) and (42) under boundary conditions (43), we get the expression for v11and hence the solutions for velocity componentsv1, w1 and the pressurep1.


## 5. Solution for Main Flow, Temperature, and Molar Concentration Fields

To solve (27), (30), and (31), we assume the following form:(44)u1(y,z)=u11(y)cos⁡πz,θ1(y,z)=θ11(y)cos⁡πz,C1(y,z)=C11(y)cos⁡πz.
On using (44) into (27), (30), and (31), we get(45)u11′′-π2u11+Re(πv11u0′+u11′)+Grθ11+GmC11+α1(2πv11′u0′′-u11′′′-πv11u0′′′+πv11′′u0′-u0′π3v11+π2u11′)=0,(46)θ11′′-π2θ11+RePr(θ11′+πv11θ0′)=0,(47)C11′′-π2C11+ReS(C11′+πv11C0′)=0,
subject to boundary conditions(48)y=0:u11=0,θ11=0,C11=0y⟶∞:u11⟶0,θ11⟶0,C11⟶0.
Solving (46) and (47) under boundary conditions (48), we get the expressions for θ11and C11,and hence the solutions for θ1and C1 have been obtained.Again, substituting(49)u11(y)=u110(y)+α1u111(y)+o(α12)
into (45) and boundary conditions (48) up to first order of α1and comparing the coefficients of like powers of α1,we obtain(50)u110′′+Reu110′-π2u110=-Re(Grθ11+GmC11+πv11u0′),(51)u111′′+Reu111′-π2u111=Re(-2πv11′u0′′+u110′′′+πv11u0′′′,-πv11′′u0′+u0′π3v11-π2u110′),
subject to boundary conditions(52)y=0:u110=0,u111=0,y⟶∞:u110⟶0,u111⟶0.
Solving (50) and (51) under boundary conditions (52), we get the expression for u110and u111, and hence the solutions for u11and u1have been obtained. The solutions of the differential equations are not presented here for the sake of brevity.


## 6. Results and Discussion

The nondimensional skin friction coefficientσxy at the plate y=0in the main flow direction is(53)σxy=σ̅xyρv02=(1Re∂u∂y)y=0+α1×(v∂2u∂y2+w∂2u∂y∂z-3∂u∂y∂v∂y-∂w∂y∂u∂z-2∂u∂z∂v∂z)y=0.
The nondimensional heat flux at the plate y=0in terms of Nusselt number Nuis(54)Nu=kρv0Cp(T̅w-T̅∞)(∂T̅∂y̅)=1PrRe[θ0′(0)+εθ11′(0)sin⁡πz].
The nondimensional mass flux at the platey=0in terms of Sherwood number Sh0 is(55)Sh0=Dv0(C̅w-C̅∞)(∂C̅∂y̅)y=0=1SRe[C0′(0)+εC11′(0)cos⁡πz].
The purpose of this study is to bring out the effects of the non-Newtonian parameter on the governing flow with the combinations of the other flow parameters as the effects of the other parameters discussed by Ahmed. The non-Newtonian effect is exhibited through the nondimensional visco-elastic parameterα1. The corresponding results for Newtonian fluid are obtained by setting α1=0.Figures1 to 5 represent the velocity profiles uagainst y to observe the visco-elastic effects for various sets of values (Table 1) of the Grashof number Gm for mass transfer, Prandtl number Pr, Reynolds number Reand Schmidt number S. It is evident from Figures 1 to 5 that the values of the velocity u increase with the increasing values of the non-Newtonian parameter |α1|,(α1=0,-0.02,-0.04)in comparison with the Newtonian fluid (α1=0)for all the cases in Table 1.Table 1
Various combinations of flow parameters.
CasesGmPrReSI233.50.1II533.50.1III553.50.1IV554.50.1V554.50.6Figure 1
Variation ofuagainst yfor case I.Figure 2
Variation ofuagainst yfor case II.Figure 3
Variation ofuagainst yfor case III.Figure 4
Variation ofuagainst yfor case IV.Figure 5
Variation ofuagainst yfor case V.Figures6 to 8 exhibit the effects of |α1| on shear stress σagainst Gmfor various sets of values of S and Pr.It is evident from the Figures 6 to 8 that the values of σdecrease with the increasing values of the non-Newtonian parameter|α1|in comparison with the Newtonian fluid for increasing values of the Schmidt number S(Figures 6 and 7) or Prandtl number Pr(Figures 7 and 8).Figure 6
Variation ofσ against Gm for s=0.1,Pr=3.Figure 7
Variation ofσ against Gm for s=0.4,Pr=3.Figure 8
Variation ofσ against Gm for s=0.4,Pr=5.Figures9 to 11 depict the shear stress σagainst Schmidt number Sfor various sets of values of the Grashof number Gm for mass transfer and Prandtl number Pr. It is observed from the figures that the shear stress decreases with the increasing values of the non-Newtonian parameter |α1|, in comparison to the Newtonian fluid for increasing values of Grashof number (Figures 9 and 10) or Prandtl number (Figures 10 and 11).Figure 9
Variation ofσ against s for Gm=2,Pr=3.Figure 10
Variation ofσ against s for Gm=5,Pr=3.Figure 11
Variation ofσ against s for Gm=5,Pr=5.It has also been observed that the heat and mass flux at the platey=0are not significantly affected by the non-Newtonian parameter.


## 7. Conclusions

The present work is an attempt to study the viscoelastic effects on free convective three-dimensional flow along a steadily moving porous vertical plate in presence of transverse sinusoidal suction velocity and uniform free stream velocity. The second-order fluid model for a viscoelastic fluid flow is assumed. The effects of viscoelastic parameter on velocity profile for differentGm,Pr,Re,and S are studied in detail. It was found that the velocity increases with the increasing values of the visco-elastic parameter in comparison to the Newtonian fluid. Also, in all the cases studied, it was found that the shear stress decreases with the increasing values of the viscoelastic parameter in comparison with the Newtonian fluid. Further, it was observed that the shear stress increases with the increasing values of Grashof number Gm for both Newtonian and non-Newtonian cases. But shear stress decreases with the increasing values of the Schmidt number for both Newtonian and non-Newtonian cases.



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