About this Journal Submit a Manuscript Table of Contents
ISRN Computational Mathematics
Volume 2012 (2012), Article ID 402037, 8 pages
http://dx.doi.org/10.5402/2012/402037
Research Article

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

1Department of Mathematics, Gauhati University, Assam, Guwahati 781014, India
2Department of Mathematics, Rajiv Gandhi University, Rono Hills, Arunachal Pradesh, Itanagar 791112, India

Received 12 September 2011; Accepted 4 October 2011

Academic Editors: G. Bella and Z. Guo

Copyright © 2012 Rita Choudhury and Utpal Jyoti Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 is𝜎=𝑝𝐼+𝜇1𝐴1+𝜇2𝐴2+𝜇3𝐴12,(1) 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 for 𝐴(1)𝑖𝑗and 𝐴(2)𝑖𝑗is given by𝐴(1)𝑖𝑗=𝑣𝑖,𝑗+𝑣𝑗,𝑖,𝐴(2)𝑖𝑗=𝑎𝑖,𝑗+𝑎𝑗,𝑖+2𝑣𝑚,𝑖𝑣𝑚,𝑗,(2) 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 in 𝑥𝑧-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 form𝑣𝑤𝑧=𝑣0𝜋1+𝜀cos𝑧𝐿,(3) where 𝜀1 and𝐿is the wave length of the periodic suction. All physical quantities are independent of𝑥 for this problem of fully developed laminar flow.

The boundary conditions relevant to the problem are𝑦=0𝑢=𝑉,𝑣=𝑣𝑤,𝑤=0,𝑇=𝑇𝑤,𝐶=𝐶𝑤,𝑦𝑢𝑈,𝑣𝑣0,𝑤0,𝑇𝑇,𝑝𝑝,𝐶𝐶.(4) We introduce the following nondimensional quantities:𝑦=𝑦𝐿,𝑧=𝑧𝐿,𝑢=𝑢𝑣0,𝑣=𝑣𝑣0,𝑤=𝑤𝑣0,𝑈=𝑈𝑣0,𝜃=𝑇𝑇𝑇𝑤𝑇,𝐶=𝐶𝐶𝐶𝑤𝐶,𝑉=𝑉𝑣0,𝜐Pr=1𝛼𝜐,S=1𝐷,𝑝=𝑝𝜌𝜐1/𝐿2,𝑝=𝑝𝜌𝜐1/𝐿2,Gr=𝐿𝑔𝛽𝑇𝑤𝑇𝑣20,Gm=𝐿𝑔𝛽𝑇𝑤𝑇𝑣20𝑣,Re=0𝐿𝜐1.(5) Here, (𝑢,𝑣,𝑤)are the velocity components along the (𝑥,𝑦,𝑧)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, 𝛽 is the co-efficient of volume expansion for mass transfer, 𝑝 is the pressure, 𝑇 is the fluid temperature, 𝐶 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, 𝑇𝑤 is the fluid temperature on the wall, and 𝑇 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𝜕𝑣+𝜕𝑦𝜕𝑤𝜕𝑧=0,(6)𝑣𝜕𝑢𝜕𝑦+𝑤𝜕𝑢1𝜕𝑧=Gr𝜃+Gm𝐶+𝜕Re2𝑢𝜕𝑦2+𝜕2𝑢𝜕𝑧2+𝛼12𝜕𝑣𝜕𝜕𝑦2𝑢𝜕𝑦2𝜕+𝑣3𝑢𝜕𝑦3+𝜕2𝑣𝜕𝑦2𝜕𝑢+𝜕𝜕𝑦2𝑤𝜕𝑦2𝜕𝑢𝜕𝑧+2𝜕𝑤𝜕𝜕𝑦2𝑢𝜕𝜕𝑦𝜕𝑧+𝑤3𝑢𝜕𝑦2+𝜕𝜕𝑧2𝑣𝜕𝑧2𝜕𝑢𝜕𝑦+2𝜕𝑣𝜕𝜕𝑧2𝑢𝜕𝜕𝑦𝜕𝑧+𝑣3𝑢𝜕𝑦𝜕𝑧2+𝜕2𝑤𝜕𝑧2𝜕𝑢𝜕𝑧+2𝜕𝑤𝜕𝜕𝑧2𝑢𝜕𝑧2𝜕+𝑤3𝑢𝜕𝑧3+𝛼2𝜕2𝑣𝜕𝑦2𝜕𝑢𝜕𝑦+2𝜕𝑣𝜕𝜕𝑦2𝑢𝜕𝑦2𝜕+22𝑢𝜕𝑦𝜕𝑧𝜕𝑤+𝜕𝑦𝜕𝑢𝜕𝜕𝑧2𝑤𝜕𝑦2𝜕+22𝑢𝜕𝑦𝜕𝑧𝜕𝑣+𝜕𝑧𝜕𝑢𝜕𝜕𝑧2𝑤𝜕𝑧2+𝜕𝑢𝜕𝜕𝑦2𝑣𝜕𝑧2𝜕+22𝑢𝜕𝑧2𝜕𝑤,𝜕𝑧(7)𝑣𝜕𝑣𝜕𝑦+𝑤𝜕𝑣1𝜕𝑧=Re2𝜕𝑝+1𝜕𝑦𝜕Re2𝑣𝜕𝑦2+𝜕2𝑣𝜕𝑧2+𝛼1𝑣𝜕3𝑣𝜕𝑦3𝜕+𝑤3𝑣𝜕𝑦2𝜕𝜕𝑧+𝑣3𝑣𝜕𝑦𝜕𝑧2𝜕+𝑤3𝑣𝜕𝑧3+2𝜕𝑤𝜕𝜕𝑦2𝑣𝜕𝑦𝜕𝑧+3𝜕𝑣𝜕𝜕𝑧2𝑣𝜕𝜕𝑦𝜕𝑧+22𝑢𝜕𝑦𝜕𝑧𝜕𝑢𝜕𝑧+13𝜕𝑣𝜕𝜕𝑦2𝑣𝜕𝑦2𝜕+32𝑤𝜕𝑦2𝜕𝑣𝜕𝑧+4𝜕𝑢𝜕𝜕𝑦2𝑢𝜕𝑦2+4𝜕𝑤𝜕𝜕𝑦2𝑤𝜕𝑦2+2𝜕𝑢𝜕𝜕𝑦2𝑢𝜕𝑧2+𝜕𝑣𝜕𝜕𝑦2𝑣𝜕𝑧2+𝛼22𝜕𝑢𝜕𝜕𝑦2𝑢𝜕𝑦2+8𝜕𝑣𝜕𝜕𝑦2𝑣𝜕𝑦2𝜕+22𝑣𝜕𝑦𝜕𝑧𝜕𝑤𝜕𝜕𝑦+22𝑤𝜕𝑦2𝜕𝑣𝜕𝑧+2𝜕𝑣𝜕𝜕𝑧2𝑣𝜕𝑦𝜕𝑧+2𝜕𝑤𝜕𝜕𝑦2𝑤𝜕𝑦2+𝜕2𝑢𝜕𝑦𝜕𝑧𝜕𝑢+𝜕𝑧𝜕𝑢𝜕𝜕𝑦2𝑢𝜕𝑧2,(8)𝑣𝜕𝑤𝜕𝑦+𝑤𝜕𝑤1𝜕𝑧=Re2𝜕𝑝+1𝜕𝑧𝜕Re2𝑤𝜕𝑦2+𝜕2𝑤𝜕𝑧2+𝛼1𝑤𝜕3𝑣𝜕𝑦3𝜕+𝑣3𝑤𝜕𝑦𝜕𝑧2𝜕+𝑤3𝑤𝜕𝑧3𝜕+𝑣3𝑤𝜕𝑦3+2𝜕𝑣𝜕𝜕𝑧2𝑤𝜕𝑦𝜕𝑧+3𝜕𝑤𝜕𝜕𝑦2𝑤𝜕𝑦𝜕𝑧+2𝜕𝑢𝜕𝜕𝑦2𝑢𝜕𝑦𝜕𝑧+13𝜕𝑤𝜕𝜕𝑧2𝑤𝜕𝑧2𝜕+32𝑣𝜕𝑧2𝜕𝑤𝜕𝑦+4𝜕𝑢𝜕𝜕𝑧2𝑢𝜕𝑧2+4𝜕𝑣𝜕𝜕𝑧2𝑣𝜕𝑧2+2𝜕𝑢𝜕𝜕𝑧2𝑢𝜕𝑦2+𝜕𝑤𝜕𝜕𝑧2𝑤𝜕𝑦2+𝛼22𝜕𝑢𝜕𝜕𝑧2𝑢𝜕𝑧2+8𝜕𝑤𝜕𝜕𝑧2𝑤𝜕𝑧2+2𝜕𝑣𝜕𝜕𝑧2𝑤𝜕𝑦𝜕𝑧+2𝜕𝑤𝜕𝜕𝑦2𝑣𝜕𝑧2+2𝜕𝑤𝜕𝜕𝑦2𝑤𝜕𝑦𝜕𝑧+2𝜕𝑣𝜕𝜕𝑧2𝑣𝜕𝑧2+𝜕2𝑢𝜕𝑦𝜕𝑧𝜕𝑢+𝜕𝑦𝜕𝑢𝜕𝜕𝑧2𝑢𝜕𝑦2,(9)𝑣𝜕𝜃𝜕𝑦+𝑤𝜕𝜃=1𝜕𝑧𝜕RePr2𝜃𝜕𝑦2+𝜕2𝜃𝜕𝑧2,(10)𝑣𝜕𝐶𝜕𝑦+𝑤𝜕𝐶=1𝜕𝑧𝜕ReS2𝐶𝜕𝑦2+𝜕2𝐶𝜕𝑧2,(11) where 𝛼1=𝜐2/𝐿2and 𝛼2=𝜐3/𝐿2.

The corresponding boundary conditions are𝑦=0𝑢=𝑉,𝑣(𝑧)=(1+𝜀cos𝜋𝑧),𝑤=0,𝜃=1,𝐶=1,𝑦𝑢𝑈,𝑣1,𝑤0,𝜃0,𝐶0,𝑝𝑝.(12)

3. Method of Solution

When the amplitude 𝜀(1) of the suction velocity is small, we assume the solutions of the nonlinear partial differential equations (7) to (11) of the form𝑢(𝑦,𝑧)=𝑢0(𝑦)+𝜀𝑢1𝜀(𝑦,𝑧)+𝑜2,𝑣(𝑦,𝑧)=𝑣0(𝑦)+𝜀𝑣1𝜀(𝑦,𝑧)+𝑜2,𝑤(𝑦,𝑧)=𝑤0(𝑦)+𝜀𝑤1𝜀(𝑦,𝑧)+𝑜2,𝑝(𝑦,𝑧)=𝑝0(𝑦)+𝜀𝑝1(𝜀𝑦,𝑧)+𝑜2,𝜃(𝑦,𝑧)=𝜃0(𝑦)+𝜀𝜃1𝜀(𝑦,𝑧)+𝑜2,𝐶(𝑦,𝑧)=𝐶0(𝑦)+𝜀𝐶1𝜀(𝑦,𝑧)+𝑜2.(13) 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 are𝑣0=0,(14)𝛼1𝑣0𝑢0+1𝑢Re0𝑣0𝑢0+Gr𝜃0+Gm𝐶0=0,(15)1Re2𝜕𝑝0+𝜕𝑦4𝛼1+2𝛼2𝜕𝑢0𝜕𝜕𝑦2𝑢0𝜕𝑦2=0,(16)𝜕𝜃0=1𝜕𝑦𝜕RePr2𝜃0𝜕𝑦2,(17)𝜕𝐶0=1𝜕𝑦𝜕ReS2𝐶0𝜕𝑦2,(18) where prime denotes differentiation with respect to 𝑦.

The corresponding boundary conditions are𝑦=0𝑢0=𝑉,𝑣0𝑤=1,0=0,𝜃0=1,𝐶0=1,𝑦𝑢0𝑈,𝑣01,𝑤0𝜃0,00,𝐶00,𝑝0𝑝.(19) Solving (14), (17), and (18) under boundary conditions (19), we get𝑣0𝜃=1,0𝐶=exp(RePr𝑦)0=exp(ReS𝑦).,(20) As |𝛼1|1 (due to small shear rate) therefore substituting 𝑢0(𝑦)=𝑢00(𝑦)+𝛼1𝑢01𝛼(𝑦)+𝑜21,(21) 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 conditions𝑢00+Re𝑢00+Re{Grexp(RePr𝑦)+Gmexp(ReS𝑦)}=0,(22)Re𝑢00𝑢01Re𝑢01=0,(23)𝑦=0𝑢00=𝑉,𝑢01=0,𝑦𝑢00𝑈,𝑢010.(24) Solving (22) and (23) under boundary conditions (24) and then substituting into (21) (neglecting 𝛼21), we get𝑢0+=𝑈+(𝑉𝑈)exp(Re𝑦)Gr+RePr(Pr1){exp(Re𝑦)exp(RePr𝑦)}Gm𝛼ReS(S1){exp(Re𝑦)exp(ReS𝑦)}1𝐴Re2+𝐴Pr(Pr1)3S(S1)exp(Re𝑦)+𝐴1𝐴Re𝑦exp(Re𝑦)2exp(RePr𝑦)𝐴Pr(Pr1)3exp(ReS𝑦),S(S1)(25) where 𝐴1=Re3{𝑈𝑉(Gr/PrRe(Pr1))(Gm/ReS(S1))},𝐴2=(GrRe2Pr2/(Pr1)),𝐴3=GmRe2S2/(S1)withPr1,S1.

Equating the coefficients of 𝜀 from both sides after substitution of (13) in (6) to (11), neglecting those of 𝜀2, using (16), and assuming 𝜕𝑝0/𝜕𝑦=0, we get𝜕𝑣1+𝜕𝑢𝜕𝑤1𝜕𝑧=0,(26)𝑣1𝜕𝑢0𝜕𝑦𝜕𝑢1𝜕𝑦=Gr𝜃1+Gm𝐶1+1𝜕Re2𝑢1𝜕𝑦2+𝜕2𝑢1𝜕𝑧2+𝛼12𝜕𝑣1𝜕𝜕𝑦2𝑢0𝜕𝑦2𝜕3𝑢1𝜕𝑦3+𝑣1𝜕3𝑢0𝜕𝑦3𝜕2𝑣1𝜕𝑦2𝜕𝑢0𝜕𝑦𝜕𝑢0𝜕𝜕𝑦2𝑣1𝜕𝑧2𝜕3𝑢1𝜕𝑦𝜕𝑧2,(27)𝜕𝑣11𝜕𝑦=Re2𝜕𝑝1+1𝜕𝑦𝜕Re2𝑣1𝜕𝑦2+𝜕2𝑣1𝜕𝑧2𝛼1𝜕3𝑣1𝜕𝑦3+𝜕3𝑣1𝜕𝑦𝜕𝑧2,(28)𝜕𝑤11𝜕𝑦=Re2𝜕𝑝1+1𝜕𝑧𝜕Re2𝑤1𝜕𝑦2+𝜕2𝑤1𝜕𝑧2𝛼1𝜕3𝑤1𝜕𝑦𝜕𝑧2+𝜕3𝑤1𝜕𝑦3,(29)𝜕𝜃1𝜕𝑦+𝑣1𝜕𝜃0=1𝜕𝑦𝜕RePr2𝜃1𝜕𝑦2+𝜕2𝜃1𝜕𝑧2,(30)𝜕𝐶1𝜕𝑦+𝑣1𝜕𝐶0=1𝜕𝑦𝜕ReS2𝐶1𝜕𝑦2+𝜕2𝐶1𝜕𝑧2,(31) with relevant boundary conditions𝑦=0𝑢1=0,𝑣1𝑤=cos𝜋𝑧,1=0,𝜃1=0,𝑝1=0,𝐶1=0,𝑦𝑢10,𝑣10,𝑤1𝜃0,10,𝑝10,𝐶10.(32) 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 that𝑣1(𝑦,𝑧)=𝜋𝑣11(𝑦)cos𝜋𝑧,(33)𝑤1(𝑦,𝑧)=𝑣11(𝑦)sin𝜋𝑧,(34)𝑝1(𝑦,𝑧)=Re2𝑝11(𝑦)cos𝜋𝑧.(35) 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:𝑣11+Re𝑣11𝜋2𝑣11+𝛼1𝜋Re2𝑣11𝑣11𝑝=11𝜋Re,(36)𝑣11+Re𝑣11𝜋2𝑣11+𝛼1𝜋Re2𝑣11𝑣IV11=𝜋𝑝11Re,(37) with the boundary conditions𝑦=0𝑣11=1𝜋,𝑣11=0,𝑦0𝑣110,𝑣110.(38) On eliminating the pressure 𝑝11from (36) and (37), we get the following differential equation in 𝑣11 as𝑣IV11+Re𝑣112𝜋2𝑣11𝜋2Re𝑣11+𝜋4𝑣11+𝛼1Re2𝜋2𝑣11𝑣V11𝜋4𝑣11=0.(39) To solve (39), we note that 𝛼1<1 for small shear, and so we can assume that𝑣11(𝑦)=𝑣110(𝑦)+𝛼1𝑣111𝛼(𝑦)+𝑜21.(40) Substituting from (40) into (39) then equating the like powers of 𝛼1 and neglecting the higher powers of 𝛼1,we get𝑣IV110+Re𝑣1102𝜋2𝑣110𝜋2Re𝑣110+𝜋4𝑣110=0,(41)𝑣IV111+Re𝑣1112𝜋2𝑣111𝜋2Re𝑣111+𝜋4𝑣111+2Re𝜋2𝑣110Re𝑣V110𝜋4Re𝑣110=0.(42)

The corresponding boundary conditions are:𝑦=0𝑣110=1𝜋,𝑣111𝑣=0,110=0,𝑣111=0,𝑦𝑣1100,𝑣111𝑣0,1100,𝑣1110.(43) 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:𝑢1(𝑦,𝑧)=𝑢11𝜃(𝑦)cos𝜋𝑧,1(𝑦,𝑧)=𝜃11𝐶(𝑦)cos𝜋𝑧,1(𝑦,𝑧)=𝐶11(𝑦)cos𝜋𝑧.(44) On using (44) into (27), (30), and (31), we get𝑢11𝜋2𝑢11+Re𝜋𝑣11𝑢0+𝑢11+Gr𝜃11+Gm𝐶11+𝛼12𝜋𝑣11𝑢0𝑢11𝜋𝑣11𝑢0+𝜋𝑣11𝑢0𝑢0𝜋3𝑣11+𝜋2𝑢11=0,(45)𝜃11𝜋2𝜃11𝜃+RePr11+𝜋𝑣11𝜃0=0,(46)𝐶11𝜋2𝐶11𝐶+ReS11+𝜋𝑣11𝐶0=0,(47) subject to boundary conditions𝑦=0𝑢11=0,𝜃11=0,𝐶11=0𝑦𝑢11𝜃0,110,𝐶110.(48) 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, substituting𝑢11(𝑦)=𝑢110(𝑦)+𝛼1𝑢111𝛼(𝑦)+𝑜21(49) into (45) and boundary conditions (48) up to first order of 𝛼1and comparing the coefficients of like powers of 𝛼1,we obtain𝑢110+Re𝑢110𝜋2𝑢110=ReGr𝜃11+Gm𝐶11+𝜋𝑣11𝑢0,(50)𝑢111+Re𝑢111𝜋2𝑢111=Re2𝜋𝑣11𝑢0+𝑢110+𝜋𝑣11𝑢0,𝜋𝑣11𝑢0+𝑢0𝜋3𝑣11𝜋2𝑢110,(51) subject to boundary conditions𝑦=0𝑢110=0,𝑢111=0,𝑦𝑢1100,𝑢1110.(52) 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 coefficient 𝜎𝑥𝑦 at the plate 𝑦=0in the main flow direction is𝜎𝑥𝑦=𝜎𝑥𝑦𝜌𝑣20=1Re𝜕𝑢𝜕𝑦𝑦=0+𝛼1×𝑣𝜕2𝑢𝜕𝑦2𝜕+𝑤2𝑢𝜕𝑦𝜕𝑧3𝜕𝑢𝜕𝑦𝜕𝑣𝜕𝑦𝜕𝑤𝜕𝑦𝜕𝑢𝜕𝑧2𝜕𝑢𝜕𝑧𝜕𝑣𝜕𝑧𝑦=0.(53) The nondimensional heat flux at the plate 𝑦=0in terms of Nusselt number Nuis𝑘Nu=𝜌𝑣0𝐶𝑝𝑇𝑤𝑇𝜕𝑇𝜕𝑦=1𝜃PrRe0(0)+𝜀𝜃11.(0)sin𝜋𝑧(54) The nondimensional mass flux at the plate𝑦=0in terms of Sherwood number Sh0 isSh0=𝐷𝑣0𝐶𝑤𝐶𝜕𝐶𝜕𝑦𝑦=0=1𝐶SRe0(0)+𝜀𝐶11.(0)cos𝜋𝑧(55) 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.

Figures 1 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.

tab1
Table 1: Various combinations of flow parameters.
402037.fig.001
Figure 1: Variation of 𝑢against 𝑦for case I.
402037.fig.002
Figure 2: Variation of 𝑢against 𝑦for case II.
402037.fig.003
Figure 3: Variation of𝑢against 𝑦for case III.
402037.fig.004
Figure 4: Variation of𝑢against 𝑦for case IV.
402037.fig.005
Figure 5: Variation of 𝑢against 𝑦for case V.

Figures 6 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).

402037.fig.006
Figure 6: Variation of 𝜎 against Gm for 𝑠=0.1,Pr=3.
402037.fig.007
Figure 7: Variation of 𝜎 against Gm for 𝑠=0.4,Pr=3.
402037.fig.008
Figure 8: Variation of 𝜎 against Gm for 𝑠=0.4,Pr=5.

Figures 9 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).

402037.fig.009
Figure 9: Variation of 𝜎 against 𝑠 for Gm=2,Pr=3.
402037.fig.0010
Figure 10: Variation of 𝜎 against 𝑠 for Gm=5,Pr=3.
402037.fig.0011
Figure 11: Variation of 𝜎 against 𝑠 for Gm=5,Pr=5.

It has also been observed that the heat and mass flux at the plate 𝑦=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 different Gm,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.

References

  1. V. M. Soundalgekar and P. D. Wavre, “Unsteady free convection flow past an infinite vertical plate with constant suction and mass transfer,” International Journal of Heat and Mass Transfer, vol. 20, no. 12, pp. 1363–1373, 1977. View at Scopus
  2. H. T. Lin and C. M. Wu, “Combined heat and mass transfer by laminar natural convection from a vertical plate,” Heat and Mass Transfer, vol. 30, no. 6, pp. 369–376, 1995. View at Publisher · View at Google Scholar · View at Scopus
  3. R. Muthucumaraswamy, P. Ganesan, and V. M. Soundalgekar, “Heat and mass transfer effects on flow past an impulsively started vertical plate,” Acta Mechanica, vol. 146, no. 1-2, pp. 1–8, 2001. View at Publisher · View at Google Scholar · View at Scopus
  4. G. V. Lachman, Boundary Layer and Flow Control, Its Principles and Application, Vols. I and II, Pergamon Press, 1961.
  5. P. Singh, V. P. Sharma, and U. N. Misra, “Fluctuating boundary layer on a heated horizontal plate,” Acta Mechanica, vol. 30, no. 1-2, pp. 111–128, 1978. View at Publisher · View at Google Scholar · View at Scopus
  6. K. D. Singh and R. Sharma, “Three dimensional Couette flow through a porous medium with heat transfer,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 12, pp. 1819–1829, 2001. View at Scopus
  7. R. C. Chaudhary and P. Kumar Sharma, “Three dimensional unsteady convection and mass transfer flow through porous medium,” Heat and Mass Transfer, vol. 39, no. 8-9, pp. 765–770, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Ahmed, “Free and forced convective three dimensional flow with heat and mass transfer,” International Journal of Applied Mathematics and Mechanics, vol. 5, no. 1, pp. 26–38, 2009.
  9. H. Markovitz and B. D. Coleman, “Incompressible second-order fluids,” Advances in Applied Mechanics, vol. 8, pp. 69–101, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. B. D. Coleman and W. Noll, “An approximation theorem for functionals, with applications in continuum mechanics,” Archive for Rational Mechanics and Analysis, vol. 6, no. 1, pp. 355–370, 1960. View at Publisher · View at Google Scholar · View at Scopus