This paper presents unsteady as well as steady-state free convection Couette flow of reactive viscous fluid in a vertical channel formed by two infinite vertical parallel porous plates. The motion of the fluid is induced due to free convection caused by the reactive nature of viscous fluid as well as the impulsive motion of one of the porous plates. The Boussinesq assumption is applied, and the nonlinear governing equations of motion and energy are developed. The time-dependent problem is solved using implicit finite difference method, and steady-state problem is solved by applying regular perturbation technique. During the course of computation, an excellent agreement was found between the well-known steady-state solutions and transient solutions at large value of time.

1. Introduction

The study of convection processes in porous channels is a well-developed field of investigation because of its importance to a variety of situations. For example, thermal insulations, geothermal system, surface catalysis of chemical reactions, solid matrix heat exchanger, microelectronic heat transfer equipment, porous flat collectors, coal and grain storage, petroleum industries, dispersion of chemical contaminants in various processes, nuclear waste material, and catalytic beds. Some of the examples mentioned above involve two or more fluids, multidimensional and unsteady flows .

The literature on the topic of unsteady as well as steady-state free convection flow problems is well surveyed by Singh et al. , Florio and Harnoy , Jha and Ajibade , Paul et al. , and Langellotto et al. . Muhuri  presented natural convection Couette flow between two vertical porous plates when one of the plates moves with uniform acceleration. Singh  investigated the motion of the fluid induced by the impulsive motion of one of the plates in the presence of convection currents due to asymmetric thermal condition on the boundaries. Abdulaziz and Hashim  considered free convection flow between porous vertical plates with asymmetric wall temperatures and concentrations and used homotopy analysis method to solve boundary-value problems. Fang [10, 11] analysed unsteady velocity and temperature profiles for Couette flow and pressure-driven flow in a channel with porous walls. Unsteady flow investigations with porous boundaries include the works of Wang et al. , Oxarango et al. , and Makinde and Ogulu . Makinde and Maserumule  presented thermal criticality and entropy analysis for a variable viscosity Couette flow.

In all previous studies the working fluid is considered as nonreactive viscous incompressible fluid. However, unsteady as well as steady-state free convection Couette flow of a reactive viscous fluid in a vertical channel formed by two vertical porous plates can be of importance for the design of equipment used in several types of engineering systems. There are many chemical reactions with important practical applications. A common configuration for such reactions is for the reactants to be made to flow over solid catalyst, with the reaction taking place on the surface of the catalyst. A full discussion of catalysis and description of many of its practical applications is given by Chaudhary and Merkin , Merkin and Chaudhary , and Chaudhary et al. . These authors studied flow configuration and assume that the reaction takes place only on the catalytic surface and can be represented schematically by the single first-order Arrhenius kinetics. According to these authors there is a three-way coupling between fluid flow, fluid/surface temperature, and reactant species concentration. Ayeni  and Dainton  studied a realistic mathematical description of thermal explosion which include the effects of Arrhenius temperature dependence with variable preexponential factor. Recently Jha et al.  investigated transient natural convection flow of reactive viscous flow in a vertical channel. Most recently Hazem Attia  studied the effect of suction and injection on unsteady Couette flow.

The objective of the present work is to analyze the effect of suction/injection on time dependent unsteady as well as steady-state free convection Couette flow of viscous reactive fluid in a vertical channel formed by two infinite vertical parallel porous plates.

2. Mathematical Analysis

Consider the transient free convective Couette flow of viscous reactive fluid in a vertical channel formed by two infinite vertical parallel porous plates. The system under consideration is sketched in Figure 1. The flow is assumed transient and fully developed. The x-axis is taken along the direction of the flow and parallel to the infinite vertical porous plates and y-direction perpendicular to the flow direction. At time t0, the fluid and the porous plates are assumed to be same temperature T0, and there is no fluid motion. At t>0, one of the porous plates (y=H) starts moving with constant velocity and fluid generate heat due to its reactive nature causes to set up fluid motion inside the channel. In addition, at the same time the flow is subjected to suction of the fluid from one porous plate and at the same rate fluid is being injected through the other porous plate.

Schematic diagram of the problem.

The fluid is assumed to be Newtonian and obeys the Boussinesq’s approximation. Under the previous assumptions the energy and momentum equations in dimensional form are (1)ut-v0uy=υ2uy2+gβ(T-T0),Tt-v0Ty=α2Ty2+QCoAρCpexp(-ERT), where T0 the initial fluid and wall temperature, T dimensional temperature of fluid, Q the heat of reaction, A the rate constant, E the activation energy, R the universal gas constant, Co the initial concentration of the reactant species, v kinematic viscosity, and H gap between the porous plates.

The initial and boundary conditions to be satisfied are  (2)t0:u=0,T=T0for0yh,t>0{u=0,T=T0,aty=0,u=U,T=T0,aty=H. Equations (1) and (2) can be nondimensionalized using the variables (3)y=yH,t=tvH2,u=uU,θ=ERT02(T-T0),λ=QCoAEH2κRT02exp(-ERT0),ε=RT0E,s=V0Hυ,δ=gβRT02h2EUυ,Pr=υα. The physical quantities used in (3) are defined in the nomenclature.

Using the expressions (3) and (1) the dimensionless momentum and energy equations are (4)ut-suy=2uy2+δθ,θt-sθy=1Pr2θy2+λPrexp(θ1+εθ), while the initial and boundary conditions in dimensionless form are  (5)t0:  u=θ=0for0y1,t>0:{u=0,θ=0at  y=0,u=1,θ=0at  y=1.

3. Analytical Solutions

The governing equations (4) presented in the previous section are highly nonlinear and exhibit no analytical solutions. The importance of analytical solutions which refers to steady free convection Couette flow of viscous reactive fluid relies on the chance to obtain nontrivial benchmarks to test the reliability of numerical codes developed for more complex physical situations. Analytical solutions are often an opportunity to inspect the internal consistency of mathematical models and of the approximations adopted, as well as to develop new theoretical results. The mathematical model representing the steady-state free convection Couette flow can be obtained by setting u/t=0 and θ/t=0 in (4) to get (6)d2udy2+sdudy+δθ=0,d2θdy2+Prsdθdy+λexp(θ1+εθ)=0. The boundary conditions are (7)u=0,θ=0at  y=0,u=1,θ=0at  y=1. In order to construct an approximate solution to (6) subject to (7), we employed a regular perturbation method by taking a power series expansion in the reactant consumption parameter λ: (8)θ=i=0θiλi,u=i=0uiλi. Substituting (8) into (6) and collecting the coefficients of like powers of λ, the solution of the governing equations is obtained as (9)θ=λ[1sPr(1-exp(-sPry))(1-exp(-sPr))-ysPr]+λ2{12sPr(y2sPr-2y(sPr)2+2(sPr)3)-y(sPr)2((1+exp(-sPry))(1-exp(-sPr)))+1(sPr)31(1-exp(-sPr))+k1sPr12sPr+k2exp(-sPry)(y2sPr-2y(sPr)2+2(sPr)3)},u=[1-exp(-sy)][1-exp(s)]+λ[δ{12sPr(y2s-2s{ys-1s2})-1z1{ys-1s2}-1z1exp(-sPry)s2Pr(1-Pr)}+c1s+c2exp(-sy){ys-1s2}{12sPr(y2s-2s{ys-1s2})]+λ2[G1s-δ6(sPr)2{y3s-3y2s2+6ys3-6s4}+δ{12(sPr)3+12(sPr)2(1-exp(-sPr))}×{y2s-2ys2+2s3}-δ(sPr)4(ys-1s2)-δ(sPr)3(1-exp(-sPr))×{ys(1-Pr)-1s2(1-Pr)2}exp(-sPry)-δ(sPr)41(1-exp(-sPr))exp(-sPry)s(1-Pr)-δ(sPr)3(1-exp(-sPr)){ys-1s2}  -k1sPr(ys-1s2)+k2exp(-sPry)s2Pr(1-Pr)+G2exp(-sy){y3s-3y2s2+6ys3-6s4}k2exp(-sPry)s2Pr(1-Pr)]. Steady-state skin frictions on the boundary plates is (10)τ0=dudy|y=0=s[1-exp(-s)]+λ[δ(Prz1s(1-Pr)-1s3Pr)-c2s]-λ2[δs5(1Pr2+1Pr3+1Pr4)+δs4Pr2(1-exp(-sPr))(1+1(1-Pr)2+1Pr)+k1s2Pr-k2s(1-Pr)-G2s1s3Pr],τ1=dudy|y=1=[sexp(-s)][1-exp(s)]+λ[k2sPrexp(-sPr)s2Pr(1-Pr)δ{12sPr(2s-2s2)+sPrz1exp(-sPr)s2Pr(1-Pr)}-c2sexp(-s){12sPr(2s-2s2)+sPrz1exp(-sPr)s2Pr(1-Pr)}12sPr][1-exp(s)]2s2[sexp(-s)][1-exp(s)]-λ2[k2sPrexp(-sPr)s2Pr(1-Pr)δ6(sPr)2{3s-6s2+6s3}+δ{12(sPr)3+12(sPr)2(1-exp(-sPr))}×{2s-2s2}-δs5(Pr)4+δsPrexp(-sPr)(sPr)3(1-exp(-sPr))1s(1-Pr)+δ(sPr)41(1-exp(-sPr))sPrexp(-sPr)s(1-Pr)-δs2(Pr)3(1-exp(-sPr))-k1s2Pr-k2sPrexp(-sPr)s2Pr(1-Pr)-G2sexp(-s)δ6(sPr)2]{12sPr(2s-2s2)+sPrz1exp(-sPr)s2Pr(1-Pr)}sPrz1. The steady-state rate of heat transfer on the boundary plates is (11)Nu0=dθdy|y=0=λ[1(1-exp(-sPr))-1sPr]-λ2[1(sPr)3+2(sPr)2(1-exp(-sPr))+sPrk2],Nu1=dθdy|y=1=λ[exp(-sPr)(1-exp(-sPr))-1sPr]+λ2[1(sPr)3(sPr-1)-1(sPr)2(1-exp(-sPr))×{1+(1-sPr)exp(-sPr)}1sPr-sPrk2exp(-sPr)1(sPr)3]. The constants c1,c2,k1,k2,A,B,A1,B1,G1,G2 are defined in the appendix section.

4. Numerical Solutions

To solve the time-dependent equation (4), the differential equations have been transformed into the corresponding finite difference equation. The procedure involves discretization of the momentum and energy equations into the finite difference equations at the grid point (i,j) in which the time derivatives are approximated by the backward difference while the spatial derivatives are replaced by the central difference formula. The above equations are solved by Thomas algorithm by manipulating into a system of linear algebraic equations in the tridiagonal form.

In each time step, firstly the temperature field has been solved and then the velocity field is evaluated using the already known values of the temperature field. The process of computation is advanced until a solution is approached by satisfying the following convergence criterion: (12)|Ai,j+1-Ai,j|M|A|max<10-5 with respect to temperature and velocity fields. Here Ai,j stands for the velocity or temperature fields, M is the number of interior grid points, and |A|max is the maximum absolute value of Ai,j.

5. Results and Discussion

Velocity profile (ε=0.01,Pr=7.0,λ=1.0).

Velocity profile (ε=0.01,Pr=7.0,s=0.5).

Velocity profile (ε=0.01,s=0.5,λ=1.0).

Temperature profile (ε=0.01,Pr=7.0,λ=1.0).

Temperature profile (ε=0.01,Pr=7.0,λ=1.0).

Temperature profile for (ε=0.01,s=0.5,Pr=7.0).

Temperature profile (ε=0.01,Pr=0.71,s=0.5).

Variation of skin friction (ε=0.01,λ=1.0) at y=0.

Variation of skin friction (ε=0.01,s=0.5,λ=1.0) at y=0.

Variation of skin friction (ε=0.01,s=0.5) at y=1.

Variation of Nusselt number (ε=0.01,λ=1.0,Pr=7.0) at y=0.

Variation of Nusselt number (ε=0.01,λ=1.0,Pr=7.0) at y=1.

Variation of Nusselt number (s=0.5,ε=0.01,Pr=7.0) at y=0.

Variation of Nusselt number (s=0.5,ε=0.01,Pr=0.71) at y=0.

Variation of Nusselt number (s=0.5,ε=0.01,Pr=7.0) at y=1.

Variation of Nusselt number (s=0.5,ε=0.01,Pr=0.71) at y=1.

6. Conclusion

The problem of unsteady as well as steady-state natural convection Couette flow of reactive viscous fluid in a vertical channel formed by two vertical porous plates has been presented. The velocity field and temperature field are obtained analytically by perturbation series method for steady free convective Couette flow of viscous reactive fluid in a vertical channel formed by two vertical porous plates and numerically by implicit finite difference technique for unsteady free convective Couette flow of viscous reactive fluid in a vertical channel formed by two vertical porous plates. Graphical results for the velocity, temperature, skin friction, and the Nusselt number variations were presented and discussed for various physical parametric values.

The main findings are as follows.

Skin friction is always higher in the case of air (Pr=0.71) than water (Pr=7.0).

It is also seen that increase in λ increases heat transfer on the porous plates.

The heat transfer is higher at left porous plate y=0 where injection takes place in comparison to right plate where suction takes place y=1.

The introduction of suction/injection has distorted the symmetric nature of the flow.

During the course of computation it is observed that at steady state the effect of Pr on the fluid flow is to increase the velocity and temperature as Pr increases when injection is considered (s<0) at y=0 while fluid velocity and temperature decreases as Pr increases in the presence of suction (s>0) at y=0, which is not true in transient case (i.e., velocity and temperature decreases as Pr increases).

Appendix

Consider (A.1)c1=δ[1-exp(-s)]{1z1+1s2Pr-12sPr-1s2Pr2(1-Pr)}-δ{1s3Pr+1z1s-1z1sPr(1-Pr)},c2=δ[1-exp(-s)]{12s2Pr+1s3Pr2(1-Pr)}-δs[1-exp(-s)]{1z1+1s2Pr},k1=-12{1sPr-2(sPr)2+2(sPr)3}1[1-exp(-sPr)]+1sPr[1+exp(-sPr)][1-exp(-sPr)]2-1(sPr)21[1-exp(-sPr)]2+exp(-sPr)(sPr)2[1-exp(-sPr)]2+1(sPr)3exp(-sPr)[1-exp(-sPr)],k2=-k1sPr-1(sPr)3[1-exp(-sPr)]-1(sPr)4,A=1s6Pr2+1s6Pr3+1s6Pr4k1s3Pr+k2s2Pr(1-Pr),B=1[1-exp(-sPr)]×{1(1-Pr)2s5Pr3+1s5Pr2-1s5Pr4(1-Pr)+1s5Pr3},A1=16(sPr)2{1s-3s2+6s3-6s4}-{12(sPr)3+12(sPr)2[1-exp(-sPr)]}×{1s-2s2+2s3},B1=1(sPr)4{1s-1s2}+1(sPr)31[1-exp(-sPr)]×{1s(1-Pr)-1s2(1-Pr)2}exp(-sPr)+1(sPr)41[1-exp(-sPr)]exp(-sPr)s(1-Pr)+1(sPr)3{1s-1s2}1[1-exp(-sPr)]+ksPr{1s-1s2}-k2exp(-sPr)s2Pr(1-Pr),G1=-δ[A+B]s-sG2,G2=δ[(A+B)+(A1+B1)][exp(-s)-1].

Nomenclature C p :

Specific heat of the fluid at constant pressure

E :

Activation energy

g :

Acceleration due to gravity

H :

Gap between the channels

N u 0 :

Nusselt number at y=0

N u 1 :

Nusselt number at y=1

Pr :

Prandtl number

Q :

Heat reaction parameter

R :

Universal gas constant

s :

Suction/injection parameter

t :

Dimensional time

t :

Dimensionless time

T :

Dimensional temperature of the fluid

T 0 :

Initial temperature of the fluid

u :

Dimensional velocity of the fluid

u :

Dimensionless velocity of the fluid

x :

Dimensional coordinate parallel to channel

y :

Dimensional coordinate perpendicular to channel

y :

Dimensionless coordinate.

Greek Letters β :

Volumetric coefficient of thermal expansion

ρ :

Density of the fluid

τ 0 :

Dimensionless skinfriction at y=0

τ 1 :

Dimensionless skin friction at y=1

α :

Thermal diffusivity

θ :

Dimensionless temperature

λ :

Reactant consumption parameter

v :

Kinematic viscosity

ε :

Activation energy parameter

δ :

Grashof number.

Acknowledgment

The author A. K. Samaila is thankful to Usmanu Danfodiyo University, Sokoto, for financial support.

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