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Effects of chemical reaction on MHD flow past an impulsively started semi-infinite vertical plate with uniform heat and mass flux.

Introduction

The influence of the magnetic field on viscous incompressible flow of electrically conducting fluid has its importance in many applications such as extrusion of plastics in the manufacture of Rayon and Nylon, purification of crude oil, pulp, paper industry, textile industry and in different geophysical cases etc. In many process industries, the cooling of threads or sheets of some polymer materials is of importance in the production line. The rate of cooling can be controlled effectively to achieve final products of desired characteristics by drawing threads, etc. in the presence of the magnetic field.

Chemical reactions can be codified as either heterogeneous or homogeneous processes. This depends on whether they occur at an interface or as a single phase volume reaction. In many chemical engineering processes, there does occur the chemical reaction between a foreign mass and the fluid in which the plate is moving. These processes take place in numerous industrial applications, e.g., polymer production, manufacturing of ceramics or glassware and food processing. Bourne ad Dixon [1] analyzed the cooling of fibres in the formation process.

Chambre and Young [3] have analyzed a first order chemical reaction in the neighbourhood of a horizontal plate. Das et al [4] have studied the effect of homogeneous first order chemical reaction on the flow past an impulsively started infinite vertical plate with uniform heat flux and mass transfer. Again, mass transfer effects on moving isothermal vertical plate in the presence of chemical reaction studied by Das et al [5]. The dimensionless governing equations were solved by the usual Laplace transform technique and the solutions are valid only at lower time level.

The effects of transversely applied magnetic field, on the flow of an electrically conducting fluid past an impulsively started infinite isothermal vertical plate was studied by Soundalgekar et al [7]. MHD effects on impulsively started vertical infinite plate with variable temperature were studied by Soundalgekar et al [8]. The dimensionless governing equations were solved using Laplace transform technique. Muthucumaraswamy and Ganesan [6] studied the effects of first order homogeneous chemical reaction on the flow past an impulsively started semi-infinite vertical plate with uniform heat flux and mass diffusion. The governing equations were solved numerically.

The problem of unsteady natural convection flow past an impulsively started semi-infinite vertical plate with uniform heat and mass flux in the presence of chemical reaction and magnetic field has not received attention of any researcher. Hence, the present study is to investigate the MHD flow past an impulsively started semi-infinite vertical plate with uniform heat and mass flux in the presence of homogeneous first order chemical reaction by an implicit finite-difference scheme of Crank-Nicolson type.

Mathematical Analysis

Here the hydromagnetic flow of a viscous incompressible fluid past an impulsively started semi-infinite vertical plate with uniform heat and mass flux is studied. It is assumed that there is a first order chemical reaction between the diffusing species and the fluid. Here, the x-axis is taken along the plate in the vertically upward direction and the y-axis is taken normal to the plate. Initially, it is assumed that the plate and the fluid are of the same temperature and concentration. At time t' > 0, the plate starts moving impulsively in the vertical direction with constant velocity u0 against gravitational filed. At the same time, the heat is supplied from the plate to the fluid at a uniform rate and the concentration level near the plate is also raised at an uniform rate. A transverse magnetic field of uniform strength B0 is assumed to be applied normal to the plate. The induced magnetic field and viscous dissipation are assumed to be negligible. Then, under the usual Boussinesq's approximation, the unsteady flow is governed by the following equations:

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The initial and boundary conditions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

On introducing the following non-dimensional quantities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

equations (1) to (4) are reduced to the following non-dimensional form

[partial derivative]U/[partial derivative]X + [partial derivative]V/[partial derivative]Y = 0 (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The corresponding initial and boundary conditions in non-dimensional form are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Numerical Technique

The unsteady, non-linear coupled equations (7) to (10) with the condition (11) are solved by employing an implicit finite difference scheme of Crank-Nicolson type. The finite difference equations corresponding to equations (7) to (10) are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The thermal boundary condition at Y = 0 in the finite difference form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

At Y = 0 (i.e., j = 0) equation (14), becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

After eliminating [T.sup.n+1.sub.i,-1] + [T.sup.n.sub.i,-1] using equation (15), equation (16) reduces to the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The boundary condition at Y = 0 for the concentration in the finite difference form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

At Y = 0 (i.e., j = 0), Equation (18) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

After eliminating [C.sup.n+1.sub.i,-1] + [C.sup.n.sub.i,-1] using equation (19), equation (20) reduces to the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

The region of integration is a rectangle with sides [X.sub.max] (= 1) and [Y.sub.max](= 14), where [Y.sub.max] corresponds to Y = [infinity] which lies very well outside the momentum, energy and concentration boundary layers. The maximum of Y was chosen as 14 after some preliminary investigations so that the last two of the boundary conditions (11) are satisfied. Here the subscript i-designates the grid point along the X-direction, j-along the Y-direction and the superscript n along the t-direction.

The computations of U, V, T and C at time level (n + 1) using the values at previous time level (n) are carried out as follows: The finite-difference equations (14), (17), (18) and (21) at every internal nodal point on a particular i-level constitute a tridiagonal system of equations. Such a system of equations are solved by using Thomas algorithm as discusses in Carnahan et al [1]. Thus, the values of C are found at every nodal point for a particular i at [(n+1).sup.th] time level. Similarly, the values of T and U are calculated from equations (17) and (13) respectively. Then the values of V are calculated explicitly using the equation (12) at every nodal point at particular i-level at [(n+1).sup.th] time level. This process is repeated for various i-levels. Thus the values of C, T, U and V are known, at all grid points in the rectangle region at [(n+1).sup.th] time level.

Computations are carried out for different time levels until the steady-state is reached. The steady-state solution is assumed to have been reached, when the absolute difference between the values of U as well as temperature T and concentration C at two consecutive time steps are less than [10.sup.-5] at all grid points.

The local truncation error is O([DELTA][t.sup.2] + [DELTA][Y.sup.2] + [increment]X) and it tends to zero as [increment]t, [increment]X and [increment]Y] tend to zero. Hence the scheme is compatible. The finite difference scheme is unconditionally stable as discussed by Muthucumaraswamy and Ganesan [6]. Stability and compatibility ensures convergence.

Results and Discussion

The effect of velocity, temperature, concentration, local as well as average skin-friction, and Nusselt number and Sherwood number are studied for different parameters during transient and steady-state period. In order to check the accuracy, the present study is compared with the available exact solution in the literature. The concentration profiles for K = 0.2, Sc = 0.7, 0.9, Gr = 2, Gc = 5 and Pr = 0.71 (corresponding to [eta] = Y /2[square root of t]) are compared with the available exact solution of Das et al [5] at t = 0.2 in Fig. 1 and they are found to be in good agreement with the available theoretical solution at lower time level.

The steady-state velocity profiles for different magnetic parameter are shown in Fig. 2. It is observed that for M = 0, 2, 5, 10, K = 2, Gr = 2, Gc = 5, Pr = 0.71 and Sc = 0.6, the velocity decreases in the presence of magnetic field than in its absence. This shows that an increase in the magnetic field parameter leads to a fall in the velocity.

The steady-state velocity profiles for different values of the chemical reaction parameter (K = -1, 0.2, 2), M = 2, Gr = 2, Gc = 5, Pr = 0.71 and (Sc = 0.16, 0.6, 2.01) are shown in Fig. 3. It is observed that the velocity increases with decreasing chemical reaction parameter. This shows that velocity increases during generative reaction and decreases in destructive reaction. It is also observed that the velocity decreases with increasing Schmidt number.

In Fig. 4, the velocity profiles for different of thermal Grashof number and mass Grashof number are shown graphically. This shows that the velocity increases with increasing thermal Grashof number or mass Grashof number.

The temperature profiles for different values of the chemical reaction parameter and Prandtl number are shown in Fig. 5. It is observed that the temperature increases with decreasing Prandtl number. This shows that the buoyancy effect on the temperature distribution is very significant in air (Pr = 0.71) compared to water (Pr = 7.0). The temperature trend is reversed with respect to the chemical reaction parameter.

The effect of the chemical reaction parameter and the Schmidt number is very important for concentration profiles. The steady-state concentration profiles for different values of the chemical reaction parameter and Schmidt number are shown in Fig. 6. There is a fall in concentration due to increasing the values of the chemical reaction parameter or Schmidt number.

Knowing the velocity, temperature and concentration field, it is customary to study the skin-friction, the rate of heat transfer and rate of concentration in steady-state conditions. The local as well as average values of the skin-friction, Nusselt number and Sherwood number are given by the following expressions:

[[tau].sub.x] = -[([partial derivative]U/[partial derivative]Y).sub.Y=0] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[Nu.sub.X] = -X[[([partial derivative]T/[partial derivative]Y).sub.Y=0]/T.sub.Y=0]] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[Sh.sub.X] = -X[[([partial derivative]C/[partial derivative]y).sub.Y=0]/C.sub.Y=0]] (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

The derivatives involved in the equations (22) to (27) are evaluated using five-point approximation formula and then the integrals are evaluated using Newton-Cotes closed integration formula.

The local values of the skin-friction are evaluated from equation (22) and plotted in Fig. 7. The local wall shear stress increases with increasing values of the magnetic field parameter. Skin-friction increases with increasing chemical reaction parameter or Schmidt number. The local Nusselt number for different values of thermal and mass Grashof number and Schmidt number are shown in Fig. 8. It is seen that the heat transfer rate increases with increasing thermal and mass Grashof numbers and decreases with increasing Schmidt number. In Fig. 9, the local Sherwood number for different values of chemical reaction parameter and Schmidt number are shown. It is seen that the local Sherwood number increases with increasing Schmidt number. The concentration rate increases during the destructive reaction and decreases during the generative reaction.

The effects of the magnetic field parameter, chemical reaction parameter, Gr, Gc and Sc on the average values of the skin-friction and Nusselt and Sherwood number are shown in Figures 10, 11 and 12 respectively. The average skin-friction decreases with decreasing magnetic field parameter or chemical reaction parameter. The average Nusselt number increases with decreasing magnetic field parameter or Schmidt number and increases with increasing thermal or mass Grashof number. The local Sherwood number increases as chemical reaction parameter increases.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

Conclusions

Numerical study has been carried out for the unsteady hydromagnetic flow past an impulsively started semi-infinite vertical plate with uniform heat and mass flux in the presence of homogeneous chemical reaction of first order. The dimensionless governing equations are solved by an implicit finite difference scheme of Crank-Nicolson type. The concentration profiles are compared with the exact solution and are found to be in good agreement. It is observed that the velocity decreases in the presence of magnetic field than in its absence. It is also observed that the velocity and concentration increases during generative reaction and decreases in destructive reaction. The study shows that the number of time steps to reach steady-state depends strongly on the chemical reaction parameter or magnetic field parameter.

Nomenclature

[B.sub.0] magnetic field strength

C' concentration

C dimensionless concentration

D mass diffusion coefficient

g accelaration due to gravity

Gr thermal Grashof number

Gc mass Grashof number

j" mass flux per unit area at the plate

k thermal conductivity of the fluid

K dimensionless chemical reaction parameter

[K.sub.1] chemical reaction parameter

M magnetic field parameter

[Nu.sub.x] dimensionless local Nusselt number

[bar.Nu] dimensionless average Nusselt number

Pr Prandtl number

q heat flux per unit area at the plate

Sc Schmidt number

[Sh.sub.x] dimensionless local Sherwood number

[bar.Sh] dimensionless average Sherwood number

T' temperature

T dimensionless temperature

t' time

t dimensionless time

[u.sub.0] velocity of the plate

u, v velocity components in x, y-directions respectively

U, V dimensionless velocity components in X, Y-directions respectively

x spatial coordinate along the plate

X dimensionless spatial coordinate along the plate

y spatial coordinate normal to the plate

Y dimensionless spatial coordinate normal to the plate

Greek symbols

[alpha] thermal diffusivity

[beta] coefficient of volume expansion

[[beta].sup.*] volumetric coefficient of expansion with concentration

[mu] coefficient of viscosity

v kinematic viscosity

[sigma] Stefan-Boltzmann constant

[[tau].sub.x]/[tau] dimensionless local skin-friction dimensionless average skin-friction

Subscripts

w conditions at the wall

[infinity] conditions in the free stream

i grid point along the X-direction

j grid point along the Y-direction

References

[1] D.E. Bourne and H. Dixon, The cooling of fibres in the formation process, Int. J. Heat and Mass Transfer, 24 (1971), 1323-1332.

[2] B. Carnahan, H.A. Luther and J.O. Wilkes, Applied Numerical Methods, John Wiley and Sons, New York, (1969).

[3] P.L Chambre and J.D. Young, On the diffusion of a chemically reactive species in a laminar boundary layer flow, The Physics of fluids, 1 (1958), 4854

[4] U.N Das, R.K. Deka, and V.M. Soundalgekar, Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, Forschung im Ingenieurwesen 60(10) (1994) 284-287

[5] U.N. Das, R.K. Deka, and V.M. Soundalgekar, Effects of mass transfer on flow past an impulsively started infinite vertical plate with chemical reaction, The Bulletin of GUMA 5 (1999) 13-20.

[6] R. Muthucumaraswamy and P. Ganesan, Effects of the chemical reaction on flow characteristics in an unsteady upward motion of an isothermal plate, Journal of applied Mechanics and Technical Physics, Vol. 42, No. 4, (2001), 665-671.

[7] V.M. Soundalgekar, S.K. Gupta and R.N. Aranake, Free convection effects on MHD Stokes problem for a vertical plate, Nuclear Engg. Des. 51, (1979) 403-407.

[8] V.M. Soundalgekar, M.R. Patil and M.D. Jahagirdar, MHD Stokes problem for a vertical plate with variable temperature, Nuclear Engg. Des. 64, (1981) 39-42.

P. Chandrakala

Department of Mathematics, Bharathi Women's College, No. 34 / 2, Ramanujam Garden Street, Pattalam, Chennai, India

E-mail: pckala05@yahoo.com
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Author:Chandrakala, P.
Publication:International Journal of Dynamics of Fluids
Date:Dec 1, 2010
Words:2713
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