Chemical reaction effects on MHD flow past an impulsively started isothermal vertical plate with uniform mass flux.
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. Many transport processes exist in nature and in industrial applications in which the simultaneous heat and mass transfer as a result of combined buoyancy effects of thermal diffusion and diffusion of chemical species. However, in nature, along with free-convection currents caused by temperature differences, the flow is also affected by the differences in concentration. Such a study is found useful in chemical processing industries such as fibre drawing, crystal pulling from the melt and polymer production.
Magnetoconvection plays an important role in various industrial applications. Examples include magnetic control of molten iron flow in the steel industry, liquid metal cooling in nuclear reactors and magnetic suppression of molten semi-conducting materials. It is of importance in connection with many engineering problems, such as sustained plasma confinement for controlled thermonuclear fusion and electromagnetic casting of metals. MHD finds applications in electromagnetic pumps, controlled fusion research, crystal growing, MHD couples and bearings, plasma jets and chemical synthesis.
Chambre and Young  have analyzed a first order chemical reaction in the neighborhood of a horizontal plate. Das et al  considered the mass transfer effects on flow past an impulsively started infinite isothermal vertical plate with uniform mass flux. Again, Das et al  studied an exact solution to the flow of a viscous incompressible fluid past an impulsively started infinite isothermal vertical plate in the presence of mass diffusion and first order chemical reaction. In all above studies, the dimensionless governing equations are solved using Laplace transform technique.
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 . Muthucumaraswamy and Ganesan  studied the effects of the first order homogeneous chemical reaction on the process of an unsteady flow past an impulsively started semi-infinite vertical plate with uniform mass flux. The governing equations are solved using implicit -finite difference scheme of Crank-Nicolson type.
The problem of unsteady natural convection flow past an impulsively started semi-infinite isothermal vertical plate with 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 isothermal vertical plate with homogeneous first order chemical reaction by an implicit finite-difference scheme of Crank-Nicolson type.
Formulation of the problem
A transient, laminar, unsteady natural convection MHD flow of a viscous incompressible fluid past an impulsively started semi-infinite isothermal vertical plate with uniform mass flux is considered. 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. The plate starts moving impulsively in the vertical direction with constant velocity [u.sub.0] against gravitational filed. The temperature of the plate is raised uniformly and the concentration level near the plate is also raised at an uniform rate. They are maintained at the same level for all time at time t' > 0. A transverse magnetic field of uniform strength [B.sub.0] is assumed to be applied normal to the plate. The induced magnetic field and viscous dissipation is assumed to be negligible. Then under the above assumptions, the governing boundary layer equations of mass, momentum and concentration for free convective flow with usual Boussinesq's approximation are as follows:
[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 quantities are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
In order to solve these unsteady, non-linear coupled equations (7) to (10) under the conditions (11), an implicit finite difference scheme of Crank-Nicolson type has been employed. 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)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
The boundary condition at Y = 0 for the concentration in the finite difference form is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
After eliminating [C.sup.n+1.sub.sub.i-1] + [C.sup.n.sub.i-1] using Equation (16), Equation (17) reduces to the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
Here the region of integration is considered as a rectangle with sides Xmax (=1) and [Y.sub.max](=14), where [Y.sub.max] corresponds to Y = [infinity] which lies very well outside both the momentum and energy 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 with in the tolerance limit [10.sup.-5].
After experimenting with a few set of mesh sizes, the mesh sizes have been fixed at the level [increment of X] = 0.05, [increment of Y] = 0.25 with time step [increment of A] = 0.01. In this case, the spatial mesh sizes are reduced by 50% in one direction, and later in both directions, and the results are compared. It is observed that, when the mesh size is reduced by 50% in the Y-direction, the results differ in the fifth decimal place while the mesh sizes are reduced by 50% in X-direction or in both directions, the results are comparable to three decimal places. Hence, the above mesh sizes have been considered as appropriate for calculation. The coefficients [U.sup.n.sub.i,j] and [V.sup.n.sub.i,j] appearing in the finite- difference equations are treated as constants in any one time step. Here i-designates the grid point along the X-direction, j along the Y-direction and the superscript n along the t-direction. The values of U, V and T are known at all grid points at t = 0 from the initial conditions.
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 (15) and (18) 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 .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 (14) 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 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, and as well as temperature T and concentration C at two consecutive time steps are less than [10.sup.-5] at all grid points.
Results and Discussion
In order to assess the accuracy of the numerical results, the present study is compared with the theoretical solution for the case K = 0 in Fig. 1. The results compared are in good agreement at lower time level. The values K > 0 and K < 0 refer to destructive and generative reactions respectively.
The steady-state velocity profiles for different magnetic parameter are shown in Fig. 2. It is observed that for (M = 0, 2, 5), K = 2, Gr =2, Gc =5, Pr = 0.71, and Sc=0.6, the velocity decreases in the presence of magnetic field than its absence. This shows that the increase in the magnetic field parameter leads to a fall in the velocity. This agrees with the expectations, since the magnetic field exerts a retarding force on the free convective flow.
The effect of velocity for different chemical reaction parameter (K = -2, 0, 2), M = 2, Gr = 2, Gc = 5, Pr = 0.71 and Sc = 0.6 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.
Fig. 4 shows the steady velocity profiles for different values of the thermal Grashof and mass Grashof numbers and the Prandtl number. The velocity increases with increasing thermal or mass Grashof number and decreases with increasing Prandtl number. As thermal Grashof number or mass Grashof number increases, the buoyancy effect becomes more significant, as expected, it implies that, more fluid is entrained from the free stream due to the strong buoyancy effects as Gr or Gc increases.
The steady-state velocity profiles for different Schmidt number are shown in Fig. 5. The velocity profiles presented are those at X = 1.0. It is observed that the velocity decreases with increasing Schmidt number. The velocity boundary layer seem to grow in the direction of motion of the plate. It is observed that near the leading edge of a semi-infinite vertical plate moving in a fluid, the boundary layer develops along the direction of the plate. However, the time required for the velocity to reach steady-state depends upon the Schmidt number. The shows that the contribution of mass diffusion to the buoyancy force increases the maximum velocity significantly.
The transient and steady-state temperature profiles for different values of the Prandtl number and chemical reaction parameter are shown in Fig. 6. The temperature increases with increasing chemical reaction parameter and decreases with increasing 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). It is known that the Prandtl number plays an important role in flow phenomena, because it is a measure of the relative magnitude of viscous boundary layer thickness to the thermal boundary layer thickness.
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. 7. 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 fields, it is customary to study the skin-friction, heat transfer rate and concentration rate in steady-state conditions. The dimensionless local as well as average values of skin-friction, Nusselt number and the Sherwood number are given by the following expressions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
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The derivatives involved in the equations (19) to (24) 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 (19) and plotted in Fig. 8 as a function of the axial coordinate. The local wall shear stress increases with increasing 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. 9. It is seen that the heat transfer rate increases with increasing thermal and mass Grashof numbers and decreases with increasing Schmidt number. In Fig. 10, 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 11, 12 and 13 respectively. The average skin-friction decreases with decreasing magnetic field parameter or chemical reaction. The average Nusselt number increases with decreasing magnetic field parameter or Schmidt number and decreases with increasing thermal or mass Grashof number. The local Sherwood number increases as chemical reaction parameter increases.
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A detailed numerical study of the incompressible viscous fluid MHD flow past an impulsively started semi-infinite isothermal vertical plate with uniform mass flux in the presence of homogeneous chemical reaction of first order, is presented in the paper. Dimensionless governing equations are solved using the implicit finite-difference scheme of the Crank-Nicolson type. The fluids considered in this study are air and water. The calculation results are in good agreement with the exact solution in the absence of the chemical reaction (K = 0). The study performed allows the following conclusions.
(1) The velocity and concentration increases during the generative reaction (K < 0) and decreases during the destructive reaction (K > 0).
(2) The temperature increases with increasing chemical reaction parameter and decreases with increasing Prandtl number.
(3) The local and averaged values of the Sherwood number increases with increasing chemical reaction parameter.
[B.sub.0]--magnetic field strength
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--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
[Sh.sub.x]--dimensionless local Sherwood number
[bar.Sh]--dimensionless average Sherwood number
[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
[beta]--coefficient of volume expansion
[[beta].sup.*]--volumetric coefficient of expansion with concentration
[mu]--coefficient of viscosity
[[tau].sub.x]--dimensionless local skin-friction
[bar.[tau]]--dimensionless average skin-friction
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
 Carnahan B., Luther H.A. and Wilkes J.O., Applied Numerical Methods, John Wiley and Sons, New York, (1969)
 Chambre, P.L. and Young, J. D, On the diffusion of a chemically reactive species in a laminar boundary layer flow, The Physics of fluids 1 (1) (1958) 48-54.
 Das, U.N., Deka, R.K. and Soundalgekar, V.M., Mass transfer effects on flow past an impulsively started infinite vertical plate with constant mass flux-an exact solution, Heat and Mass Transfer, Vol. 31, (1996) pp. 163-167.
 Das, U.N., Deka, R.K. and Soundalgekar, V.M., Effects of mass transfer on flow past an impulsively started infinite vertical plate with chemical reaction, The Bulletin of GUMA 5 (1999) 13-20.
 Muthucumaraswamy R. and Ganesan P., 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) pp. 665-671.
 Soundalgekar V.M., Gupta S.K. and Aranake R.N. Free convection effects on MHD Stokes problem for a vertical plate, Nuclear Engg. Des. 51, (1979) 403407.
Department of Mathematics, Bharathi Women's College, No. 34/2, Ramanujam Garden Street, Pattalam, Chennai-600 012, INDIA E-mail: firstname.lastname@example.org
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|Publication:||International Journal of Dynamics of Fluids|
|Date:||Dec 1, 2010|
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