_{1}

^{*}

Analytical solutions for the effect of chemical reaction on the unsteady free convection flow past an infinite vertical permeable moving plate with variable temperature has been studied. The plate is assumed to move with a constant velocity in the direction of fluid flow. The highly nonlinear coupled differential equations governing the boundary layer flow, heat and mass transfer are solved using two-term harmonic and non-harmonic functions. The parameters that arise in the perturbation analysis are Prandtl number (thermal diffusivity), Schmidt number (mass diffusivity), Grashof number (free convection), modified Grashof number, Chemical reaction parameter (rate constant), Skin friction coefficient and Sherwood number (wall mass transfer coefficient). The study has been compared with available exact solution in the literature and they are found to be in good agreement. It is observed that: The concentration increases during generative reaction and decreases in destructive reaction. The concentration increases with decreasing Schmidt number. The effect of increasing values of K leads to a fall in velocity profiles. The velocity decreases with increasing values of the Schmidt number. An increase in modified Grashof number leads to an increase in velocity profiles. The skin friction increases with decreasing Schmidt number. In generative reaction the skin friction decreases and in destructive reaction the skin friction increases.

Chemical heat and mass transfer in natural convection flows on vertical cylinders has a wide range of applications in the field of science and technology [

This example represents a situation of particulate suspension where a pure fluid assumption does not accurately represent the reality. The outlook for a direct coal fired magnetohydrodynamic (MHD) power generator as potentially significant source of energy seems promising in view of its efficiently, its effect on environment and the availability of needed natural resources. These studies are useful in understanding the effect of the presence of a slag layer on heat transfer characteristics of a coalfired magnetohydrodynamic (MHD) generator. A fiew representative fields of interest in which combined heat and mass transfer along with chemical reaction play an important role in chemical process industries such as food processing and polymer production. Bottemanne [_{r} = 0.71)

and Schmidt number (S_{c} = 0.63) under the uniform wall temperature/concentration condition. Bottemanne’s [

Again, Soundalgekar et al. [^{}

It is proposed to study, the flow past an impulsively started infinite vertical plate with variable temperature and uniform mass diffusion in the presence of a homogeneous chemical reaction. The main reason for the lack of study of this problem is due to difficult mathematical and numerical procedures in dealing with the non-similar boundary layers. The highly non-linear coupled differenttial equations governing the boundary layer flow, heat and mass transfer are solved using two-term harmonic and non harmonic functions. Details of the velocities, temperature and concentration fields as well as the local skin friction and the local Sherwood number for the various values of the parameters of the problem are presented.

Consider unsteady two-dimensional flow of a laminar, viscous, and heat absorbing fluid past an infinite vertical permeable moving plate. The axial coordinate x' is measured vertically upward along the plate, and the y' axis is taken normal to the plate. At time t' ≤ 0 the plate and fluid are at the same temperature and concentration. At time, the plate is given an impulsive motion in the vertical direction against the gravitational field with uniform velocity u_{0}, the plate temperature is made to raise linearly with time. Also the level of the species concentration is raised to. It is also assumed that there exists a homogeneous first order chemical reaction between the fluid and species concentration. But here we assume the level of species concentration to be very low and hence heat generated during chemical reaction can be neglected. In this reaction the reactive component given off by the surface, occurs only in very dilute form. Hence, any convective mass transport to or from the surface due to a net viscous dissipation effects in the energy equation are assumed to be negligible. Under these assumptions, the boundary layer flow with Boussinesq’s approximation is governed by:

where, u' is the velocity of the fluid in the x' direction, g is the acceleration due to gravity, β is the volumetric coefficient of thermal expansion, β^{*} is the volumetric coefficient of expansion with concentration, T' is the temperature of the fluid near the plate, is the temperature of the fluid away from the plate, is the surface temperature t' time, t is the dimensionless time, ν is the kinematic viscosity, C' species concentration, C dimensionless species concentration, species concentration away from the plate, the surface species concentration, D mass diffusion coefficient, K thermal conductiveity, K_{l} chemical reaction parameter, ρ density of the fluid, C_{P} specific heat at constant pressure, u dimensionless velocity, y' coordinate axis normal to the plate, y dimensionless coordinate axis normal to the plate.

With the following initial and boundary conditions:

for all

at

as (4)

where, , u_{0 }velocity of the plate.

We now introduce the following non-dimensional quantities:

where, G Grashof number, G_{0} modified Grashof number, S_{c} Schmidt number, μ coefficient of viscosity, θ dimensionless temperature.

In Equations (1)-(4), which leads to

Therefore,

Equations (6)-(8), represent a set of partial differential equations that can not be solved in enclosed form. However, it can reduced to a set of ordinary differential equations in dimensional form that can be solved analytically, this can be done by representing the velocity, temperature and the concentration as:

Substituting Equations (9)-(11) into Equations (6)-(8), equating the harmonic and non harmonic terms and neglecting the higher order of (ε^{3}), and simplifying we obtain the following set of differential equations for u, θ and C.

In the above equations, the primes denote differentiation with respect to y.

The boundary conditions (4) after substitution Equations (9)-(11) are reduced toat

at

Hence from Equations (18)-(20) under the respective boundary conditions (21), and substituting the solutions into Equation (11) the solution for concentration distribution is given by:

Also, by solving the differential Equations (15)-(17), under the boundary conditions (21), and substituting the solutions into Equation (10). We have the temperature distribution is given by:

and either from Equations (12)-(14) under the respective boundary conditions (21), and substituting the solutions into Equation (9). We have the velocity distribution.

by knowing velocity, temperature, and concentration profiles, it is interesting to study about local and average values of skin friction. In non-dimensional quantities, the skin friction

where,

In order to get a physical understanding of the problem and for purpose of discussing the results, numerical calculations have been performed for the concentrationvelocity, temperature, rate of mass transfer, skin friction and rate of heat transfer. The results are represented graphically in Figures 1-10. The Prandtl number, P_{r} = 0.71 corresponds to air. The Grashof number, G > 0 corresponds to cooling of the plate by free convection currents, and Grashof number G < 0 corresponds to heating of the plate by free convection currents [3-6,12-19]. The mass diffusion Equation (8) can be adjusted to meet these circumstances if one takes, k > 0 for the destructive reaction, k = 0 for no reaction and k < 0 for the generative reaction. The effect of Prandtl number is very important in temperature profiles. There is a decrease in temperature due to increasing values of the Prandtl number.

The numerical values of the concentration profiles are computed and plotted in

The velocity profiles for different values of Grashof number G are described in _{0} are described in _{0} leads to a decrease in the values of velocity. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the plate as modified Grashof number decreases, and then decays to the relevant free stream velocity.

The transient concentration profiles for different Schmidt number is shown in

The effects of buoyancy ratio parameter for both aiding (G/G_{0} > 0) as well as opposing (G/G_{0} < 0) are shown in _{c}. Increasing values of S_{c} and P_{r} give rise to lower shear stress. Since increasing S_{c} and P_{r} gives thicker velocity profiles which in turn give lower skin friction values. As the buoyancy ratio parameter increases, higher skin friction is observed. For generative reaction, shear stress decreases as reaction parameter decreases. A similar situation is noted for destructive reaction.

Local skin friction values are plotted in Figures 9, 10 against the Grashof number G. They are increasing for

decreasing values of K. As the buoyancy ratio parameter increases, higher skin friction is observed. For generative reaction, shear stress decreases, as reaction parameter decreases. A similar situation is noted for destructive reaction.

A detailed numerical study has been carried out for the flow past an impulsively started infinite vertical plate with variable temperature and mass diffusion. These circumstances are of interest in several manufacturing processes.

The dimensionless governing equations are solved by a perturbation technique. Numerical evaluations of the closed form results were performed and some graphical results were obtained to illustrate the details of the flow and heat and mass transfer characteristics and their dependence on some of physical parameters. The study has been compared with available exact solution the literature and they are found to be in good agreement. It is observed that, the concentration increases during generative reaction and decreases in destructive reaction. The concentration increases with decreasing Schmidt number. The effect of increasing values of K leads to a fall in velocity profiles. The velocity decreases with increasing values of the Schmidt number. An increase in modified Grashof number leads to an increase in velocity profiles. The skin friction increases with decreasing Schmidt number. In generative reaction the skin friction decreases and in destructive reaction the skin friction increases.

Appreciation is extended to the referees for their constructive and helpful comments. These led to improvements in the revised paper.