NONLINEAR THERMAL RADIATION AND TEMPERATURE DEPENDENT VISCOSITY EFFECTS ON MHD HEAT AND MASS TRANSFER IN A THIN LIQUID FILM OVER A STRETCHING SURFACE

This paper describes nonlinear thermal radiation effects on MHD heat and mass transfer in a thin liquid film over a permeable unsteady stretching surface taking temperature-dependent fluid viscosity with convective boundary condition. For the non-linearity of the momentum, energy and mass diffusion equations, the problem is solved numerically. At first, Similarity transformations is used to the governing equations to reduce the equations into a set of ordinary differential equations. Then the resulting nonlinear ordinary differential equations are solved using Runge-Kutta-Felberg method with shooting technique. Different physical parameters effects on heat and mass transfer in a thin liquid film are presented graphically. It is found that increase in the unsteadiness parameter leads to increase in the velocity distribution, temperature and concentration gradient. Further, increase in the value of magnetic parameter results in a decrease in the velocity profile and increase in the temperature and concentration gradient. For enhancement of thermal radiation decreases the temperature gradient of the thin film flow. Also, for increase in viscosity variation parameter is to decrease velocity distribution but reverse effects shown in case of temperature and concentration gradient.


INTRODUCTION
In recent years magnetohydrodynamic heat and mass transfer on a thin liquid film over stretching surface have become more important in number of engineering application, science and technology such as wire and fiber coating, metal and polymer extrusion, cooling of metallic plates, drawing of polymer sheets and thining of copper wires, aerodynamic extrusion of plastic sheet, artificial fibers, glass fiber, continuous stretching of plastic films.
First, Crane [1] gives an exact solution for the problem of steady two-dimensional boundary layer flow caused by the stretching of a sheet. Wang [2] studied the flow within a thin liquid film over an unsteady stretching surface. Later, Andersson et al. [3] extended Wangs problem to study heat transfer. The effect of variable thermal properties on flow and heat transfer in a liquid film for viscous Newtonian fluid over a unsteady stretching sheet was studied by Dandapat et. al. [4]. Lai and Kulacki [5] analyzed the effects of variable viscosity on mixed convection heat transfer along a vertical surface in a saturated porous medium considering Newtonian fluid. The heat transfer in a liquid film on an unsteady stretching surface with viscous dissipation in the presence of external magnetic field was investigated by Abel et al. [6]. Siti et al. [7] investigated hydromagnetic boundary layer flow over stretching surface with thermal radiation. Hazarika et al. [8] studied the effect of variable viscosity and thermal conductivity on MHD flow past a vertical plate. Mohebujjaman et al. [9] considered MHD heat transfer mixed convection flow along a vertical stretching sheet in the presence of magnetic field with heat generation. Agrawal et. al. [10] studied MHD flow past a stretching surface embedded in porous medium using lie similarity analysis along with variable viscosity. Ali [11] observed the effect of variable viscosity on mixed convection heat transfer along a moving surface. Pantokratoras [12] made a theoretical study to investigate the effect of variable viscosity on flow and heat transfer on a continuous moving plate. Mukhopadhaya et al. [13] studied the effect of variable viscosity on the boundary layer flow through a porous medium towards a stretching sheet in the presence of heat generation or absorption. Heat transfer in a thin liquid film over a unsteady stretching sheet in the presence of thermal radiation subject to variable surface heat flux conditions was studied by Liu and Megahed [14]. Cortell [15] analyzed heat transfer and viscoelastic fluid flow over a stretching sheet under the effect of a non uniform heat source, viscous dissipation and thermal radiation. Pantokratorus and Fung [16] used the Rosseland diffusion approximation to study radiative non-linear heat transfer in different geometries. Aziz et. al. [17] studied heat transfer in a liquid film over a permeable stretching sheet. The variable viscosity with magnetic field on flow and heat transfer to a continuous moving flat plate were reported in Seddek and Salem [18]. Nadeem and Akbar [19] observed the effects of heat transfer on MHD Newtonian fluid with variable viscosity. Benazir and Sivaraj [20] observed the effect of unsteady MHD casson fluid over a vertical cone and flat plate saturated with porous medium and non-uniform heat source/sink. Kumar and Sivaraj [21] investigated heat and mass transfer in MHD viscoelastic fluid flow over a vertical cone and flat plate with variable viscosity.
The motivation of present study is to investigate the influence of non-linear thermal radiation on MHD heat and mass transfer with temperature dependent variable viscosity in a thin liquid film on a permeable unsteady stretching sheet with convective boundary condition. This problem may have useful applications such as wire coating and food processing.

FORMULATION OF THE PROBLEM
Consider two dimensional unsteady fluid flow of a Newtonian fluid in a thin liquid film over a permeable stretching surface with variable viscosity and magnetic parameter. It is assumed that the elastic sheet emerges from a narrow slit at the origin of a Cartesian co-ordinate system.
The continuous surface aligned with the x-axis at y=0 moves in its own plane with a velocity U(x,t) (see Fig. 1). A thin liquid film of uniform thickness h(t) lies on the horizontal surface.
The surface heat flux q t (x,t) at the stretching sheet varies with the power of distance x from the slit and with the inverse power of time factor t as [22] (1) The surface mass flux q m (x,t) at the stretching sheet varies with the power of distance x from the slit and with the inverse power of time factor t as [23] (2) where k is the thermal conductivity, T re f is reference temperature, C re f is reference concentration, d is a constant. The applied transverse magnetic field B 1 (t) is defined by [24] where B 0 is uniform magnetic field. The boundary layer equations mass, momentum and for energy conservation are given by, where u and v are components of velocity along the direction of x and y respectively. ρ is the fluid density, t is time, µ is the variable viscosity of the fluid, q r called the radiative heat flux and c p is the specific heat at constant pressure. The term Q is the heat generated(> 0) per unit volume and absorbed (< 0) per unit volume is defined as (Liu and Megahed [22]): where µ h is constant viscosity, B * denotes the temperature dependent heat generation or absorption. That is for the generation of heat B * is positive and for the absorption of heat B * is negative within the fluid system. Thus for the present problem the corresponding boundary conditions are: where U(x,t) is the surface velocity of the stretching sheet, h be the thickness of the liquid film.
The stretching elastic surface at y = 0 moves continuously in x−direction with the velocity: where b and a are both positive constant with dimension per time. The elastic sheet's temperature is assumed to vary both along the sheet and with time accordance with where T re f is the constant reference temperature.
The radiative heat flux q r is taken according to Rosseland approximation as where σ * is the Stefan-Boltzman constant, k * be the mean absorption coefficient.
Now the system of partial differential equations transformed into a system of nonlinear ordinary differential equation by using the similarity transformations which are given as follows The dimensionless thin film thickness β is defined by The temperature dependent fluid viscosity is given by (Batchelor [25]), where µ h is the constant value of the coefficient of viscosity far away from sheet and m, n are constants and n(> 0).
This relation can be written in expanded form as, where A = n(T s − T 0 ), being viscosity variation parameter.
The transformed set of ordinary differential equations are: f subject to the boundary conditions: where prime represent differentiation with respect to η, S = a b be the unsteadiness parameter, Pr = µ h c p k be the Prandtl number, β be the dimensionless thin film thickness, Nr = 16σ * T 3 0 3k * k be the radiation parameter, θ w = T s T 0 be the temperature ratio parameter, ρb be the magnetic parameter, f w being the permeability parameter.

NUMERICAL METHOD
The Runge-Kutta-Fehlberg method (RKF45) use to solve initial value problem (19) dy It has a procedure to determine if the proper step size is being used. At each step, two different approximations are made and compared. If the two results are in close agreement, the approximation is accepted. If the two result do not give the specified accuracy, the step size is reduced.
If the result agree to more significant digits than required, the step size is increased. Each step requires the use of following six values since this is fifth order method with six stages that uses all the points of the first one. Then an approximation to the solution of the initial value problem (IVP) is made using a Runge-Kutta method of order 4; where the four functional values K 1 , K 3 , K 4 and K 5 are used. A better value for the solution is determine using Runge-Kutta method of order 5; where, The optimal step size sh is determined by multiplying the scaler s times the current step size h, where the scaler s can be determined from; where Tol is the specified error control tolerance.
The non-linear differential Eqs. (15), (16) and (17) with appropriate boundary conditions (18) are solved numerically by using Runge-Kutta-Fehlberg (RKF) fifth order technique along with shooting method. At a very first step, the higher order non-linear differential equations (15), (16) and (17) are converted into simultaneous differential equation of first order and further they are transformed into initial value problem by applying the shooting technique.
Then initial value problem is solved by Runge-Kutta-Fehlberg fifth order method. The ordinary differential equations (15) to (17)  the desired accuracy then the iteration process is stopped. The governing non-linear ordinary differential equations are reduced to a set of simultaneous first order differential equation as follows, The boundary condition becomes

RESULTS AND DISCUSSION
The system of highly non-linear differential equations (15)   the temperature gradient of the flow also increases, due to reduction in the dimensionless thin film thickness. So, the heat transfer rate increases within the thin liquid film. Fig. 4 shows the effect of unsteadiness parameter S on the concentration gradient of the thin film. It shows that increasing value of S, the concentration gradient of the flow also increase, due to the reduction in dimensionless thin film thickness. So mass transfer rate increases within the thin film liquid. proportional to T 0 , so when θ w increase then the value of T 0 is decrease i.e. fluid flow system remain cool, due to the decrease of heat transfer rate. Fig. 6 displays the temperature gradient profile with η for different values of thermal radiation parameter Nr in the presence magnetic field. It is found that the effect of thermal radiation is to decrease the the temperature gradient in the thermal boundary layer. This is due to the fact that when radiation parameter Nr increases then from the expression for Nr = 16σ * T 3 0 3k * k , the Rosseland radiative absorption coefficient k * decreases and therefore the heat flux q r (= − 16σ * 3k * T 3 ∂ T ∂ y ) decreases.   A. Fig. 13 shows the effect of the viscosity variation parameter A on the temperature gradient.
By increasing A temperature gradient slowly increase. Also, the thin film thickness decreases.

CONCLUSION
In this paper the effects of temperature dependent viscosity and external magnetic field on thin liquid film flow of Newtonian fluid over a permeable unsteady stretching sheet in the presence of variable heat flux, non-linear thermal radiation are investigated and following conclusions are drawn.
(i) The effect of the thermal radiation is to decrease the cooling rate of the thin liquid film, but reverse effect is true with the Prandtl number.
(ii) Increasing in the magnetic parameter results in decrease in the velocity distribution and increase in the temperature gradients distribution and concentration gradients.
(iii) The effect of viscosity variation parameter is to decrease velocity distribution in the momentum boundary layer. Also, increase in viscosity variation parameter results in increase in the temperature gradient and concentration gradient.
(iv) The mass transfer rate increase with increase in the value of the Schmidt number.
(v) Increase in the unsteadiness parameter results in increase in the velocity distribution due to decrease in the thin film thickness.