a b s t r a c tIn this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.
a b s t r a c tThis article presents a numerical solution for the steady two-dimensional mixed convection MHD flow of an electrically conducting viscous fluid over a vertical stretching sheet, in its own plane. The stretching velocity and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent fluid properties, namely, the fluid viscosity and the thermal conductivity are assumed to vary, respectively, as an inverse function of the temperature and a linear function of the temperature. A generalized similarity transformation is introduced to study the influence of temperature dependent fluid properties. The transformed boundary layer equations are solved numerically, using a finite difference scheme known as Keller Box method, for several sets of values of the physical parameters, namely, the stretching parameter, the temperature dependent viscosity parameter, the magnetic parameter, the mixed convection parameter, the temperature dependent thermal conductivity parameter and the Prandtl number. The numerical results thus obtained for the flow and heat transfer characteristics reveal many interesting behaviors. These behaviors warrant further study of the effects of the physical parameters on the flow and heat transfer characteristics. Here it may be noted that, in the case of the classical Navier-Stokes fluid flowing past a horizontal stretching sheet, McLeod and Rajagopal (1987)  showed that there exist an unique solution to the problem. This may not be true in the present case. Hence we would like to explore the non-uniqueness of the solution and present the findings in the subsequent paper.
a b s t r a c tThe influence of temperature-dependent fluid properties on the hydro-magnetic flow and heat transfer over a stretching surface is studied. The stretching velocity and the transverse magnetic field are assumed to vary as a power of the distance from the origin. It is assumed that the fluid viscosity and the thermal conductivity vary as an inverse function and linear function of temperature, respectively. Using the similarity transformation, the governing coupled non-linear partial differential equations are transformed into coupled non-linear ordinary differential equations and are solved numerically by the Keller-Box method. The governing equations of the problem show that the flow and heat transfer characteristics depend on five parameters, namely the stretching parameter, viscosity parameter, magnetic parameter, variable thermal conductivity parameter, and the Prandtl number. The numerical values obtained for the velocity, temperature, skin friction, and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of the parameters on the flow and heat transfer characteristics are discussed.
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