Dynamics of steady, unsteady flows and heat transfer in Casson fluid over a free stretching surface: stability analysis

Abstract

The dynamics of the steady flow, heat transfer, and MHD in Casson fluid (CF) were currently studied in the literature. But studies on unsteady flow are rare. In those studies, the boundary conditions at the boundary layer, for the directed and perpendicular velocities, and the temperature field, were only taken constants. Or they were taken to depend linearly on the directed distance. The study of CF over a free-stretching surface is too complicated and was not considered in the literature. Our objective, here, is to study the dynamics of CF with boundary conditions at the free surface, where similar solutions of the the steady flow, unsteady flow and temperature are found. This is completely a novel problem. The exact solutions for the steady flow, unsteady flow and temperature field dynamic equations are found via the extended unified method. The only existing tool, to deal with this problem, is the use of Lie symmetry which requires a hierarchy of long steps. The solutions obtained are evaluated numerically and represented in graphs. It is found that the steady-state flow exhibits two, upper and lower, layers with variant behavior which agree with what is found in the literature. It is, also, observed that in the unsteady state case that the directed and perpendicular velocities tend to zero asymptotically when the lateral space variable and the time tend to infinity. While it is observed that the temperature attains a constant value. An approach is presented to study the stability of the steady-state flow and temperature. It is found that the steady-state temperature is unstable for varying the magnetic field strength, viscosity and mass density. While, when varying the Casson parameter Λ, it is stable when and it is unstable for . This is in contrast to the case of steady-state flow which is shown to be stable.