# Similarity solutions of the steady-state MHD flow with in-homogeneous viscosity over stretching surfaces with free boundary. stability analysis

### Abstract

The similarity solutions of the steady-state (SS) magneto-hydrodynamics (MHD) flow over a semi infinite surface were currently studied in the literature. It is worthy to mention that the works done in the literature were concerned with studying the cases when the boundary conditions of the directed and perpendicular velocities at the surface are taken constants or linear in the space variable.. Here, the objective of present work is to investigate the behavior of similarity solutions of the SS-MHD equation for in-homogeneous kinematic viscosity and arbitrary boundary conditions at the stretching surface.The exact solutions are obtained by implementing the extended and generalized unified methods. It is found that, multiple SS-MHD flow structures, over an infinite and semi-infinite surfaces, occur. This arises from the presence of arbitrary functions which are embedded in the solutions. In the case of semi-infinite surface the boundary condition is used. Highly oscillatory flow over an infinite surface and zig-zag flow over semi infinite surface are observed. The effects of varying the induced magnetic field and the viscosity of the stretching and perpendicular velocities are investigated. It is found that, over semi-infinite surface, the magnitude of the stretching velocity raises, while the perpendicular velocity decreases, significantly with the strength of the field. The inverse statement hods when varying the magnitude of the viscosity. A novel approach for analyzing the stability of

the steady flow solution is proposed and used here.