Some Exacts Solutions of Two Dimensional Unsteady Incompressible Magnetohydrodynamic Flow of Viscous Fluid by Ansatz Method

Development of geometrical and topological methods in fluid flows, will give more insight into wide range of problems. Our purpose in this paper is to study and obtain certain invariants related to vorticity field and magnetic field in the case of fluid and magneto hydrodynamic flows. This paper aims at presenting some exact solutions for two dimensional incompressible hydromagnetic flow of viscous fluid. The governing nonlinear partial differential equations for an incompressible MHD flow of viscous fluid and topology conservation equation are obtained. The equations are solved by assuming appropriate vector potential. Velocity and vorticity vector are obtained for particular cases and verified the topology conservation equation. The topology of vorticity vector field is conserved even when Lorentz force term exist in magnetohydrodynamic flow under the condition that the magnetic field is the extremal field for the magnetic energy functional throughout the flow. Velocity vectors are demonstrated graphically for each case and graphs are given for various low and high viscous fluids.