HODOGRAPH TRANSFORM INSIGHTS INTO MAGNETOHYDRODYNAMIC FLOW WITH INCLINATION VARIATIONS IN POROUS MEDIA

A magnetohydrodynamic (MHD) fluid flow with variable inclination is one in which the velocity and magnetic fields are coplanar, and the angle between the vector fields may be altered by all the flow parameters. This work extends the use of hodograph transformation to address variably inclined flows that occur via porous formation that are incompressible, stable, and viscous without being aligned with the goal of getting an exact solution. Deriving exact solutions is difficult because the flow’s governing equations are essentially nonlinear partial differential equations. As a result, many transformation methods are thus used to put these equations in a form that can be solved. A specific method is the hodograph transformation, in which the governing equations are linearized by changing the roles of dependent and independent variables, thereby allowing analytic solutions. The present study uses the hodograph-Legendre transformation method, which combines the linearizing features of the hodograph approach with the functional properties of Legendre transformations. Using this hybrid framework, exact solutions for MHD planar flows through a porous structure under varying angles are obtained. Specifically, two forms of the Legendre function representing radial flow and hyperbolic flow are considered as applications with corresponding exact solutions for velocity, magnetic field, vorticity, stream function, and pressure for variably inclined MHD flows in porous media. The results function as benchmark solutions, establishing an objective point of reference for validating the accuracy of approximation models and mathematical simulations of MHD porous flows. The streamlines and magnetic field lines for each case are graphically depicted. Additionally, the variations in pressure concerning the parameter for porous formation with a fixed density and concerning density at a constant porous media parameter are graphically analyzed for both forms of the Legendre function, leading to suitable conclusions.