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Exact and FDM solutions of 1D MHD flow between parallel electrically conducting and slipping plates

In this study, the steady, laminar, and fully developed magnetohydrodynamic (MHD) flow is considered in a long channel along with the z-axis under an external magnetic field which is perpendicular to the channel axis. The fluid velocity u and the induced magnetic field b depend on the plane coordinates x and y on the cross-section of the channel. When the lateral channel walls are extended to infinity, the problem turns out to be MHD flow between two parallel plates (Hartmann flow). Now, the variations of u and b are only with respect to y-coordinate. The finite difference method (FDM) is used to solve the governing MHD equations with the wall conditions which include both the slip and the conductivity of the plates. The numerical results obtained from FDM discretized equations are compared with the exact solution derived here for the 1D MHD flow with Robin's type boundary conditions. The fluid velocity and the induced magnetic field are simulated for each special case of boundary conditions on the plates including no-slip to highly slipping and insulated to perfectly conducting plates. The well-known characteristics of the MHD flow are observed. It is found that the increase in the slip length weakens the formation of boundary layers. Thus, the FDM which is simple to implement, enables one to depict the effects of Hartmann number, conductivity parameter, and the slip length on the behavior of both the velocity and the induced magnetic field at a small expense.