Finite difference method solution of magnetohydrodynamic flow in channels with electrically conducting and slipping walls

Arslan, Sinem
In this thesis, the laminar, steady and fully developed magnetohydrodynamic (MHD) flow is considered in a pipe (channel) along with the z-axis under an external magnetic field applied perpendicular to the pipe. The velocity and the induced magnetic field depend only on the plane coordinates x and y on the cross-section of the pipe (duct) when the flow reaches to fully-developed case. This results in two-dimensional MHD duct flow. When the lateral channel walls are extended to infinity the flow is considered between two parallel plates (Hartmann flow). Then, the variations of the velocity and the induced magnetic field are only with respect to the coordinate y between the plates which are perpendicular to the external magnetic field and the problem becomes one-dimensional MHD flow between parallel plates. The finite difference method (FDM) is used to solve the governing equations of 1D and 2D MHD flow problems with the boundary conditions which include both the slip and the varying conductivity of the walls. The numerical results obtained from FDM discretized equations are compared with the exact solution derived for the 1D MHD flow between parallel plates with the most general case of slipping and variably conducting boundary conditions. On the other hand, for the validation of the numerical results obtained from the FDM for the 2D MHD flow in a square duct with the exact solution, the case of no-slip and insulated duct walls is considered and the agreement is obtained. Also, for both of the 1D and the 2D MHD flow problems, the velocity of the fluid and the induced magnetic field are simulated for each special case of boundary conditions including no-slip to highly slipping and insulated to perfectly conducting plates. The well-known characteristics of the MHD flow and the influences of slipping and electrically conducting plates on the flow and the induced magnetic field are observed. Thus, the FDM which is simple to implement, enables one to depict the effects of Hartmann number, conductivity parameter and the slip parameter on the behavior of both the velocity of the fluid and the induced magnetic field at a small expense.