European Numerical Mathematics and Advanced Applications Conference(ENUMATH), Eindhoven, Netherlands, 30 September - 04 October 2019
In this study, the MHD flow of an electrically conducting fluid is considered in a long channel (pipe) of rectangular cross-section along with the z-axis. The fluid is driven by a pressure gradient along the z-axis. The flow is steady, laminar, fully-developed and is influenced by an external magnetic field applied perpendicular to the channel axis. So, the velocity field V=(0,0,V) and the magnetic field B=(0, B_{0}, B) have only channel-axis components V and B depending only on the plane coordinates x and y on the cross-section of the channel which is a rectangular duct. The finite difference method (FDM) is used to solve the governing equations with several type of boundary conditions such as slip or no-slip velocity V(x,y) and conducting, insulated or partly conducting/partly insulated side walls for B(x,y). The numerical solutions for each case of boundary conditions are simulated in terms of equivelocity contours and current lines. The effects of the slip and the wall conductivities on the behavior of the velocity V and the induced magnetic field B are investigated to see their physical effects on the solution mostly. Also, the numerical results obtained from the FDM discretized equations and the exact solution values are shown on the same figure for no-slip and insulated walls to see the coincidence with the exact results and we obtain the accuracy at least 10^(-2). Thus, the FDM which is simple to implement, enables one to depict the physical effects of the slip and wall conductivities on the behavior of both the velocity and the induced magnetic field at a small expense.