Stabilized FEM solution of MHD duct flow with conducting cracks in the insulation

Abstract

In this paper, the numerical solution of the fully developed liquid–metal magnetohydrodynamic (MHD) flow is given in a rectangular duct under an external oblique magnetic field with no-slip and insulated walls containing crack regions. The coupled MHD flow equations are transformed first into decoupled convection–diffusion equations in terms of the velocity and induced magnetic field. Thus, we apply the SUPG stabilization in the finite element method (FEM) solution procedure for high values of Hartmann number which determine the convection dominant case. Numerical solutions for high values of Hartmann number and several orientation angles of external magnetic field as well as different crack configurations depict the effects of these parameters on the flow, flowrate and induced current. The SUPG in the FEM solution of MHD equations enables one to display the well-known characteristics of MHD pipe flow for large values of Hartmann number in which wall insulation is faced to several cracks. It is found that, the flowrate drops with an increase in the Hartmann number, and the increase in the number and lengths of the cracks which are located on one Hartmann wall. The flowrate is minimum if the crack is located at the center of the Hartmann wall. If the crack is located on the side layer, it does not significantly affect the flowrate. The case of each Hartmann wall contains one crack, significantly drops the flowrate compared to two cracks on the same Hartmann wall. Also, the breakdowns and peaks are sharpened when the Hartmann wall is a curved boundary. The FEM with SUPG stabilization is capable of capturing flow changes for high values of Hartmann number even in the vicinity of small-sized crack regions.