Effects of wall perturbations on the stabilized FEM solution of steady MHD flow in a duct

In this study, the effects of curved boundary perturbations on the solution of steady magnetohydrodynamic (MHD) duct flow are investigated. Hartmann (upper and bottom) walls are perturbly curved and perfectly conducting while the side walls are insulated and plane. The velocity of the flow and induced magnetic field are obtained numerically by solving the steady MHD flow coupled equations using the finite element method (FEM with Streamline Upwind Petrov Galerkin (SUPG)) stabilization to inhibit instabilities in the flow. The results are obtained for Hartmann number ($Ha$) values up to $500$, for several definitions of the curved upper and bottom walls, and for several values of perturbation parameters of the curved walls ($0 \le \epsilon_u , \epsilon_b \le 0.3$). The velocity and the induced magnetic field sensitivity to the curved wall shapes are visualized in terms of equivelocity and current lines. It is found that the flow and the induced magnetic field are affected by the curved boundary shapes especially near those boundaries and also, to some extent, in the whole duct. It is also observed that increasing the Hartman number pushes the flow near the upper boundary even if both upper and bottom walls are perturbed since the external magnetic field applies vertically. The increase in the perturbation parameter of the curved upper boundary forces the flow to move through this wall and the induced magnetic field reaches its maximum value near the maximum points of the perturbed curve. Further, an increase in ${\rm Ha}$ delays the effect of the curved boundaries and gives rise to flattened flow with side layers and stagnant fluid at the central part of the duct overwhelming the effects of curved boundaries.