A global particular solution meshless approach for the four-sided lid-driven cavity flow problem in the presence of magnetic fields

The global meshless method of approximate Stokes particular solutions (MASPS) is used to solve a two-dimensional incompressible fluid flow in the presence of a uniform magnetic field, i.e. the Navier–Stokes equations with the Lorentz force as a source term in the momentum equations. Magnetohydrodynamic (MHD) problems at low magnetic Reynolds number (Re_m) but finite flow Reynolds number (Re) are considered, so the fluid flow is affected by the magnetic field which remains unaltered by the fluid flow. The base functions to approximate the variables of the problem are the particular solutions of an auxiliary Stokes flow field in which a multiquadric (MQ) radial basis function (RBF) is applied as source term. The nonlinear equations resulting from the discretization in a fully implicit finite-difference form are solved by a variable step Newton–Raphson method. The capability of the numerical scheme to simulate MHD problems for different geometries is shown by solving the one-sided lid-driven cavity and the backward facing step flows in the presence of horizontal and vertical magnetic fields, respectively. The existence of simultaneous steady-state solutions in the four-sided lid-driven cavity (4S-LDC) problem is studied with the MASPS for Re between 0 and 1000 and Hartmann numbers (Ha) up to 10. Critical Reynolds numbers (Re_c), corresponding to stationary and Hopf bifurcations, are evidenced. Bifurcation diagrams are constructed based on simultaneous solutions and their stability analyses. The increase of Ha modifies the bifurcation diagram and causes the displacement of bifurcation points towards higher Re. Three types of bifurcation, detected by the MASPS in the 4S-LDC flow, are classified based on the stability state analyses. Vertical and oblique magnetic fields are imposed on the flow to study their influence on the bifurcation maps. The effects of the vertical magnetic field on the map are stronger than those of the oblique field.