SUMMARYThis paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier-Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function-vorticity-magnetic induction-current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson-type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time-dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward-facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number.

SUMMARYThe two-dimensional convection-diffusion-type equations are solved by using the boundary element method (BEM) based on the time-dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady-state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time-domain BEM solution procedure is tested on some convection-diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time-dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter.