Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection-diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well-known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall

Magnetohydrodynamic Flow in a Rectangular Duct

The magnetohydrodynamic (MHD) flow of an incompressible, viscous and electrically conducting fluid in a rectangular duct with insulated and perfectly conducting walls is investigated numerically in the presence of hydrodynamic slip. The flow is fully developed and driven by a constant pressure gradient in the axial direction under the effect of an externally applied uniform and inclined magnetic field. A direct boundary element method (BEM) using a fundamental solution which enables to treat the governing MHD flow equations in their original coupled form is employed and the validity of the code is also ascertained. The numerical simulations are carried out for several values of slip length, Hartmann number and the inclination angle of the external magnetic field. It is well-observed from the equivelocity and induced current lines that the velocity increases through the duct and the Hartmann layers weaken while the side layers become thicker with an increase in slip length especially at low values of Hartmann number irrespective of the conductivity of the walls