Boundary element solution of unsteady magnetohydrodynamic duct flow with differential quadrature time integration scheme

Abstract

SUMMARYA numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the di erential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) ow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic ÿeld are transformed ÿrst into the decoupled time-dependent convection-di usion-type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step-by-step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M 6 50) at transient and the steady-state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic ÿeld and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady-state solution.

Solution of magnetohydrodynamic flow in a rectangular duct by Chebyshev collocation method

“…Solution of the other equation was obtained easily by changing the sign in front of Hartmann number in the solution of the first equation. All the methods used in [10,[16][17][18][19] Hartmann numbers have taken H 300. Hence, the collocation with the Chebyshev polynomials has high potential as an alternative to other solution techniques mentioned above with Hartmann number H 1000.…”

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

“…In this study DQM is employed to discretize the time derivative of w in Equation (25) [26,27]. The DQM analogue of the first-order derivative of a function f (t) at a grid point t i can be expressed as…”

“…These time blocks are discretized with very small number of Gauss-Chebyshev-Lobatto (GCL) points since it is known that it gives better accuracy than the use of equally spaced points [25]. The use of the DQM for the time derivative has been used with success in solving transient convection-diffusion and unsteady MHD equations [26,27]. We extend this idea now for the solution to unsteady Navier-Stokes and energy equations.…”

Meshless Local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity

“…The stabilized FEM for solution of the 3-dimensional time-dependent MHD equations was given by Salah et al [26]. Bozkaya and Sezgin employed the dual reciprocity boundary element method (DRBEM) [11,12] for non-conducting walls and also time-domain BEM [13] for arbitrary wall conductivity unsteady MHD flow.…”