The DRBEM solution of Cauchy MHD duct flow with a slipping and variably conducting wall using the well-posed iterations

The present study focuses on the numerical investigation of the Cauchy Magnetohyrodynamic (MHD) duct flow in the presence of an externally applied oblique magnetic field, with a slipping and variably conducting wall portion of the duct walls. The underspecified and overspecified boundary informations for the velocity of the fluid and the induced magnetic field on both slipping and variably conducting duct wall and its opposite part, respectively, constitutes the Cauchy MHD duct flow problem. This study aims to recompute the velocity of the fluid and induced magnetic field with specified slip length and conductivity constant, respectively, on the underspecified wall which is both slipping and variably conducting. The governing coupled convection-diffusion type MHD equations for the direct and inverse formulations are solved in one stroke using the dual reciprocity boundary element method (DRBEM). Both the velocity and induced magnetic field and their normal derivatives to be used as overspecified boundary conditions for the construction of Cauchy problem are obtained through the direct formulation of the problem. The well-posed iterations are used in the regularization of the ill-conditioned systems of linear algebraic equations resulting from the DRBEM discretization of Cauchy problem (inverse problem). Numerical solutions for the slip velocity and induced magnetic field are obtained for Hartmann number values $M$=5, 10, 50. The main advantages of the DRBEM are its boundary only nature and the capability of providing both the unknowns and their normal derivatives on the underspecified walls so that the conductivity constant and the slip length between them can be recovered at a low computational expense.