For MHD flows in a rectangular duct with unsymmetrical walls, two analytical solutions have been obtained by solving the governing equations in the liquid and in the walls coupled with the boundary conditions at fluid-wall interface. One solution of 'Case I' is for MHD flows in a duct with side walls insulated and unsymmetrical Hartmann walls of arbitrary conductivity, and another one of 'Case II' is for the flows with unsymmetrical side walls of arbitrary conductivity and Hartmann walls perfectly conductive.The walls are unsymmetrical with either the conductivity or the thickness different from each other. The solutions, which include three parts, well reveal the wall effects on MHD. The first part represents the contribution from insulated walls, the second part represents the contribution from the conductivity of the walls and the third part represents the contribution from the unsymmetrical walls. The solution is reduced to the Hunt's analytical solutions when the walls are symmetrical and thin enough. With wall thickness runs from 0 to∞, there exist many solutions for a fixed conductance ratio. The unsymmetrical walls have great effects on velocity distribution. Unsymmetrical jets may form with a stronger one near the low conductive wall, which may introduce stronger MHD instability. The pressure gradient distributions as a function of Hartmann number are given, in which the wall effects on the distributions are well illustrated.
A benchmark solution is of great importance in validating algorithms and codes for magnetohydrodynamic(MHD) flows.Hunt and Shercliff's solutions are usually employed as benchmarks for MHD flows in a duct with insulated walls or with thin conductive walls,in which wall effects on MHD are represented by the wall conductance ratio.With wall thickness resolved,it is stressed that the solution of Sloan and Smith's and the solution of Butler's can be used to check the error of the thin wall approximation condition used for Hunt's solutions.It is noted that Tao and Ni's solutions can be used as a benchmark for MHD flows in a duct with wall symmetrical or unsymmetrical,thick or thin.When the walls are symmetrical,Tao and Ni's solutions are reduced to Sloan and Smith's solution and Butler's solution,respectively.
