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Magnetohydrodynamic (MHD) flow of liquid metals through conduits play animportant role in the proposed systems for harnessing fusion energy, andvarious other engineering and scientific problems. The interplay between themagnetic fields and the fluid motion gives rise to complex flow physics, whichdepend on the electrical conductivity of the bounding walls. An effectiveapproach to represent the latter is via the Shercliff boundary condition forthin conducting walls relating the induced magnetic field and its wall normalgradient at the boundary via a parameter referred to as the wall conductanceratio (Shercliff, JA, J. Fluid Mech. 1, 644 (1956)). Within the framework ofthe highly parallelizable lattice Boltzmann (LB) method, a lattice kineticscheme for MHD involving a vector distribution function for the magnetic fieldswas proposed by Dellar (Dellar, PJ, J. Comp. Phys. 179, 95 (2002)). However,the prior LB algorithms only accounted for limiting special cases involvingperfectly insulated boundaries and did not consider the finite conductivityeffects. In this paper, we present two new boundary schemes that enforce theShercliff boundary condition in the LB schemes for MHD. It allows for thespecification of the wall conductance ratio to any desired value based on theactual conductivities and length scales of the container and the wallthicknesses. One approach is constructed using a link-based formulationinvolving a weighted combination of the bounce back and anti-bounce back of thedistribution function for the magnetic field and the other approach involves anon-node moment-based implementation. Moreover, their extensions to representingmoving walls are also presented. Numerical validations of the boundary schemesfor body force or shear driven MHD flows for a wide range of the values of thewall conductance ratio and their second order grid convergence aredemonstrated.

On Boundary Conditions for Lattice Kinetic Schemes for Magnetohydrodynamics Bounded by Walls with Finite Electrical Conductivities