  Magnetohydrodynamic (MHD) flow of liquid metals through conduits play an
important role in the proposed systems for harnessing fusion energy, and
various other engineering and scientific problems. The interplay between the
magnetic fields and the fluid motion gives rise to complex flow physics, which
depend on the electrical conductivity of the bounding walls. An effective
approach to represent the latter is via the Shercliff boundary condition for
thin conducting walls relating the induced magnetic field and its wall normal
gradient at the boundary via a parameter referred to as the wall conductance
ratio (Shercliff, JA, J. Fluid Mech. 1, 644 (1956)). Within the framework of
the highly parallelizable lattice Boltzmann (LB) method, a lattice kinetic
scheme for MHD involving a vector distribution function for the magnetic fields
was proposed by Dellar (Dellar, PJ, J. Comp. Phys. 179, 95 (2002)). However,
the prior LB algorithms only accounted for limiting special cases involving
perfectly insulated boundaries and did not consider the finite conductivity
effects. In this paper, we present two new boundary schemes that enforce the
Shercliff boundary condition in the LB schemes for MHD. It allows for the
specification of the wall conductance ratio to any desired value based on the
actual conductivities and length scales of the container and the wall
thicknesses. One approach is constructed using a link-based formulation
involving a weighted combination of the bounce back and anti-bounce back of the
distribution function for the magnetic field and the other approach involves an
on-node moment-based implementation. Moreover, their extensions to representing
moving walls are also presented. Numerical validations of the boundary schemes
for body force or shear driven MHD flows for a wide range of the values of the
wall conductance ratio and their second order grid convergence are
demonstrated.
