Boundary conditions for a rapidly rotating hydromagnetic system in a cylindrical container

Abstract

A conducting fluid is contained in a cylindrical annulus with rigid, perfectly conducting boundaries and is permeated by a magnetic field B₀ = B₀ (s)ϕ (where (s,φ,z) are cylindrical polar coordinates). The equations governing the linear stability of this system are separable in φz, and time t, and form a tenth-order boundary-value problem in s. The five conditions which must be satisfied at each boundary are u · ŝ = 0, u × ŝ=0 and e× ŝ=0 where u is the fluid velocity and e the electric field. In the limit where the cylinder is rapidly rotating about its axis, viscous effects can be neglected in the body of the fluid. The system of equations reduces to sixth order and the no-slip conditions can no longer be applied. A complication arises when waves on a timescale long compared with the rotation period are of interest, because then the equations reduce to fourth order and it is unclear what are the correct boundary conditions to be applied. A boundary-layer analysis shows that the normal magnetic field and a linear combination of the normal velocity and tangential current must vanish at the edge of the mainstream. This was checked by a numerical solution of the full, tenth-order system. The boundary condition derived is applicable when the Lorentz force is a leading order effect in the momentum equation.