XLVI. On the inhibition of convection by a magnetic field

Abstract

In this paper the theory of the stability of a layer of fluid heated below which has been developed by Rayleigh, Jeffreys, Pellew and Southwell is extended to the case when the fluid considered is an electrical conductor and an external magnetic field is impressed on the fluid. The problem is first investigated on the assumption that the principle of the exchange of stabilities (i.e. the equations governing marginal stability are those which are obtained from the general time dependent equations by setting ∂/∂t=0) is valid. A differential equation of order six for the normal component of the velocity is derived. Suitable boundary conditions are formulated; they depend on whether the bounding surfaces are free or rigid. The case when the magnetic field and gravity act in the same direction is investigated in some detail. A variational procedure for solving the relevant equations satisfying the necessary boundary conditions and determining the critical Rayleigh numbers for the onset of convection are described. Tables of the critical Rayleigh numbers are provided for the three cases (i) both bounding surfaces free, (ii) both bounding surfaces rigid and (iii) one bounding surface free and the other rigid. Finally, the principle of the exchange of stabilities is examined and it is shown that it is valid if 4 πμσκH ²/4πρ₀)^1·2).