The character of the equilibrium of an incompressible fluid sphere of variable density and viscosity subject to radial acceleration

Abstract

This paper is devoted to a consideration of the following problem: An incompressible fluid sphere, in which the density and the viscosity are functions of the distance r from the centre only, is subject to a radial acceleration -γr, where γ is a function of r: to determine the manner of initial development of an infinitesimal disturbance. By analysing the disturbance in spherical harmonics, the mathematical problem is reduced to one in characteristic values in a fourth-order differential equation and a variational principle characterizing the solution is enunciated. The particular case of a sphere of radius R and density p₁ embedded in a medium of a different density p₂ (but of the same kinematic viscosity v) is considered in some detail; and it is shown that the character of the equilibrium depends on the sign of γR(p₂-p₁) and the magnitude of = γ_RR⁴/v². If γ^R(p₂-p₁) > 0, the situation is unstable and the mode of maximum instability is l = 1 for all < 230; for larger values of it shifts progressively to higher harmonics. In the case γ_R(p₂-p₁) > 0 the results of both an exact calculation and an approximate calculation (based on the variational principle) are given and contrasted. In the case γ_R(p₂-p₁) < 0 when the situation is stable, the manner of decay of the disturbance is briefly discussed in terms of an approximate theory only.