The character of the equilibrium of an incompressible heavy viscous fluid of variable density: an approximate theory

Abstract

The equilibrium of a layer of a viscous incompressible fluid of density ρ variable in the vertical (z) direction, the fluid extending indefinitely in the horizontal directions and being confined between rigid horizontal boundaries, is examined by the usual method of studying the initial behaviour of a small disturbance. Diffusion effects are ignored. Chandra-sekhar has shown quite generally that a variational procedure for solving the resulting characteristic value problem is admissible, suggesting a method whereby an approximate solution can be obtained analytically. In this paper, by making use of appropriate approximate methods of solution, three types of density configuration, ρ(z), are studied. The first corresponds to two superposed fluids of great depth, the upper and lower fluids haying densities and coefficients of viscosity ρ₂ and μ₂, and ρ₁ and μ₁, respectively. Chandrasekhar has given an exact numerical solution to this problem for the case where μ₂/ρ₂ = μ₁/ρ₁ The results of the approximate theory, by which not all the boundary conditions can be satisfied, and the exact theory are in good agreement. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (ρ₂ > ρ₁) are found. The stable case (ρ₂ > ρ₁) is also discussed in detail; the formulae obtained for the phase and group velocities of gravity waves in deep viscous fluids are especially interesting. There is an upper limit to the natural frequency of oscillation of the system which would be well within the range of experimental investigation.

The second problem considered is that of two superposed fluids of small depth. Formulae for the phase and group velocities of ‘long gravity waves’ (which are not equal when viscous effects are included) are obtained.

Finally, a continuously stratified fluid of finite depth d in which ρ(z) = ρ₀^eβz is investigated. The character of the equilibrium depends on a Grashof number G = (g^βd4)/(π⁴ν²) where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (G > 0) are obtained. In the stable case (G < 0) all modes are aperiodically damped if – G < . Otherwise, the possibility of a disturbance being propagated as a horizontal wave does arise, but only within a limited wave-length range.