THE CHARACTER OF THE EQUILIBRIUM OF A HEAVY, VISCOUS, INCOMPRESSIBLE, ROTATING FLUID OF VARIABLE DENSITY

Abstract

In a previous paper, the equilibrium of a plane horizontal layer of a viscous incompressible fluid of variable density po in the vertical (z) direction, which rotates uniformly at Ω rad./sec. about an axis making an angle θ with the vertical, was examined by the usual method of studying the initial behaviour of a small disturbance. The general properties of the ensuing hydrodynamical motion for any ρ0(z) and μ0(z) (μ0 being the coefficient of viscosity) were discussed. It was shown that in some circumstances, including the case θ = 0, the solution is characterized by a variational principle. In the present paper, approximate methods suggested by the variational principle are applied to two special problems, in each of which θ = 0. The first is that of a continuously stratified fluid of finite depth in which ρ0(z) = ρ1 exp(βz), β > 0. The properties of the mode of maximum instability, characterized by its growth rate, nm, and its total wave number, km, depend only on two dimensionless parameters G = (gβδ4/v2) and T = (16Ω2δ4/π4v2) if nm and km are measured in suitable units, where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Values of nm and km are calculated for many assigned values of G and T. These results show that instability is inhibited by the combined influence of viscosity and rotation. For a given rate of rotation, the inhibition is most pronounced when ν has a finite, non-zero value. The second case examined is that of two very deep superposed fluids of density ρ1 and ρ2, the subscripts referring to the lower and upper fluids respectively. Again, ν is assumed to be constant, and only the unstable case (ρ2 > ρ1) is considered. The properties of the mode of maximum instability are influenced by rotation to an extent measured by the dimensionless parameter 4Ω2v(ρ1+ρ2)2/(g2(ρ2-ρ1)2)