On the instability of inviscid, rigidly rotating immiscible fluids in zero gravity

Abstract.

The linear stability of an axisymmetric rotating two-fluid column in zero gravity is investigated. The inner core fluid of radius a and density $\rho_1$ is surrounded by fluid of density $\rho_2$ itself bounded by an outer impermeable cylinder of radius b. The surface tension coefficient at the immiscible interface is $\gamma$. The parameters governing instability are the density ratio $\lambda = \rho _1/ \rho _2$, the radius ratio $\kappa = b/a$ and the generalized Hocking parameter $L_i = \gamma / \rho _i\Omega^2 a^3$. While the necessary and sufficient condition for stability $L_2 \le (1 - \lambda )$ is known, the preferred modes and wavenumbers at onset of instability have not been determined over the space of dimensionless parameters. Analytically and numerically determined maximum growth rates and corresponding preferred instability wavelengths for a rotating liquid column show that axisymmetric disturbances are most unstable for $L_1 > 0.1053$ while planar disturbances determine instability when $L_1 < 0.1053$. The rotating two-fluid system bounded by an outer cylinder with statically stable density ratios $0 \le \lambda \le 1$ is unstable only to axisymmetric disturbances. Maximum growth rates and preferred wavelengths are computed as a function of $\lambda$ and $L_2$ at radius ratios $\kappa = 1.2$ and $\kappa = 5.0$, respectively corresponding to relatively large and relatively small fluid cores. Complete results for the hollow-core vortex $(\lambda = 0)$ and a stationary two-fluid system $(L_2 \to \infty)$ are also presented.