On the instability of inviscid, rigidly rotating immiscible fluids in zero gravity
The paper deals with the linear instability of the rigid rotation of a two-fluid column in zero gravity against three-dimensional normal-mode perturbations. Both fluids are inviscid and incompressible, and the rotation is uniform. The surface tension coefficient at the immiscible interface is constant. The inner core fluid is surrounded by another fluid bounded by an outer impermeable right cylinder, and the radially distant fluid is of higher density than the core fluid. The nondimensionalized problem consists of the Euler and continuity equations, kinematic and dynamic conditions on the interface, the slip condition on the outer wall, a condition for the pressure on the cylinder axis, and a pressure jump relation on the interface. The basic state is the zero velocity field in each fluid and parabolic pressure distributions depending on the radial coordinate only. The perturbation problem is an eigenvalue three-point problem for a system of eight ordinary differential equations with variable coefficients defined on two adjacent segments of the real axis and containing seven real parameters. The authors derive the eigenvalue relation in terms of the modified Bessel functions of the first and second kind. Their investigation is performed by the package Mathematica and mainly deals with the maximum growth rates and with the corresponding preferred instability wavelengths. Finally, the authors consider the limiting cases of no outer fluid (or the hollow-core vortex), and of the stationary two-fluid system.
three-dimensional normal-mode perturbations
system of eight ordinary differential equations
scientific article; zbMATH DE number 1097771