Stability of couette flow between free cylinders with allowance for self-gravitation

The free two-dimensional motion of a gravitating system consisting of two unfixed rotating cylinders and the incompressible fluid between them is considered. The undisturbed state of the system is determined by the viscous Couette flow between the coaxial cylinders. The stability of the central position of the cylinders is investigated within the framework of the equations of an inviscid fluid in a rotating coordinate system in which the flow is assumed to be potential. The critical values of the parameters and the spectral characteristics of the system are found. The qualitative form of the neutral curves obtained is determined by the value of the unique parameter of the problem ρ^r, which is equal to the ratio of the density of the inner cylinder to the density of the fluid. It is noted that if ρ^r→ 1, then however small the values of the angular velocities of the cylinders, generally speaking, they cannot be neglected in calculating the natural vibrations of the system. It is shown that if ρ^r > 1, then in the absence of gravitation the radial play between the cylinder axes leads to the growth of disturbances in the region stable with respect to the Rayleigh criterion.