Application of the viscosity-expansion method to a rotating thin fluid disk bound by central gravity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 2D rotation of a thin fluid disk with a porous sink around the center is studied within the Navier–Stokes and Euler equations. The viscosity (ν)-expansion method is applied to the viscous fluid bound to the central mass via gravity. The Navier–Stokes equations yield various types of rotation curve, including a flat one, depending on the choice of the pressure function that is not determined within the fluid dynamics. Stationary flow is achieved through the balance of the pressure gradient, gravity, and the centrifugal force. These features of the stationary flow survive in the inviscid limit. The stability of the inviscid flow is examined by the Euler equations for the perturbations. At large distances, the real part of eigenfrequencies (EFs) are dominantly positive and decreasing with distance for flat and rising rotation curves, meaning that the spiral pattern of the perturbations is trailing. One real increasing EF exists for the decaying rotation curve, for which the spiral pattern is leading. Complex frequencies always emerge when the disk has m-fold rotational symmetry with m ≥ 2. The shape of the perturbed rotation curve has azimuthal as well as temporal dependences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .