Unstable perturbations in a swirling flow of an ideal incompressible fluid in the regions with open boundaries

This paper examines the linearized Euler equations determining the axisymmetric swirling flow of an ideal incompressible fluid. The main flow is characterized by a single nontrivial (azimuthal) velocity component that depends arbitrarily on the radial cylindrical coordinate. Based on a perturbed steady-state solution, explicit expressions for vortex helices and spiral-shaped stream surfaces are obtained. Exact solutions are constructed that incorporate arbitrary functions in their structure and describe perturbations of velocity and/or pressure growing linearly in time. A hydrodynamic interpretation of these solutions is given in terms of the fluid flow through a given domain. An analytical description is provided for the unstable fluid states in the regions with open boundary segments. Three geometric configurations of such domains are analyzed. The first is the flow outside a vortex cylinder attached to an impermeable surface with a non-monotonic profile in the radial cylindrical coordinate. The second is the flow inside an impermeable cylinder with movable permeable spherical segments at its ends; the boundary conditions on these segments represent a coupling between velocity and pressure perturbations and the velocity of the segment. The third configuration is the flow in a gap between coaxial cylinders with movable permeable segments located at their ends. The effect of the pressure gradient on the spatial structure of the vortex field is studied. The effect of geometric parameters of the open regions on the flow properties is determined.

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