Exact solutions of time-dependent oscillations in multipolar spherical vortices

Abstract

Exact solutions of the time-dependent three-dimensional nonlinear vorticity equation for Euler flows with spherical geometry are provided. The velocity solution is the sum of a multipolar oscillatory function and a rigid cylindrical motion with swirl. The multipolar oscillation is a velocity mode whose radial and angular dependencies are given by the spherical Bessel functions and vector spherical harmonics, respectively. The local frequency of the velocity oscillations equals the angular speed of the rigid flow times the angular azimuthal wavenumber of the oscillating flow. The unsteady motion corresponds to inertial oscillations in multipolar flows with spatial azimuthal waves (non-vanishing azimuthal wavenumber) in the presence of a background flow with constant axial vorticity. In these nonlinear solutions, the curl of the Lamb vector has a linear dependence with the oscillation velocity, a property that makes it possible for the oscillating motion to satisfy different linear wave equations. Based on these inviscid time-dependent velocity modes, new exact solutions to the time-dependent Navier–Stokes equation are also provided.

On the Superposition of Multipolar Spherical and Cylindrical Oscillations in Swirling Rigid Flow

“…Despise their physical relevance there is a lack of exact mathematical solutions to the three-dimensional time-dependent nonlinear fluid equations that may account for the coexistence or superposition of normal oscillating modes in background rigid flows. In inviscid and incompressible fluids a particular class of time-dependent spherical velocity oscillations, in the presence of background swirling rigid flow, was recently found as an exact solution to the nonlinear vorticity equation [1]. These velocity oscillations u p,q (x, t) ≡ U p,q (x, t) + ūq (x) (1) consist in the sum of an oscillating, both in time and space, flow U p,q and a rigid cylindrical swirling flow…”