Isochoric unsteady multipolar spherical oscillations in compressible radial and cylindrical background flows

The multipolar spherical vortex solutions to the Euler and Navier–Stokes equations in background cylindrical flow with swirl admit an additional background divergent radial flow with arbitrary time-dependent amplitude. In this case the radial wavenumber $k$, fundamental frequency $\omega$ and overall amplitude $U$ of the multipolar mode superposition become time-dependent and related functions. Assumption of an additional constraint, as a constitutive equation defining the time evolution of the spatially homogeneous divergence of the background flow, is required for the time evolution of the total flow to be completely evaluated from the initial conditions. It is found that flow compression implies an increase of the absolute values of the fundamental frequency $\omega$ and overall velocity amplitude $U$ of the oscillations.