Streaming Fluid Motion Within a Laterally Oscillating Sphere With Density Stratification

Summary In this investigation, we have examined the fundamental problem of streaming motion in a liquid-filled sphere undergoing lateral oscillations. Such motion can be externally created, or exists in spacecrafts where $ g $-jitter is well known. The important point here is that for spatially constant liquid density, such an internal problem is degenerate, when no non-trivial oscillatory flow and hence no streaming occur. However, this totally changes in the presence of some density stratification, which may be due to compositional and/or thermal non-uniformities. To clarify the phenomenon in simplest possible terms, we here just consider a constant volumetric heating source within the liquid and isothermal container walls. Proceeding with oscillatory displacement of the spherical shell, we assume a high-frequency limit relative to the viscous and thermal times, and a small displacement amplitude relative to the container (sphere) size. Treating the oscillations as a perturbation to an otherwise stationary shell, two steady streaming contributions are revealed. One is driven in the bulk of the liquid, which is atypical in the incompressible-liquid limit. The other classically originates in the Stokes layer at the boundary also engaging the bulk by viscosity. Even if the former is asymptotically greater here, it does not turn out to be practical to outright disregard the latter. The reason is the particularly low prefactor values arising in the former, which is typical for the internal problem. The streaming pattern consists of two or four axially symmetric vortices, which are dependent on the result of the competition between the two contributions.