Oscillatory and streaming flow between two spheres due to combined oscillations

Abstract

The flow induced by the combined torsional and transverse oscillations of a sphere with amplitude ratio $\unicode[STIX]{x1D6FC}$ and phase difference $\unicode[STIX]{x1D6FD}$ in a concentric spherical container is examined. Analytical solutions of the leading-order flow field and shear stress profiles have been obtained. Steady streaming flows are also analysed not only for the case of unrestricted Womersley number $|M|$, but also in the low-frequency $(|M|\ll 1)$ and high-frequency ($|M|\gg 1$) limits. At high frequency, the flow field has been divided into three regions: two boundary layers and the outer region. The streaming flow field is determined for the limiting case of the streaming Reynolds number $R_{s}\ll 1$. The results are compared with those of single torsional or transverse oscillation, and found to match very well. The amplitude ratio $\unicode[STIX]{x1D6FC}$ and phase difference $\unicode[STIX]{x1D6FD}$, in determining the streaming, are also discussed. The number and direction of steady streaming recirculation on the $r$–$\unicode[STIX]{x1D703}$ plane depend on value of the amplitude ratio $\unicode[STIX]{x1D6FC}$. The phase difference $\unicode[STIX]{x1D6FD}$ plays a dominant role in the azimuthal velocity $u_{1\unicode[STIX]{x1D719}}^{(s)}$ of steady streaming. When $\unicode[STIX]{x1D6FD}$ is approximately $(2n+1)\unicode[STIX]{x03C0}/2$, $u_{1\unicode[STIX]{x1D719}}^{(s)}$ vanishes under low-frequency oscillation, while steady streaming has a recirculation on the $r$–$\unicode[STIX]{x1D719}$ plane under higher-frequency oscillation.