Oscillatory and streaming flow between two spheres due to combined oscillations

The flow induced by the combined torsional and transverse oscillations of a sphere with amplitude ratio and phase difference 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 , but also in the low-frequency and high-frequency ( ) 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 . The results are compared with those of single torsional or transverse oscillation, and found to match very well. The amplitude ratio and phase difference , in determining the streaming, are also discussed. The number and direction of steady streaming recirculation on the – plane depend on value of the amplitude ratio . The phase difference plays a dominant role in the azimuthal velocity of steady streaming. When is approximately , vanishes under low-frequency oscillation, while steady streaming has a recirculation on the – plane under higher-frequency oscillation.