Bifurcations of a creeping air–water flow in a conical container

Abstract

This numerical study describes the eddy emergence and transformations in a slow steady axisymmetric air–water flow, driven by a rotating top disk in a vertical conical container. As water height $$H_{\mathrm{w}}$$ and cone half-angle $$\beta $$ vary, numerous flow metamorphoses occur. They are investigated for $$\beta =30^{\circ }, 45^{\circ }$$ , and $$60^{\circ }$$ . For small $$H_{\mathrm{w}}$$ , the air flow is multi-cellular with clockwise meridional circulation near the disk. The air flow becomes one cellular as $$H_{\mathrm{w}}$$ exceeds a threshold depending on $$\beta $$ . For all $$\beta $$ , the water flow has an unbounded number of eddies whose size and strength diminish as the cone apex is approached. As the water level becomes close to the disk, the outmost water eddy with clockwise meridional circulation expands, reaches the interface, and induces a thin layer with anticlockwise circulation in the air. Then this layer expands and occupies the entire air domain. The physical reasons for the flow transformations are provided. The results are of fundamental interest and can be relevant for aerial bioreactors.