Axisymmetric jet falling from a vertical nozzle and bubble rising in a tube of circular cross section

Abstract

The motion of a bubble rising at a constant velocity U in a vertical tube of circular cross section is considered. The problem is formulated by using the Stokes’ streamfunction as one of the independent variables. The resulting equations are solved by finite differences. It is found that there is a flow for each value of the Froude number F =U/√gD. Here D is the diameter of the tube and g is the acceleration of gravity. For F≳Fd∼0.49, there is a cusp at the apex of the bubble. For F<Fd the slope of the free surface profile is continuous at the apex. For F=Fd, the solution is a pointed bubble with a 130° angle at the apex. The results for F<Fd agree with the solutions previously computed by Levine and Yang. In addition, the problem of an axisymmetric jet falling from a vertical nozzle is considered. It is shown that there is a flow for each value of the Froude number.