Bubbles rising in a tube and jets falling from a nozzle

The shape of a two-dimensional bubble rising at a constant velocity U in a tube of width h is computed. The flow is assumed to be inviscid and incompressible. The problem is solved numerically by collocation. The results confirm Garabedian’s findings. There exists a unique solution for each value of the Froude number F=U/( gh)1/2 smaller than a critical value Fc. Here g denotes the acceleration of gravity. It is found that Fc =0.36. In addition the problem of a jet emerging from a vertical nozzle is considered. It is shown that the slope of the free surface at the separation points is horizontal for F<Fc and vertical for F>Fc. Graphs and tables of the results are included.