Axisymmetric bubble or drop in a uniform flow

The deformation of an axisymmetric bubble or drop in a uniform flow of constant velocity U is computed numerically. The flow is assumed to be inviscid and incompressible. The problem is formulated as a nonlinear integrodifferential system of equations for the bubble surface and for the potential function on the surface. These equations are discretized and the resulting algebraic system is solved by Newton’s method. For U = 0 the bubble is a sphere. The results show that as U increases the bubble becomes oblate, spreading out in the direction normal to the flow and contracting in the direction of the flow. Then the poles get pushed in and ultimately they touch each other. The results also show that there is a maximum value of the Weber number above which there is no steady axially symmetric bubble. This value is somewhat smaller than the approximate value obtained by Moore (1965) but close to that found by El Sawi (1974). We also compute the added mass, the drag on the bubble, and its terminal velocity in a gravitational field, for large Reynolds numbers.