On steady-state bubbles generated by Taylor instability

Abstract

A theory of the steady motion of a long bubble rising through an infinite plane vertical tube of liquid is presented. It is shown that, contrary to current belief, the flow is not uniquely determined by the width h of the tube and the acceleration of gravity g alone, but that the speed U of the bubble can also be prescribed. However, a criterion of stability singles out the unique physically significant flow of this type as the one which maximizes the velocity of the bubble. With the aid of a difference-differential equation derived from the free-boundary condition, the Froude number U/(gh)$^{\frac{1}{2}}$ for this latter case is estimated to exceed 0$\cdot $2363, a value slightly higher than that indicated by earlier work on the problem.