A Bubble Rising in Viscous Fluid: Lagrange's Equations For Motion at a High Reynolds Number

Abstract

A gas bubble rising steadily in a pure liquid otherwise at rest at a moderate Weber number is, to a good approximation, of oblate spheroidal shape. Previous analytical calculations of that shape at high Reynolds numbers have ignored viscosity. This paper shows that if one includes viscosity by incorporating Rayleigh's dissipation integral in Lagrange's equations, then the speed of rise is that given by Moore, and the shape is that found for inviscid flow by El Sawi using the virial integral and by Benjamin using Hamiltonian theory.