The slow spreading of a viscous fluid film over a deep viscous pool

A model is introduced for the spreading of an isolated viscous foam film over a deep viscous fluid pool. The effects of the bubbles rising in the foam are neglected, but the O(1) alteration to the density and viscosity of the foam due to the bubbles are accounted for. It is assumed that the foam phase is well modelled by the Stokes-flow equations. By exploiting the slenderness of the spreading layer, asymptotic techniques are used to analyse the flow in the foam. It is shown that in this regime the dominant horizontal force balance is between the hydrostatic pressure and the tangential stress induced in the layer by the underlying pool. Boundary-integral techniques are used to determine the form of the flow in the pool and from this an expression for the evolution of the spreading foam is given analytically. In this parameter regime it is found that this expression does not depend upon the viscosity of the foam.