The slow spreading of several viscous films over a deep viscous pool

In this study, a previously derived two-dimensional model is used to describe the slow spreading of viscous films on the surface of a quiescent deep viscous pool due to gravity. It is assumed that the densities and viscosities of the fluids in the films and pool are comparable, but may be different. It is also assumed that surface tension effects are negligible. The fluid in the films and in the pool are both modelled using the Stokes flow equations. By exploiting the slenderness of the spreading films, asymptotic techniques are used to analyse the flow. It is shown that the dominant forces controlling the spreading are gravity and the tangential stress induced in the films by the underlying pool. As a consequence, the rate of spreading of the films is independent of their viscosity. For the case special of a symmetric configuration of films on the surface of the pool, the flow is studied by assuming that the solution becomes self-similar and hence the problem is recast in a self-similar coordinate system. Stokeslet analysis is then used to derive a singular integral equation for the stresses on the interfaces between the films and the pool. The form of this integral equation depends on the configuration of spreading films that are to be considered. A number of different cases are then studied, namely, a single film, two films, and an infinite periodic array of films. Finally, some results are derived that apply to a general symmetric configuration of films. It is shown that the profile of a spreading film close to its front (where the film thickness becomes zero) is proportional to x1/4. It is also shown that fronts move, and hence, the distance between adjacent fronts increases proportional to t1/3.