Numerical solution of thin-film flow equations using adaptive moving mesh methods

Abstract

Thin liquid films are found everywhere in nature. Their flows play a fundamental role in a wide range of applications and processes. They are central to a number
of biological, industrial, chemical, geophysical and environmental applications. Thin films driven by external forces are susceptible to instabilities leading to the break-up of the film into fingering-type patterns. These fingering-type patterns are usually undesirable as they lead to imperfections and dry spots. This behaviour has motivated theoreticians to try to understand the behaviour of the flow and the mechanisms by which these instabilities occur. In the physically relevant case when surface tension is large, the film’s free surface exhibits internal layers where there is rapid spatial variation in the film’s curvature over very short lengthscales and away from these internal layers the film’s curvature is almost negligible. This provides the main motivation for this thesis which is to develop adaptive numerical solution techniques for thin film flow equations that fully resolve such internal layers in order to obtain accurate numerical solutions. We consider two thin film flow problems in one and two-dimensions to test the adaptive numerical solution techniques developed in this thesis. The first problem we consider is related to a liquid sheet or drop spreading down an inclined pre-wetted plane due to influence of gravity. The second problem we consider is also related to the spreading of a liquid sheet or drop down an inclined pre-wetted plane including surfactant-related effects in addition to gravity. We follow the r-adaptive moving mesh technique which uses moving mesh partial differential equations (MMPDEs) to adapt and move the mesh coupled to the underlying PDE(s). We show how this technique can accurately resolve the various one and two-dimensional structures observed in the above test problems as well as reduce the computational effort in comparison to numerical solutions using a uniform mesh.