Extensional thin layer flows

In this thesis we derive and solve equations governing the flow of slender threads and sheets of viscous fluid. Our method is to solve the Navier-Stokes equations and free surface conditions in the form of asymptotic expansions in powers of the inverse aspect ratio of the fluid, i.e. the ratio of a typical thickness to a typical length. In the first chapter we describe some of the many industrial processes in which such flows are important, and summarise some of the related work which has been carried out by other authors. We introduce the basic asymptotic methods which are employed throughout this thesis in the second chapter, while deriving models for two-dimensional viscous sheets and axisymmetric viscous fibres. In chapter 3 we show that when these equations govern the straightening or buckling of a curved viscous sheet, simplification may be made via the use of a suitable short timescale. In the following four chapters, we derive models for nonaxisymmetric viscous fibres and fully three-dimensional viscous sheets; for each we consider separately the cases where the dimensionless curvature is small and where the dimensionless curvature is of order one. We find that the models which result bear a marked similarity to the theories of elastic rods, plates and shells. In chapter 8 we explain in some detail why the Trouton ratio - the ratio between the extensional viscosity and the shear viscosity - is 3 for a slender viscous fibre and 4 for a slender viscous sheet. We draw our conclusions and suggest further work in the final chapter