Modelling the spreading and draining of viscous films

Abstract

The focus of the work in this thesis is to gain new insight into the fluid behaviour observed in a float glass furnace by means of simplified mathematical models. In particular, the models explore the dynamics of films of foam, known as logs, that spread across the surface of a pool of molten glass. The model employed throughout is the two dimensional Navier-Stokes equations, in the limit of zero Reynolds number, together with appropriate conditions at moving boundaries. Throughout the thesis, the slender geometry of the films is exploited using asymptotic techniques to simplify the models. In the introductory chapter, the motivating float glass manufacturing process is described, then the mathematical techniques and modelling assumptions that are used throughout the thesis are introduced. In the first technical chapter a model for the spreading of viscous films on the surface of a deep viscous pool is considered. Although this model neglects the effects of drainage it enables analytical progress to be made. As such, insight is gained into how the spreading logs interact with one another as they spread across the surface of the underlying pool. Analytic expressions for the evolution of a single spreading film, two spreading films and an infinite array of films are obtained. In addition, some comments on a general configuration of films are made. In the next technical chapter a model for the spreading and draining of a viscous film on a flat surface is considered. Although the model is simplistic, and neglects the interaction of logs via the underlying pool, it does allow some initial ideas on the effects of drainage to be explored. The model is systematically reduced to a nonlinear diffusion PDE. The subsequent analysis is applicable to a broad family of PDEs, hence the analysis is presented in some generality. Solutions to the PDEs under consideration exhibit an interesting behaviour in which the front of a compactly supported solution changes its direction of propagation. To explore this phenomenon, the behaviour of the front of the film as it advances (due to gravity driven spreading) and recedes (due to drainage) is examined. In particular, asymptotic solutions local to a time at which the front of the film changes its direction of propagation are obtained and their implications discussed. In the final technical chapter, the ideas from the previous two chapters are drawn together. A model is considered that incorporates both drainage, and allows the spreading logs to interact via the molten glass pool. It is shown that the model can be systematically reduced to a singular integro-differential equation (SIDE). In a special case, a steady state solution to this SIDE is obtained using a combination of asymptotic and numerical techniques. To complement the analysis in the previous chapters, advancing and receding fronts of solutions to the model are also examined. In the final chapter, the results of the previous chapters are summarised, and the practical implementation of the modelling is discussed. The work not only gives rise to a number of novel mathematical results, but also provides new understanding on the behaviour of spreading viscous films and the industrial float glass process

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