Symmetry and Variational Analyses of Fluid Interface Equations in the Thin Film Limit

Citation

Nicolaou, Zachary George (2017) Symmetry and Variational Analyses of Fluid Interface Equations in the Thin Film Limit. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9R20ZBS. https://resolver.caltech.edu/CaltechTHESIS:09082016-145215432

Abstract

This thesis concerns a class of nonlinear partial differential equations up to fourth order in spatial derivatives that models thin viscous films. In Chapter 1, we review the derivations of thin film equations from the fundamental transport equations. Section 1.1 contains the derivation for a thermocapillary driven film to familiarize the reader with the key long-wavelength approximation that has been successful in modeling a myriad of thin viscous films. In Section 1.2, we consider the coupling between a thin viscous layer and a much thicker fluid layer with much larger viscosity and conductivity and show how a novel, non-local thermocapillary thin film equation can be derived to model such a system. We then review the wider class of thin film equations in Section 1.3, note the important Cahn-Hilliard variational form of these equations, and demonstrate that classic mathematical results concerning the inverse problem of the calculus of variations permit an algorithmic procedure for discovering Lyapunov functionals. In Chapter 2, we review applications of symmetry methods to partial differential equations. Section 2.1 contains an original geometrical motivation for the study of self-similar reductions which draws an analogy with the fixed points of dynamical systems. In Section 2.2, we derive for the first time the full set of symmetries of the fully two-dimensional thin film equations. We then enumerate the possible symmetry reductions of the thin film equations, and discover several which have not been previously recognized. In Chapter 3, we consider rotationally invariant, steady droplet solutions and their stability. In Section 3.1, we derive stability criteria for thermocapillary-driven droplets, and show a novel correspondence between droplet stability, droplet volume, and droplet Lyapunov energy. We consider thin films under other forces in Section 3.2 and make new predictions about conditions under which such films develop into droplets, columns, or jets of fluid. In Chapter 4, we consider the scale invariant symmetry reductions of thin film equations. In Section 4.1 we describe the extraordinarily rich variety of such solutions in the spreading of a insoluble surfactant on a thin viscous film, identify previously unrecognized scale invariant solutions which are well-behaved at the origin, and demonstrate their relevance with finite element simulations. Lastly, in Section 4.2, we illustrate for capillary driven films some numerical solutions to the novel reductions we uncovered in Chapter 2. Each chapter concludes with a Notes section which summarizes the new results contained therein and relates them to the wider literature.

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