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  • Thin Viscous Films on Curved Geometries

    Dissertation

    zur Erlangung des Doktorgrades (Dr. rer. nat.)

    der Mathematisch-Naturwissenschaftlichen Fakultät

    der Rheinischen Friedrich-Wilhelms-Universität Bonn

    vorgelegt von Orestis Vantzos

    aus Athen

    Bonn 2014

  • Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn am Institut für Numerische Simulation.

    1. Gutachter: Prof. Dr. Martin Rumpf 2. Gutachter: Prof. Dr. Nikolaos Alikakos Tag der Promotion:

    Erscheinungsjahr: 2014

  • Dedicated to the Light of my Eyes

  • Abstract

    The topic of this thesis is the evolution of thin viscous films on curved substrates. Using techniques from differential geometry, namely the exterior calculus of differential forms, and from optimization theory, in particular the theory of saddle point problems and the shape calculus, we reduce a variational form of the Stoke equations, which govern the flow, to a two dimensional optimization problem with a PDE constraint on the substrate. This reduction is analogous to the lubrication approximation of the classic thin film equation. We study the well-posedness of a, suitably regularised, version of this reduced model of the flow, using variational techniques. Furthermore, we study the well-posedness and convergence of time- and space-discrete versions of the model. The time discretization is based on the idea of the natural time discretization of a gradient flow, whereas the spatial discretization is done via suitably chosen finite element spaces. Finally, we present a particular implementation of the discrete scheme on subdivision surfaces, together with relevant numerical results.

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  • Contents

    Introduction v

    1. Exterior Calculus on Thin Domains 1 1.1. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. The cylindrical manifold K . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3. Musical isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4. Hodge star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5. Exterior & Lie derivatives and the interior product . . . . . . . . . . . . . 15 1.6. Pullback and pushforward . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7. Vector calculus with forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.8. Tensor algebra and tensor calculus in Rn . . . . . . . . . . . . . . . . . . . 32

    2. A Reduced Model of Thin Film Motion 39 2.1. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2. Constrained optimization and saddle-point problems . . . . . . . . . . . . 41 2.3. Shape calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4. Variational form of the Stokes equations . . . . . . . . . . . . . . . . . . . 51 2.5. Flow in thin domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.6. Reduced energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.7. Reduced dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.8. Optimal velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.9. The reduced model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    3. Evolution and Variational Discretization of the Model 79 3.1. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3. Regularization of the mobility . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4. Well-posedness of the model . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5. Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.6. Galerkin approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    4. Numerical Implementation with Subdivision Surfaces 119 4.1. Introduction and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2. Galerkin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

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  • Contents

    4.3. H2-conforming elements on subdivision surfaces . . . . . . . . . . . . . . . 123 4.4. Convergence tests on level sets . . . . . . . . . . . . . . . . . . . . . . . . 128

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  • Introduction

    In recent years, the investigation of the dynamics of liquid thin films has attracted in- creased attention in the field of physics, engineering and mathematics. In many applica- tions in materials science and biology, liquid thin films do not reside on a flat Euclidean domain but on curved surfaces (Howell[How03], Roy, Roberts and Simpson[RRS02], Schwartz and Weidner[SW95], Wang[Wan84]). Examples are the spreading of liquid coatings on surfaces, the surfactant-driven thin film flow on the interior of the lung alve- oli (Xu et al.[XLLZ06]) and the tear film on the cornea of the eye (Braun et al.[BUM+]). The evolution of the film thickness is often of greater interest than the actual velocity or pressure field within the fluid volume. In that case, a lubrication approximation dating back already to Reynolds[Rey86] allows us to replace the governing Navier-Stokes and moving free boundary model by with an evolution model expressed solely in terms of the film height or a related quantity. For a thin film deposited on a planar substrate, and in the limit of vanishing thickness-to-length ratio, one can derive through the well-known lubrication theory (Oron, Davis and Bankoff[ODB97]) a limit model in the form of a fourth order nonlinear parabolic problem for the evolution of the film height h (Bernis and Friedman[BF90], Bertozzi and Pugh[BP96], Bernis[Ber95], Beretta, Bertsch and Dal Paso[BBDP95]). We refer to Oron, Davis, and Bankoff[ODB97] for the derivation of the model and to Myers[Mye98] for an overview of the mathematical treatment of surface- tension-driven thin fluid films. A recent review by Craster and Matar[CM09] discusses the dynamics and stability of thin liquid films involving external forcing, thermal effects and intermolecular forces.

    Already in ’84, Wang[Wan84] presented a lubrication model for the evolution of a thin film flowing down a curved surface. Schwartz and Weidner[SW95] discussed the additional forcing effect due to the surface curvature. A lubrication model for the dy- namics of the film, in the form of a PDE for the evolution of the film thickness, has been derived by Roy, Roberts and Simpson[RRS02]. Unlike the case of a flat substrate, their lubrication model is an approximation of the Navier-Stokes equations, rather than the limit model for vanishing film thickness. The approximation is based on a second order expansion in �, where � is the scale ratio between the characteristic height of the film and the characteristic length of the surface. Roberts and Li[RL06] extended this model to include inertial effects, by adding an evolution law for the average lateral velocity. In Thiffeault and Kamhawi[TK06] gravity-driven thin film flows on curved substrates are studied from a dynamical systems point of view. A related gravity-driven shallow water model on curved geometries, namely topographic maps, was investigated by Boutounet

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  • Introduction

    et al.[BCNV08] Kalliadasis and Bielarz[KB00] directly applied a thin film model on topographic maps to analyze the impact of topological features on the formation of cap- illary ridges. Jensen et al.[JCK04] studied the flow of a thin, homogeneous liquid layer induced by a sudden change in the shape of the substrate. Thin film flow on moving curved surfaces was investigated by Howell[How03], who explored the behavior for large, non-uniform curvature, whose gradient dominates the flow and leads in the limit to a hyperbolic equation with shock formation at specific regions of the substrate. The flow of a thin film on a flat, but non-linearly stretching, sheet was discussed by Santra and Dandapat[SD09].

    There are two main challenges in modelling the thin film flow on a curved substrate. The first one is that, contrary to the flat case, the anisotropic nature of the mobility can not be ignored and therefore it needs to be taken as a tensor, rather than a scalar, function of the film thickness. The second difficulty is that the free energy of the film is dominated by curvature- and gravity-driven transport-like terms, whereas the surface tension-driven Dirichlet energy is a first order correction. Since the regularizing effects of the Dirichlet energy are vital to the proper modelling of the problem, we can not limit ourselves to a leading order approximation, as in the classic lubrication approximation. The first chapter of the thesis lays down the foundations for simultaneously dealing with both of these issues. We use the exterior calculus of differential forms (presented with particular emphasis in physical applications in Frankel[Fra04]) to explore the differential calculus of curved thin structures. The main result (Prop. 1.55) is a set of expressions for the gradient, curl and divergence that feature

    1. a decomposition into normal-tangential components,

    2. a natural expansion into terms of different order in the thickness parameter �, and

    3. transparent inclusion of the effects of both the scalar and tensor curvatures of the substrate.

    In the second chapter, we combine the results of the first chapter with tools from the variational theory of saddle point problems (as developed by Brezzi[Bre74] and Babuška[Bab73]) and shape calculus (as presented in