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Instabilities in gravity driven flow of thin fluid films
 SIAM Rev
, 2003
"... Abstract. This paper presents theoretical, computational, and experimental aspects of the instability development in the flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents derivation of the thin film equation using lubrication approximation. A simplified versi ..."
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Abstract. This paper presents theoretical, computational, and experimental aspects of the instability development in the flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents derivation of the thin film equation using lubrication approximation. A simplified version of this equation is then analyzed analytically using linear stability analysis, and also numerically. The results are then compared directly to experiments. The experimental part outlines the setup, as well as data acquisition and analysis. This immediate comparison to experiments is very useful for gaining better insight into the interpretation of various theoretical and computational results. Key words. nonlinear partial differential equations, perturbation theory, finite difference methods, fluid dynamics
Convergence of a stepdoubling Galerkin method for parabolic problems
 Math. Comp
, 2005
"... Abstract. We analyze a single step method for solving secondorder parabolic initial–boundary value problems. The method uses a stepdoubling extrapolation scheme in time based on backward Euler and a Galerkin approximation in space. The technique is shown to be a secondorder correct approximation ..."
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Cited by 3 (1 self)
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Abstract. We analyze a single step method for solving secondorder parabolic initial–boundary value problems. The method uses a stepdoubling extrapolation scheme in time based on backward Euler and a Galerkin approximation in space. The technique is shown to be a secondorder correct approximation in time. Since stepdoubling can be used as a mechanism for stepsize control, the analysis is done for variable time steps. The stability properties of stepdoubling are contrasted with those of CrankNicolson, as well as those of more general extrapolated thetaweighted schemes. We provide an example computation that illustrates both the use of stepdoubling for adaptive time step control and the application of stepdoubling to a nonlinear system. 1.
Dynamics of cooling viscoplastic domes
 J. Fluid Mech
, 2004
"... A variety of problems in engineering and geology involve spreading cooling nonNewtonian fluids. If the fluid is relatively shallow and spreads slowly, lubricationstyle asymptotic approximations can be used to build reduced models for the spreading dynamics. The centrepiece of such models is a nonl ..."
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Cited by 3 (1 self)
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A variety of problems in engineering and geology involve spreading cooling nonNewtonian fluids. If the fluid is relatively shallow and spreads slowly, lubricationstyle asymptotic approximations can be used to build reduced models for the spreading dynamics. The centrepiece of such models is a nonlinear diffusion equation for the local fluid thickness, and ideally this should become coupled to a correspondingly simple equation determining the local temperature field. However, when heat diffuses relatively slowly as the fluid flows, we cannot usefully reduce the temperature equation, and the asymptotic reduction couples the local thickness equation to an advection diffusion equation that crucially involves diffusion in the vertical. We present an efficient computational algorithm for numerically solving this more complicated type of lubrication model, and describe a suite of solutions that illustrate the dynamics captured by the model in the case of an expanding Bingham fluid with a temperaturedependent viscosity. Based on these solutions, we evaluate two simpler models that further approximate the temperature equation: a vertically isothermal theory, and a ‘skin theory’. The latter is based on the integralbalance method of heattransfer
Evaporation of a thin film: Diffusion of the vapour and Marangoni instabilities
, 2008
"... The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffu ..."
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The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. A nonlocal lubrication equation is obtained; this nonlocal nature comes from the Laplace equation associated with quasistatic diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to moderately thick films for which it is shown that evaporation is diffusion limited and that the gas phase is saturated in vapour in the vicinity of the interface. The stability depends only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilising whereas capillarity and evaporation are stabilising processes. The results of the linear stability analysis are compared with the experiments of Poulard et al. (2003) performed in a different geometry. In order to study the resulting patterns, the amplitude equation is obtained through a systematic multiplescale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the nonlocal nature of the diffusion problem unusually affects the amplitude equation. 1.
DOI: 10.1016/j.colsurfa.2012.04.014 Stability of vertical films of molten glass due to evaporation
"... First, we report observations achieved on a gravitationallydriven film drainage with molten glass pointing out a stabilizing effect when temperature is larger than 1250 ◦ C. ..."
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First, we report observations achieved on a gravitationallydriven film drainage with molten glass pointing out a stabilizing effect when temperature is larger than 1250 ◦ C.
Research Proposal: Ridge Coarsening in EvaporativelyDriven Climbing Films
"... When the evolution of a thin film mixture of alcohol and water on an inclined plate is examined several interesting instabilities form. Where the base of the thin film rejoins the larger reservoir, small ridges form which slowly drift to the side and merge with others, coarsening into larger ridges ..."
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When the evolution of a thin film mixture of alcohol and water on an inclined plate is examined several interesting instabilities form. Where the base of the thin film rejoins the larger reservoir, small ridges form which slowly drift to the side and merge with others, coarsening into larger ridges. These instabilities are driven primarily by the alcohol evaporating more