Three-dimensional direct numerical simulation of coating flows

Abstract

Many industrial processes and natural phenomena involve flows of thin viscous liquid films, such as the coating process of a solid substrate with paint or lubricant, the flow of condensed water on a clothes wire, or the flow of lava. The thinness and slowness of these flows allow us to simplify the governing partial differential equations by utilizing the lubrication approximation. These approximations simplify the numerical solution of the problem while retaining the most important nonlinear features, so that the basis of the problem can be understood.
We first consider surface-tension-gradient effects on the leveling of a drying multi-component liquid. We develop a mathematical model based on the lubrication approximation, which describes the time evolution of an evaporating film. As the film dries the physical properties of the film change, therefore, we include the dependence of viscosity, surface tension, diffusivity, and evaporation rate on the resin concentration in the mathematical model. Using nonlinear numerical simulations, we demonstrate that an initially doubly-modulated sinusoidal profile can reverse its shape during the leveling process.
We then consider two closely related thin-film flow problems: downhill drainage under the influence of gravity, and surface-shear-stress-driven climbing films. The stability and nonlinear finger growth mechanisms of these problems are studied in detail. The most unstable transverse wave numbers are found from the numerical solution, and are compared with published experimental results. The influence of finite contact angle is also included in the mathematical model through a disjoining-pressure model. In this study, we show that disjoining pressure is the main mechanism controlling the sideways spreading of fingers, resulting in infinitely long rivulets.
Next, the spraying and spreading process on a moving substrate is considered. For this problem, we develop a mathematical model which not only captures the effects of surface tension and gravity, but also includes the moving substrate, and models different spray patterns. Under certain assumptions we find a closed-form similarity solution, which agrees well with the numerical solution.
The gravitational instabilities of thin liquid films leading to drop formation are also considered. The mathematical model captures the essentials of the flow. The linear growth rates which are extracted from the nonlinear three-dimensional simulation agree very well with the analytical dispersion characteristics of the problem.
Finally, we present a two-dimensional higher order lubrication approximation for the surface tension and gravity driven leveling problem. The zeroth order lubrication approximation for any flow problem is valid in the limit when the thickness of the coating is small compared to the characteristic length scale of the problem. We present the derivation of a fourth-order lubrication approximation, and show an appropriate numerical solution technique.

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ORCID for Murat Hakki Eres: orcid.org/0000-0003-4967-0833

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