A mathematical analysis of foam films

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The filling process of a car tank is complicated by foam formation, which can hinder or delay the tank's complete filling. This dissertation focuses on optimizing tank geometry through numerical simulation to minimize foam's impact. We analyze foam behavior mathematically on the mezoscopic scale, specifically examining single lamellae. Our primary objectives are to deepen the understanding of the interactions among relevant physical effects and to develop a model for simulating lamella decay that can be integrated into a global foam model. The first part introduces the physical properties of foam, identifying the Marangoni effect as a key factor in its stability. We then create a mathematical model for the dynamic behavior of a lamella, employing asymptotic analysis based on its unique geometry. This leads to a system of nonlinear partial differential equations (PDE) of third order in two spatial dimensions and one time dimension. In the second part, we analyze this system mathematically, proving existence and uniqueness for a simplified case. Certain parameter domains allow for further simplification, enabling the derivation of explicit solutions in some instances. The final part involves solving the system using a finite element approach, with a detailed discussion of the results.