A Hybrid Finite Difference Level Set–Implicit Mesh Discontinuous Galerkin Method for Multi-Layer Coating Flow Problems

Industrial painting operations consume significant amounts of energy, owing in part to theindividual application and curing of multiple layers of paint. Energy-efficient manufacturing lines that co-cure (i.e., simultaneously bake) multiple film-layers have the potential to reduce energy consumption by 30%. However, achieving a smooth, defect-free film of paint is often the biggest technical hurdle to commercializing these energy-efficient coating systems. In this thesis, we develop high-fidelity mathematical and numerical frameworks to model the complex multi-physics underlying multi-layer coating flow dynamics, with applications to the leveling of multi-layer paint films, i.e., the coupled evaporation, solidification, fluid flow, and settling dynamics of multiple layers of liquid paint./nOur mathematical model captures a coupled set of multi-physics that includes multi-phasequasi-Newtonian fluid dynamics; the transport, diffusion, and mixing of multiple dissolved species; mass transfer and interface recession from solvent evaporation; intricate interfacial forces of surface tension and Marangoni stresses on paint-gas and paint-paint interfaces and their coupling; and substrate roughness and the pull of gravity. Using this model, we study the highly complex and intricate dynamics of “watching paint dry”, capturing several experimental findings and studying the formation of Marangoni plumes and B´enard cells, the impact of long-wave deformational surface modes on immersed interfaces, and the emergence of the final multi-layer film profile./nThis thesis presents a hybrid numerical framework for the multi-layer coating flow problemthat consists of: finite difference level set methods and high-order accurate sharp interface implicit mesh discontinuous Galerkin methods; newly developed local discontinuous Galerkin solvers for Poisson problems with Robin boundary and jump conditions on implicitly-defined domains, to capture solvent evaporation; state-of-the-art Stokes solvers that integrate concentration-dependent rheological parameters for quasi-Newtonian interface dynamics; high-order accurate methods to couple the transport, diffusion, and evaporation of multiple dissolved species while also tracking interface recession; a tailored finite difference projection algorithm that calculates surface gradients, to robustly and accurately incorporate Marangoni stresses; and a coupled multi-physics time stepping approach that incorporates all the different solvers at play, among a host of additional numerical algorithms. Several components of our hybrid numerical framework are high-order accurate and the algorithm is applicable to an arbitrary number of layers and dissolved species. Our particular implementation of the fully coupled numerical algorithm for the multi-layer coating flow problem is 2nd order accurate in space and 1st order in time. A new high-order accurate local discontinuous Galerkin formulation for Stokes problems with Navier-slip boundary conditions on implicitly-defined domains is also presented in the appendix./nThe framework is designed, in part, to predict the ultimate surface roughness of the multilayersystem; here, we apply it to a variety of settings, including multi-solvent evaporative paint dynamics, the flow and leveling of multi-layer automobile paint coatings in both 2D and 3D—presenting the results of a 2D parametric study performed at industrially-relevant conditions, and an examination of “interfacial turbulence” within a multi-layer matter cascade. This work revealed many of the driving mechanisms underlying multi-layer coating flow dynamics, including: the creation, characteristics, and impact of short- and long-wave Marangoni hydrodynamic instabilities; the impact of basecoat deformation and telegraphing of substrate roughness on the clearcoat surface profile; conjectures concerning the role of interfacial forces exhibited by the immersed paint-paint interface; and the overall dynamic’s significant sensitivity to mass diffusion coefficients. The model and the developed numerical framework presented in this thesis provide opportunities to develop new coating formulas that can be co-cured with a single, lower-temperature bake and to identify specific features critical to achieving a smooth paint surface.

Advisor: Sethian, James A