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.

Our 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.

This 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.

The 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.

The Brownian motion of particles can be decomposed into translational and rotational components, characterized by translational and rotational diffusion coefficients respectively. Measuring these two coefficients plays an important role in determining the structure and understanding the dynamic properties of materials, with benefit to research in such areas as molecular biology, material sciences. One of the promising tools for investigation of Brownian motion is the emerging X-ray Photon Correlation Spectroscopy (XPCS), which can capture dynamics of samples comprising large groups of particles in a broad range of time scales and length scales. Methods for estimating translational diffusion coefficients on the basis of the temporal auto-correlation analysis of XPCS images are used widely. However, to the best of our knowledge, there is no XPCS-based algorithm for assessing the rotational diffusion coefficients from such temporal auto-correlation data. In this thesis, we take a different route, and propose exploiting an angular-temporal cross-correlation function whose values are approximated by an estimator based on the collected experimental images. We prove the consistency of this estimator by deriving a tail bound. A numerical algorithm, MTECS, for estimating the rotational diffusion coefficients from the cross-correlation is designed and implemented. We demonstrate the capability of this algorithm by testing it on a range of simulated data.

X-ray nanocrystallography is an emerging technique for imaging nanoscale objects that alleviates the large crystallization requirement of conventional crystallography by collecting diffraction patterns from a large ensemble of smaller and easier to build nanocrystals, which are typically delivered to the x-ray beam via a liquid jet. In order to determine the structure of an imaged object, several parameters must first be determined, including the crystal sizes, incident photon flux densities, and crystal orientations. Autoindexing techniques, which have been used extensively to orient conventional crystals, only determine the orientation of the nanocrystals up to symmetry of the crystal lattice, which is often greater than the symmetry of the diffraction information, resulting in what is known as the twinning problem. In addition, the image data is corrupted by large degrees of shot noise due to low collected signal, background signal due to the liquid jet and detector electronics, as well as other sources of noise. Furthermore, diffraction only measures the magnitudes of the Fourier transform of the object and, thus, one must recover phase information in order to invert the data and recover a three-dimensional reconstruction of the constituent molecular structure. Previous approaches for handling the twinning problem have mainly relied on having a known similar structure available, which may not be present for fundamentally new structures. We present a series of techniques to determine the crystal sizes, incident photon flux densities, and crystal orientations in the presence of large amounts of noise common in experiments. Additionally, by using a new sampling strategy, we demonstrate that phase information can be computed from nanocrystallographic diffraction images using only Fourier magnitude information, via a compressive phase retrieval algorithm. We demonstrate the feasibility of this new approach by testing it on simulated data with parameters and noise levels common in current experiments.

The simulation of the dynamics of thin elastic surfaces is an essential component in applications ranging from cardiovascular medicine to flapping wing aerospace engineering. In this thesis we introduce a fully Eulerian method for accurately simulating elastic surfaces. Our approach can be applied to membranes and shells with nonlinear elastic properties, either moving under their own dynamics or immersed in a Fluid. By representing the surface with a level set function and a set of advected reference coordinates, we are able to evaluate the full elastic surface forces. Our approach is compatible with compatible with implicit interface representations, including level set techniques, and can make use of high-order finite difference stencils or more sophisticated techniques. We introduce our method and its implementation, including new solution techniques for several problems associated with immersed interface representations. We demonstrate second order accurate numerical results for a range of model problems in two and three dimensions.

The volume fraction data structure describes the location of a given material within space. It partitions a domain into a set of discrete cells, and for each cell stores the ratio of material contained within it to the total volume of the cell. The task of transforming a grid of volume fractions into a surface representation of the boundary is called the Material Interface Reconstruction Problem. We investigate a variational formulation to this problem, implemented via a level set method, that produces a surface representation of the boundary. We start with an initial guess and iteratively refine the surface by computing a surface-area minimizing curvature flow, followed by an approximate L2 projection of that flow onto a volume-preserving space. We find that the method yields satisfactory results for both two-phase and multiphase data. In the cases where there are C0 but not C1 boundaries between only two materials, the exact solution produces oscillations with period equal to the volume fraction grid size, and amplitude exponentially decreasing away from the location of the loss of smoothness. To reduce oscillations, an alternative method is proposed where in place of the volume fraction constraint, we minimize a weighted sum of the total surface area, plus the sum of squared error in reconstruction. We discuss the results and compare with the projection algorithm.