As digital microfluidics has continued to mature since its advent in the early 1980's, an increase in new and novel applications of this technology have been developed. However, even as this technology has become more common place, a consensus on the physics and force models of the motion of the contact line between the fluid, substrate, and ambient has not been reached. This uncertainty along with the dependence of the droplet geometry on the force to cause its motion has directed much of the research at specific geometries and droplet actuation methods. The goal of this thesis is to help characterize the components of the friction force which opposes droplet motion as a one dimensional system model based upon simple system parameters independent from the actuation method. To this end, the force opposing the motion of a droplet under a thin rectangular glass cover slip was measured for varying cover slip dimensions (widths, length), gap height between the cover slip and substrate, and bulk droplet velocity. The stiffness of the droplet before droplet motion began, the force at which the motion initiated, and the steady-state force opposing the droplet motion were measured. The data was then correlated to hypothesized equations and compared to simple models accounting for the forces due to the contact angle hysteresis, contact line friction, and viscous losses. It was found that the stiffness, breakaway force, and steady-state force of the droplet could be correlated to with an error standard deviation of 8 %, 14%, and 10 % respectively. Much of the error was due to an unexpected height dependence for the breakaway and steady-state forces and testing error associated with the velocity. The models for the stiffness and breakaway force over predicted the results by 36% and 16% respectively. During testing, viii stability issues with the cover slip were observed and simple dye testing was conducted to visualize the droplet flow field.

In this work, we develop an interface-preserving level-set method in the finite-element framework for interfacial flows with moving contact lines. In our method, the contact line is advected naturally by the flow field. Contact angle hysteresis can be easily implemented without explicit calculation of the contact angle or the contact line velocity, and meshindependent results can be obtained following a simple computational strategy. We have implemented the method in three dimensions and provide numerical studies that compare well with analytical solutions to verify our algorithm. We first develop a high-order numerical method for interface-preserving level-set reinitialization. Within the interface cells, the gradient of the level set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved. For the non-interface cells, we compute the gradient of the level set function by solving a Hamilton-Jacobi equation as a conservation law system using the discontinuous Galerkin method. This follows the work by Hu and Shu [SIAM J. Sci. Comput. 21 (1999) 660-690]. The missing constant for these cells is recovered using the continuity of the level set function while taking into account the characteristics. To treat highly distorted initial conditions, we develop a hybrid numerical flux that combines the Lax-Friedrichs flux and a penalty flux. Our method is accurate for non-trivial test cases and handles singularities away from the interface very well. When derivative singularities are present on the interface, a second-derivative limiter is designed to suppress the oscillations. At least (N + 1)th order accuracy in the interface cells and Nth order accuracy in the whole domain are observed for smooth solutions when Nth degree polynomials are used. Two dimensional test cases are presented to demonstrate superior properties such as accuracy, long-term stability, interface-preserving capability, and easy treatment of contact lines. We then develop a level-set method in the finite-element framework. The contact line singularity is removed by the slip boundary condition proposed by Ren and E [Phys. Fluids, vol. 19, p. 022101, 2007], which has two friction coefficients: βN that controls the slip between the bulk fluids and the solid wall and βCL that controls the deviation of the microscopic dynamic contact angle from the static one. The predicted contact line dynamics from our method matches the Cox theory very well. We further find that the same slip length in the Cox theory can be reproduced by different combinations of (βN; βCL). This combination leads to a computational strategy for mesh-independent results that can match the experiments. There is no need to impose the contact angle condition geometrically, and the dynamic contact angle automatically emerges as part of the numerical solution. With a little modification, our method can also be used to compute contact angle hysteresis, where the tendency of contact line motion is readily available from the level-set function. Different test cases, including code validation and mesh-convergence study, are provided to demonstrate the efficiency and capability of our method. Lastly, we extend our method to three-dimensional simulations, where an extension equation is solved on the wall boundary to obtain the boundary condition for level-set reinitializaiton with contact lines. Reinitialization of ellipsoidal interfaces is presented to show the accuracy and stability of our method. In addition, simulations of a drop on an inclined wall are presented that are in agreement with theoretical results. / Doctor of Philosophy / When a liquid droplet is sliding along a solid surface, a moving contact line is formed at the intersection of the three phases: liquid, air and solid. This work develops a numerical method to study problems with moving contact lines. The partial differential equations describing the problem are solved by finite element methods. Our numerical method is validated against experiments and theories. Furthermore, we have implemented our method in three-dimensional problems.