Moving contact line problems appear in many natural and industrial processes. A contact line is formed where the interface between two immiscible fluids meets a solid wall. Examples from everyday life include raindrops falling on a window and water bugs resting on water surfaces. In many cases the dynamics of the contact line affects the overall behavior of the system. Industrial applications where the contact line behavior is important include gas and oil recovery in porous media, lubrication, inkjet printing and microfluidics. Computer simulations are fundamental tools to understand and predict the behavior.  

In this thesis we look at numerical simulations of dynamic contact line problems. Despite their importance, the physics of moving contact lines is poorly understood. The standard Navier-Stokes equations together with the conventional no-slip boundary condition predicts a singularity in the shear stresses at the contact line. Atomistic processes at the contact line come into play, and it is necessary to include these processes in the model to resolve the singularity. In the case of capillary driven flows for example, it has been observed that the microscopic contact line dynamics has a large impact on the overall macroscopic flow.

In Paper I we present a new multiscale model for numerical simulation of flow of two immiscible and incompressible fluids in the presence of moving contact points (i.e. two-dimensional problems). The paper presents a new boundary methodology based on combining a relation between the apparent contact angle and the contact point velocity, and a similarity solution for Stokes flow at a planar interface (the analytic Huh and Scriven velocity). The relation between the angle and the velocity is determined by performing separate microscopic simulations.

The classical Huh and Scriven solution is only valid for flow over flat walls. In Paper II we use perturbation analysis to extend the solution to flow over curved walls. Paper III presents the parallel finite element solver that is used to perform the numerical experiments presented in this thesis. Finally, the new multiscale model (presented in Paper I) is applied to a relevant microfluidic research problem in Paper IV. For this problem it is very important to have a model that accurately takes the atomistic effects at contact lines into account.