The present thesis treats the numerical simulation of immiscible incompressible two-phase flows with moving contact lines. The conventional Navier–Stokes equations combined with a no-slip boundary condition leads to a non-integrable stress singularity at the contact line. The singularity in the model can be avoided by allowing the contact line to slip. Implementing slip conditions in an accurate way is not straight-forward and different regularization techniques exist where ad-hoc procedures are common. This thesis presents the first steps in developing the macroscopic part of an accurate multiscale model for a moving contact line problem in two space dimensions. It is assumed that a micro model has been used to determine a relation between the contact angle and the contact line velocity. An intermediate region is introduced where an analytical expression for the velocity field exists, assuming the solid wall is perfectly flat. This expression is used to implement boundary conditions for the moving contact line, at the macroscopic scale, along a fictitious boundary located a small distance away from the physical boundary. Model problems where the shape of the interface is constant throughout the simulation are introduced. For these problems, experiments show that the errors in the resulting contact line velocities converge with the grid size h at a rate of convergence p ≈ 2. Further, an analytical expression for the velocity field in the intermediate region for the case with a curved solid wall is derived. The derivation is based on perturbation analysis.

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.