Computational methods for microfluidics

This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this. The first paper of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an explicit interface tracking method, wherein the interface between two fluids is decomposed into segments, which are represented locally on an Eulerian grid. The framework of this method provides a natural setting for solving the advection-diffusion equation governing the surfactant concentration on the interface. Open interfaces and moving contact lines are also incorporated into the method in a natural way, though we show that care must be taken when regularizing interface forces to the grid near the boundary of the computational domain. In the second paper we present a boundary integral formulation for sedimenting particles in periodic Stokes flow, using the completed double layer boundary integral formulation. The long-range nature of the particle-particle interactions lead to the formulation containing sums which are not absolutely convergent if computed directly. This is solved by applying the method of Ewald summation, which in turn is computed in a fast manner by using the FFT-based spectral Ewald method. The complexity of the resulting method is O(N log N), as the system size is scaled up with the number of discretization points N . We apply the method to systems of sedimenting spheroids, which are discretized using the Nyström method and a basic quadrature rule. The Ewald summation method used in the boundary integral method of the second paper requires a decomposition of the potential being summed. In the introductory chapters of the thesis we present an overview of the available methods for creating Ewald decompositions, and show how the methods and decompositions can be related to each other.