Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting

摘要

Digital microfluidics is a rapidly growing field wherein droplets are manipulated for use in small-scale applications such as variable focus lenses, display technology, fiber optics, and lab-on-a-chip devices. There has been considerable interest in digital microfluidics and the various methods for liquid actuation by thermal, chemical, and electrical means, where each of the actuation methods make use of the favorable scaling relationship of surface tension forces at the micro scale.Another increasingly important field is addressing the ever growing need for improved heat transfer techniques in the next generation of electronic devices. As device size decreases and device efficiency increases, high heat flux removal capabilities (100 - 1000 W/cm2) are critical to achieve the lower device operating temperatures necessary to ensure reliably and performance.In this thesis, we investigate the nature of the forcing that occurs in the transport of liquid drops by electrical means. The effects of system parameters on the force density and its net integral are considered in the case of dielectrophoresis (insulating fluids) and electrowetting-on-dielectric (conductive fluids). Moreover, we explore the effectiveness of a new heat transfer technique called digitized heat transfer (DHT), where droplets are utilized to enhance the removal of heat from electronic devices. Numerical computations of the Nusselt number for these types of flows provide strong evidence of the effectiveness of DHT in comparison to continuous flows.These two physical phenomena are but two examples that illustrate the growing need for numerical techniques that simply and efficiently handle problems on irregular domains. We present two algorithms appropriate in this environment. The first extends the recently introduced Immersed Boundary Projection Method (IBPM), originally developed for the incompressible Navier-Stokes equations, to elliptic and parabolic problems on irregular domains in a second-order accurate manner. The second algorithm employs a boundary integral approach to the solution of elliptic problems in three-dimensional axisymmetric domains with non-axisymmetric boundary conditions. By using Fourier transforms to reduce the three-dimensional problem to a series of problems defined on the generating curve of the surface, a Nyström discretization employing generalized Gaussian quadratures can be applied to rapidly compute the solution with high accuracy. We demonstrate the high order nature of the discretization. An accelerated technique for computing the kernels of the reduced integral equations is developed for those kernels arising from Laplaceu27s equation, overcoming what was previously the major obstacle in the solution to such problems. We extend this technique to a wide class of kernels, with a particular emphasis on those arising from the Helmholtz equation, and provide strong numerical evidence of the efficiency of this approach. By combining the above approach with the Fast Multipole Method, we develop an efficient and accurate technique for solving boundary integral equations on multiply connected domains.