| The cumulative goal of the works presented throughout this document is to recognize when and how the partial differential equations that describe the physics of moving fluids and deforming solids can be simplified to produce faster, but accurate, numerical solutions. The main vehicle for these simplifications is the assumption that the fluid and/or solid is a “thin film.” A film is body with a geometry that can be described by two surfaces, be it curved or planar, in close proximity to one another. The length and width of the two surfaces are, or are near, an order of magnitude greater than the distance separating them. A thin film is a film whose governing physics can be simplified from a mathematical description in three dimensions ( x, y, and y in a Cartesian coordinate system) to two dimensions (x and y), or from two dimensions to one dimension, i.e. it is dimensionally reduced. The effectiveness of dimensional reduction is demonstrated for pressure driven flows in both a rigid and deformable-ceiling surface-micromachined channel (fabricated at Sandia National Laboratories, Albuquerque, NM), high Peclét number mixing in a T-shaped microchannel, roll coating, and aqueous humor flow through a Ahmed™ glaucoma drainage device. The theoretical results from the last example are validated by published experimental results. Reynolds, or lubrication, theory and von Karman plate theory are used extensively and are solved with the Galerkin finite element method. |
