We investigate the use of the Method of Fundamental Solutions (MFS) for the numerical solution of elliptic problems arising in engineering. In particular, we examine harmonic and biharmonic problems with boundary singularities, certain steady-state free boundary flow problems and inhomogeneous problems. The MFS can be viewed as an indirect boundary method with an auxiliary boundary. The solution is approximated by a linear combination of fundamental solutions of the governing equation which are expressed in terms of sources located outside the domain of the problem. The unknown coefficients in the linear combination of fundamental solutions and the location of the sources are determined so that the boundary conditions are satisfied in a least squares sense. The MFS shares the same advantages of the boundary methods over domain discretisation methods. Moreover, it is relatively easy to implement, it is adaptive in the sense that it takes into account sharp changes in the solution and/or in the geometry of the domain and it can easily incorporate complicated boundary conditions.

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We study a random homogenization problem concerning the flow of a viscous fluid through a permeable membrane with a highly oscillatory geometry and nonlinear boundary condition on it. Along an interface we consider a periodic distribution of small permeable obstacles with a random geometry. Leak boundary conditions of threshold type are considered on the obstacle part of the membrane: the normal velocity of the fluid is zero until the jump of the normal component of the stress acting on it reaches a certain limit, and then the fluid may pass freely. The problem is studied first in the deterministic case, and then in the random case, for which assumptions on the randomness of the solid obstacles are needed in order to obtain a limiting behaviour. The description of the obstacles is given in terms of a random set-valued variable defined on a probability space and a dynamical system acting on it. Effective boundary conditions for the fluid are derived, and these depend on the relative size of the obstacles. We establish two major cases, in one of them we obtain an effective permeability across the membrane and in the critical case a slip boundary condition of Navier type. If the dynamical system is assumed to be ergodic, the limiting behaviour of the fluid is deterministic. The approach is based on the Mosco convergence, which also allows us to pass from the stationary case to the time dependent case via the convergence of the associated semigroups.

The flow of fluids over solid surfaces is an integral part of many technologies, and the analysis of such flows is important to the design and operation of these technologies. Solid surfaces, however, are generally rough at some scale, and analyzing the effects of such roughness on fluid flows represents a significant challenge. There are two fluid flow situations in which roughness is particularly important, because the fluid shear layers they create can be very thin, of order the height of the roughness. These are very high Reynolds number turbulent wall-bounded flows (the viscous wall layer is very thin), and very low Reynolds number lubrication flows (the lubrication layer between moving surfaces is very thin). Analysis in both of these flow domains has long accounted for roughness through empirical adjustments to the smooth-wall analysis, with empirical parameters describing the fluid dynamic roughness effects. The ability to determine these effects from a topographic description of the roughness is limited (lubrication) or non-existent (turbulence). The commonly used parameter, the equivalent sand grain roughness, can be determined in terms of the change in the rate of viscous energy dissipation caused by the roughness and is generally obtained by measuring the effects on a fluid flow. However, determining fluid dynamic effects from roughness characteristics is critical to effective engineering analysis. Characterization of this mapping from roughness topography to fluid dynamic impact is the main topic of the dissertation. Using the mathematical tools of shape calculus, we construct this mapping by defining the roughness functional and derive its first- and second- order shape derivatives, i.e., the derivatives of the roughness functional with respect to the roughness topography. The results of the shape gradient and complete spectrum of the shape Hessian are presented for the low Reynolds number lubrication flows. Flow predictions based on this derivative information is shown to be very accurate for small roughness. However, for the study of high Reynolds number turbulent flows, the direct extension of the current approach fails due to the chaotic nature of turbulent flows. Challenges and possible approaches are discussed for the turbulence problem as well as a model problem, the sensitivity analysis of the Lorenz system. / text

A single light sphere is placed inside of a drum completely filled with a viscous fluid. the drum rotates perpendicular to gravity such that the Reynolds number remains small. In this flow configuration there is a wide range of behaviours. We have measured eccentric fixed points, circular orbits and asymmetric orbits by the walls of the drum. A full description of these phenomena is given as a function of the size of the sphere. We have also studied the case of a porous sphere and compared it the solid case. Finally we have studied the effect of roughness on this system.

Microorganisms swimming in their natural habitat interact with debris and boundaries, which can modify their swimming characteristics. However, the boundary effect on swimming microorganisms have not been completely understood yet, and is one of most active areas of research. Amongst microorganisms, unicellular ciliates are the fastest swimmers and also respond to a variety of external cues. We choose Paramecium multimicronucleatum as a model system to understand the locomotion of ciliates. First, we explore the effects of boundaries on swimming modes of Paramecium multimicronu- cleatum by introducing them in 2D films and 1D channels. The geometric confinements cause the Paramecia to transition between: a directed, a meandering and a self-bending behaviors. During the self-bending mode the cell body exerts forces on the walls; which is quantified by using a beam bending analogy and measuring the elasticity of the cell body. The first inves- tigation reveals the complicated swimming patterns of Paramecium caused by boundaries. In the second study we investigate the directed swimming of Paramecium in cylindrical capillaries, which mimics the swimming of ciliates in the pores of soil. A finite-sized cell lo- comoting in extreme confinements creates a pressure gradient across its ends. By developing a modified envelop model incorporating the confinements and pressure gradient effects, we are able to predict the swimming speed of the organisms in confined channels. Finally we study how Paramecium can swim and feed efficiently by stirring the fluid around its body. We experimentally employ "-Particle Image Velocimetry to characterize flows around the freely swimming Parameicum and numerically use Boundary Element Method to quantify the effect of body shapes on the swimming and feeding process. Results show that the body shape of Paramecium (slender anterior and bulky posterior) is hydrodynamically optimized to swim as well as feed efficiently. The dissertation makes significant advances in both experimentally characterizing and the- oretically understanding the flow field and locomotion patterns of ciliates near solid bound- aries. / Ph. D.

Fluid phenomena dominated by viscous effects can, in many cases, be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the three-dimensional problem to two-dimensional integral equations to be discretized over the boundaries of the domain. Hence for the study of objects immersed in a fluid, such as drops or elastic/solid particles, integral equations are to be discretized over the surfaces of these objects only. As outer boundaries or confinements are added these must also be included in the formulation. An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations, e.g. the so called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?1/%7C%5Cmathbf%7Bx%7D%7C" />. To assess the convergence properties of this quadrature rule a theoretical analysis has been performed. The slightly more complicated singularity of the Stokeslet required certain modifications of the integration rule developed for <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?1/%7C%5Cmathbf%7Bx%7D%7C" />. An extension of this type of quadrature rule to a cylindrical surface is also developed. These quadrature rules are tested also on physical problems that have an analytic solution in the literature. Another difficulty associated with boundary integral problems is introduced by periodic boundary conditions. For a set of particles in a periodic domain periodicity is imposed by requiring that the motion of each particle has an added contribution from all periodic images of all particles all the way up to infinity. This leads to an infinite sum which is not absolutely convergent, and an additional physical constraint which removes the divergence needs to be imposed. The sum is decomposed into two fast converging sums, one that handles the short range interactions in real space and the other that sums up the long range interactions in Fourier space. Such decompositions are already available in the literature for kernels that are commonly used in boundary integral formulations. Here a decomposition in faster decaying sums than the ones present in the literature is derived for the periodic kernel of the stress tensor. However the computational complexity of the sums, regardless of the decomposition they stem from, is <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BO%7D(N%5E%7B2%7D)" />. This complexity can be lowered using a fast summation method as we introduced here for simulating a sedimenting fiber suspension. The fast summation method was initially designed for point particles, which could be used for fibers discretized numerically almost without any changes. However, when two fibers are very close to each other, analytical integration is used to eliminate numerical inaccuracies due to the nearly singular behavior of the kernel and the real space part in the fast summation method was modified to allow for this analytical treatment. The method we have developed for sedimenting fiber suspensions allows for simulations in large periodic domains and we have performed a set of such simulations at a larger scale (larger domain/more fibers) than previously feasible. / <p>QC 20121122</p>