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

We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid with inertial forces. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. We introduce a new equation for the local Lagrange multiplier whose solution essentially provides a harmonic extension of the multiplier off the interface while maintaining the local inextensibility constraint near the interface. We also develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. Asymptotic analysis is presented that shows that our new system converges to a relaxed version of the inextensible sharp interface model. This is also verified numerically. To solve the equations, we use an adaptive finite element method with implicit coupling between the Navier-Stokes and the diffuse interface inextensibility equations. Numerical simulations of a single vesicle in a shear flow at different Reynolds numbers demonstrate that errors in enforcing local inextensibility may accumulate and lead to large differences in the dynamics in the tumbling regime and smaller differences in the inclination angle of vesicles in the tank-treading regime. The local relaxation algorithm is shown to prevent the accumulation of stretching and compression errors very effectively. Simulations of two vesicles in an extensional flow show that local inextensibility plays an important role when vesicles are in close proximity by inhibiting fluid drainage in the near contact region.