Existence and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system over moving domains

This dissertation concerns the well-posedness of the Navier-Stokes-Smoluchowski system. The system models a mixture of fluid and particles in the so-called bubbling regime. The compressible Navier-Stokes equations governing the evolution of the fluid are coupled to the Smoluchowski equation for the particle density at a continuum level. First, working on fixed domains, the existence of weak solutions is established using a three-level approximation scheme and based largely on the Lions-Feireisl theory of compressible fluids. The system is then posed over a moving domain. By utilizing a Brinkman-type penalization as well as penalization of the viscosity, the existence of weak solutions of the Navier-Stokes-Smoluchowski system is proved over moving domains. As a corollary the convergence of the Brinkman penalization is proved. Finally, a suitable relative entropy is defined. This relative entropy is used to establish a weak-strong uniqueness result for the Navier-Stokes-Smoluchowski system over moving domains, ensuring that strong solutions are unique in the class of weak solutions.

THE STOCHASTIC NAVIER STOKES EQUATIONS FOR HEAT CONDUCTING, COMPRESSIBLE FLUIDS

This dissertation is devoted to the equations of motion governing the evolution of a fluid or gas at the macroscopic scale. The classical model is a PDE description known as the Navier-Stokes equations. The behavior of solutions is notoriously complex, leading many in the scientific community to describe fluid mechanics using a statistical language. In the physics literature, this is often done in an ad-hoc manner with limited precision about the sense in which the randomness enters the evolution equation. The stochastic PDE community has begun proposing precise models, where a random perturbation appears explicitly in the evolution equation. Although this has been an active area of study in recent years, the existing literature is almost entirely devoted to incompressible fluids. The purpose of this thesis is to take a step forward in addressing this statistical perspective in the setting of compressible fluids. In particular, we study the well posedness for the corresponding system of Stochastic Navier Stokes equations, satisfied by the density, velocity, and temperature. The evolution of the momentum involves a random forcing which is Brownian in time and colored in space. We allow for multiplicative noise, meaning that spatial correlations may depend locally on the fluid variables. Our main result is a proof of global existence of weak martingale solutions to the Cauchy problem set within a bounded domain, emanating from large initial datum. The proof involves a mix of deterministic and stochastic analysis tools. Fundamentally, the approach is based on weak compactness techniques from the deterministic theory combined with martingale methods. Four layers of approximate stochastic PDE's are built and analyzed. A careful study of the probability laws of our approximating sequences is required. We prove appropriate tightness results and appeal to a recent generalization of the Skorohod theorem. This ultimately allows us to deduce analogues of the weak compactness tools of Lions and Feireisl, appropriately interpreted in the stochastic setting.

Mesh Adaption for Tracking Vortex Structures in OVERTURNS Simulation of the S-76 Rotor in Hover

The constant need to improve helicopter performance requires the optimization of existing and future rotor designs. A crucial indicator of rotor capability is hover performance, which depends on the near-body flow as well as the structure and strength of the tip vortices formed at the trailing edge of the blades. Computational Fluid Dynamics (CFD) solvers must balance computational expenses with preservation of the flow, and to limit computational expenses the mesh is often coarsened in the outer regions of the computational domain. This can lead to degradation of the vortex structures which compose the rotor wake. The current work conducts three-dimensional simulations using OVERTURNS, a three-dimensional structured grid solver that models the flow field using the Reynolds-Averaged Navier-Stokes equations. The S-76 rotor in hover was chosen as the test case for evaluating the OVERTURNS solver, focusing on methods to better preserve the rotor wake. Using the hover condition, various computational domains, spatial schemes, and boundary conditions were tested. Furthermore, a mesh adaption routine was implemented, allowing for the increased refinement of the mesh in areas of turbulent flow without the need to add points to the mesh. The adapted mesh was employed to conduct a sweep of collective pitch angles, comparing the resolved wake and integrated forces to existing computational and experimental results. The integrated thrust values saw very close agreement across all tested pitch angles, while the power was slightly over predicted, resulting in under prediction of the Figure of Merit. Meanwhile, the tip vortices have been preserved for multiple blade passages, indicating an improvement in vortex preservation when compared with previous work. Finally, further results from a single collective pitch case were presented to provide a more complete picture of the solver results.

Site Report for Phase III Archaeological Investigations at Reynolds Tavern (18AP23), 4 Church Circle, Annapolis, Maryland. 1982-1984

This report details the archaeology completed at Reynolds Tavern in the years 1982,1983, and 1984. It was completed in 2013, nearly 30 years after the excavation took place, using archival materials such as the draft interim reports, unit summary forms, original notes and photographs which are currently stored in the University Archives at Hornbake Library, at the University of Maryland, College Park. This report has been a collaboration across time and space, drawing from preliminary reports written by Anne Yenstch and Susan Mira in 1982 and Joe Dent and Beth Ford in 1983, as well as original notes from students of the field schools held there during those years, various analyses by scholars from many universities (including the University of Maryland, University of Georgia, and the College of William and Mary), and historical research by Nancy Baker. Thomas Cuddy began the writing of this report in 2002, completing the first three chapters in addition to the artifact analysis that led to the postexcavation identification of the African bundles in the Reynolds Tavern basement. This remarkable discovery was made along with Mark Leone of the University of Maryland, founder and director of Archaeology in Annapolis, who also served as the Principle Investigator during all three years of the Reynolds Tavern excavations. Dr. Leone contributed the fifth and final chapter to this report, the Conclusions and Recommendations, during its final compilation in 2013. The final report, including the fourth chapter on the archaeology itself, was written in part and compiled by Patricia Markert of the University of Maryland in the spring of 2013. Reynolds Tavern has been part of the landscape of Annapolis for two-hundred and fifty five years (at the time of the publication of this report). It sits on Church Circle facing St. Anne’s Church, and is a beautiful example of 18th century Georgian architecture as well one of the defining features of Historic Annapolis today. It currently operates as a popular restaurant and pub, but has served variously as a hat shop, a tavern, an inn, a library and a bank over time, among other things. Its long history contributes to its significance as an archaeological site, and also as a historic marker in present day Annapolis. The archaeology conducted at Reynolds Tavern shed light on life in 18th and 19th century Annapolis, illuminating details of the occupants’ lives through the material traces they left behind. These include an 18th century cobblestone road that ran diagonally through the Tavern’s yard, telling of the movement through early Annapolis; a large and intact well, which was found ii to contain a 19 foot wooden pipe; a large, ovular privy containing many of the objects used on a day to day basis at the Tavern or the structures around it; a subterranean brick storage feature in the basement of the Tavern, which may have been used by Reynolds during his days operating a hat shop; and also in the basement, two African caches of objects, providing a glimpse into West African spiritual practices alive in historic Annapolis and the presence of African American individuals at the Tavern in the 18th and 19th centuries. The purpose of this report is to detail these archaeological investigations and their findings, so that a public record will be available and the archaeology completed at Reynolds Tavern can continue to contribute to the history of Annapolis.