ANALYSIS OF DETERMINISTIC AND STOCHASTIC IMPLICIT INTERFACE MODELS OF FLUID-INTERFACE INTERACTIONS

Open Access
Author:
Li, Manlin
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
February 05, 2010
Committee Members:
  • Qiang Du, Dissertation Advisor
  • Qiang Du, Committee Chair
  • Chun Liu, Committee Member
  • Ludmil Tomov Zikatanov, Committee Member
  • Xiantao Li, Committee Member
  • John Fricks, Committee Member
Keywords:
  • SIIM
  • phase field
  • implict interface
Abstract:
The work presented in this thesis focuses on the analysis of fluid implicit interface interaction models. Some rigorous theory is presented for a phase field Navier-Stokes vesicle-fluid interaction model in Chapter 2. The existence and uniqueness theorems of global weak solutions are proved. In Chapter 3, a consistent and rigorous derivation of some stochastic fluid-structure interaction models based on an implicit interface formulation of the stochastic immersed boundary method is presented. As dictated by the fluctuation-dissipation theorem, a proper noise has been derived to be incorporated into the deterministic hydrodynamic fluid-structure interaction models in either the phase field or level-set framework. The resulting stochastic system is referred to as stochastic implicit interface model, which not only captures the fluctuation effect near equilibrium but also provides an effective tool to model the complex interfacial morphology. Furthermore, the stochastic implicit interface model is also considered with a quasi-steady flow, which reduces the stochastic implicit interface model to a Langevin type equation of phase field (or level set) function with multiplicative noise. The mathematical analysis of the stochastic implicit interface models is presented in Chapter 4. The well-posedness of pathwise solutions are established and also a uniform control over solutions in probability is provided.