Analysis of Two Partial Differential Equation Models in Fluid Mechanics: Nonlinear Spectral Eddy-Viscosity Model of Turbulence and Infinite-Prandtl-Number Model of Mantle Convection

• Other Authors: Saka, Yuki, 1980- (authoraut)
• Format: Others
• Language: English; English
• Published: Florida State University
• Subjects: Mathematics
• Online Access: http://purl.flvc.org/fsu/fd/FSU_migr_etd-2108

This thesis presents two problems in the mathematical and numerical analysis of partial differential equations modeling fluids. The first is related to modeling of turbulence phenomena. One of the objectives in simulating turbulence is to capture the large scale structures in the flow without explicitly resolving the small scales numerically. This is generally accomplished by adding regularization terms to the Navier-Stokes equations. In this thesis, we examine the spectral viscosity models in which only the high-frequency spectral modes are regularized. The objective is to retain the large-scale dynamics while modeling the turbulent fluctuations accurately. The spectral regularization introduces a host of parameters to the model. In this thesis, we rigorously justify effective choices of parameters. The other problem is related to modeling of the mantle flow in the Earth's interior. We study a model equation derived from the Boussinesq equation where the Prandtl number is taken to infinity. This essentially models the flow under the assumption of a large viscosity limit. The novelty in our problem formulation is that the viscosity depends on the temperature field, which makes the mathematical analysis non-trivial. Compared to the constant viscosity case, variable viscosity introduces a second-order nonlinearity which makes the mathematical question of well-posedness more challenging. Here, we prove this using tools from the regularity theory of parabolic partial differential equations. === A Dissertation Submitted to the Department of Mathematics in Partial FulfiLlment of the Requirements for the Degree of Doctor of Philosophy.. === Fall Semester, 2007. === August 9, 2007. === Partial Differential Equations, Turbulence, Fourier Analysis, Navier-Stokes Equations === Includes bibliographical references. === Max D. Gunzburger, Professor Co-Directing Dissertation; Xiaoming Wang, Professor Co-Directing Dissertation; Anter El-Azab, Outside Committee Member; Janet Peterson, Committee Member; Xiaoqiang Wang, Committee Member.