Title page for ETD etd-08212007-125009

Type of Document

Dissertation

Author

Saka, Yuki

Author's Email Address

saka@scs.fsu.edu

URN

etd-08212007-125009

Title

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.

Degree

Doctor of Philosophy

Department

Mathematics, Department of

Advisory Committee

Advisor Name	Title
Max D. Gunzburger	Committee Chair
Xiaoming Wang	Committee Co-Chair
Anter El-Azab	Committee Member
Janet Peterson	Committee Member
Xiaoqiang Wang	Committee Member

Keywords

Partial Differential Equations
Turbulence
Fourier Analysis
Navier-Stokes Equations

Date of Defense

2007-08-09

Availability

unrestricted

Abstract

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.

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