Buoyancy-Driven Flows With Navier-Slip Boundary Conditions

TL;DR

This dissertation investigates buoyancy-driven flows with Navier-slip boundary conditions, deriving bounds on heat transfer and analyzing long-term behavior, including convergence to equilibrium, for Rayleigh-Bénard convection and related systems.

Contribution

It introduces the analysis of buoyancy-driven flows with Navier-slip conditions, deriving bounds on heat transfer and establishing long-time regularity and convergence results.

Findings

Bounds on Nusselt number for various boundary conditions.

Uniform regularity estimates for non-diffusive systems.

Solutions converge to hydrostatic equilibrium over time.

Abstract

In this dissertation two-dimensional buoyancy-driven flows are investigated. While usually the Navier-Stokes equations are equipped with no-slip boundary conditions here we focus on the Navier-slip conditions that, depending on the system at hand, better reflect the physical behavior. In particular, we study two systems, Rayleigh-B\'enard convection and a closely related problem without thermal diffusion. In the former, bounds on the vertical heat transfer, given by the Nusselt number, with respect to the strength of the buoyancy force, characterized by the Rayleigh number, are derived. These bounds hold for a broad range of applications, allowing for non-flat boundaries, any sufficiently smooth positive slip coefficient, and are valid over all ranges of the Prandtl number, a system parameter determined by the fluid. For the thermally non-diffusive system, regularity estimates are proven. Up to a certain order, these bounds hold uniformly in time, which, combined with estimates for their growth, provide insight into the long-time behavior. In particular, solutions converge to the hydrostatic equilibrium, where the fluid's velocity vanishes and the buoyancy force is balanced by the pressure gradient.