Well-Posedness of the Infinite Prandtl Number Model for Convection With Temperature-Dependent Viscosity

Abstract

We establish the well-posedness of the infinite Prandtl number model for convection with temperature-dependent viscosity, free-slip boundary condition and zero horizontal fluxes.

Spatial decay bounds in a channel flow of 2‐D Boussinesq fluid with variable thermal diffusivity

“…In this paper, we consider the following Boussinesq equations, which take the form (see Diaz and Galiano, Gunzburger et al, and Turcotte and Schubert) where u = ( u 1 , u 2 ) and θ are the velocity and the temperature of the fluid, respectively. p is the hydrostatic pressures.…”

“…In this paper, we consider the following Boussinesq equations, which take the form (see Diaz and Galiano, 9 Gunzburger et al, 10 and Turcotte and Schubert 11 ) ∂ t u − μdivð∇uÞ þ u∇u − γgθ þ ∇p ¼ 0; divu ¼ 0;…”

Global regularity for a two‐dimensional nonlinear Boussinesq system

“…The Boussinesq system is a system of nonlinear partial differential equations that models the thermal convection and geophysical flows and plays an important role in the atmospheric sciences and oceanographic turbulence (e.g., ). In the two‐dimensional case, the standard Boussinesq system takes the form (e.g., ) where u = ( u 1 , u 2 ) is the velocity vector field, , θ ( x , t ), and P ( x , t ) denote the scalar temperature and pressure of the fluid, respectively. The unknown function θ e 2 represents the buoyancy force.…”

“…The Boussinesq system is a system of nonlinear partial differential equations that models the thermal convection and geophysical flows and plays an important role in the atmospheric sciences and oceanographic turbulence (e.g., [1][2][3]). In the two-dimensional case, the standard Boussinesq system takes the form (e.g., [4][5][6]) 8 < : u t div. ru/ C u ru C rP D Âe 2 , e 2 D .0, 1/, div u D 0, and further proved the global existence and large time asymptotic behavior.…”

Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

“…From the mathematical point of view it has recently been proven that convection problems in which viscosity is a function of temperature is a well posed problem [16,17] for dependences which are smooth bounded positive analytical functions, so it stands a good chance of solution on a computer using a stable algorithm. The variability in viscosity introduces strong couplings between the momentum and heat equations, as well as introducing important nonlinearities into the whole problem.…”