Theoretical Results on Steady Convective Flows between Horizontal Coaxial Cylinders

Exact Poincaré constants in two‐dimensional annuli

“…see, e.g., the appendix of [10]. After the transformation to σ we find the equivalent zeros κ 1,L (σ ) and κ 1,S (σ ) as the smallest positive solutions of…”

“…An analytic approach to narrow down possible regions for bifurcations can be found in and references therein. Moreover, using the theory in , for small , a lower bound for the smallest critical Rayleigh number can be immediately estimated for the Stokes problems as treated, e.g., in . As is to be expected, the Poincaré constant defined in Eq.…”

Bénard Problem for Slightly Compressible Fluids: Existence and Nonlinear Stability in 3D

“…As is well known, one of the most accepted models in the study of convection problems in fluid mechanics is the socalled Oberbeck-Boussinesq approximation (hereafter denoted by O-B) [1][2][3][4][5]. e latter is characterized by the fact that one keeps the fluid incompressible and allows for density changes only in the buoyancy term of the linear momentum equation, by assuming a linear dependence on temperature.…”

Existence and nonlinear stability of convective solutions for almost compressible fluids in Bénard problem

“…In modeling convective phenomena, almost all available results of existence, stability etc., are achieved in the Oberbeck-Boussinesq (O-B) approximation [1,8,11,18,15,17,7]. As is well known, in spite of the need of compressibility to produce convection , the peculiarity of this approximation consists in keeping the incompressibility hypothesis ∇ • v = 0 (with v velocity field of the fluid), while allowing for (linear) variation of density with temperature only in the term involving the external force (gravity).…”

“…18) with Neumann conditions at z = 0, 1 and x−periodic conditions: if w • n = 0 and w ∈ W1,2 (Ω 0 ), then Π 2 ≤ c(β) w 2 . (2.19)where c(β) is a constant increasing with β and bounded from below.…”