Abstract
In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if T' is of size at most nu ^{1/3} in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh–Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.
Highlights
The Boussinesq equations are a standard approximate model of heat transfer in fluids and are given by a coupled system of the Navier–Stokes equations and a dissipative transport equation for the temperature density:∂t v + v · ∇v + ∇ p =v + θ e2, ∂t θ + v · ∇θ =θ, ∇ · v = 0.Communicated by Charles R
We further show that for affine T and full vertical dissipation the same stability results hold
Building on the results of Lemma 2.2 for a combination of Couette flow and an unstable affine temperature profile, we consider the problem with partial dissipation
Summary
The Boussinesq equations are a standard approximate model of heat transfer in (viscous) fluids and are given by a coupled system of the Navier–Stokes equations and a dissipative transport equation for the temperature density:. ∂t v + v · ∇v + ∇ p = (νx ∂x2 + νy∂y2)v + θ e2, ∂t θ + v · ∇θ = (μx ∂x2 + μy∂y2)θ, ∇ · v = 0.
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