Abstract
In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain mathbb {T}times mathbb {R}. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their L^2 norm grows as t^{1/2} and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order nu ^{-1/6} (with nu ^{-1} being proportional to the Reynolds number) on a time-scale nu ^{-1/3}, after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.
Highlights
We consider the isentropic compressible Navier-Stokes system∂t ρ + div(ρu) = 0, for (x, y) ∈ T × R, t ≥ 0, (1.1) ∂t + div(ρu ⊗ u) 1 M2 ∇ p(ρ)
Trefethen et al [58] observed that a common feature in these problems is the non-normality of the operators involved. This implies the possibility of large transient growths that can take out the dynamics from the linear regime before the stability mechanisms takes over
Due to the observed transient growths, a question of great interest in the viscous problem is a quantification of transition thresholds, namely how small the initial perturbation has to be with respect to the viscosity parameter
Summary
Trefethen et al [58] observed that a common feature in these problems is the non-normality of the operators involved This implies the possibility of large transient growths (which are not captured via a pure eigenvalue analysis) that can take out the dynamics from the linear regime before the stability mechanisms takes over. Due to the observed transient growths, a question of great interest in the viscous problem is a quantification of transition thresholds, namely how small the initial perturbation has to be with respect to the viscosity parameter On this side, several numerical studies predicted a power law dependence and estimated the exponents below which stability is possible. From this equation, appealing to some formal approximation, they deduce an instability phenomenon that appears due to the compressibility of the flow
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