Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

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 T × 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. Instead, the solenoidal component of the velocity field experience inviscid damping, meaning that 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 ν −1/6 (with ν −1 being proportional to the Reynolds number) on a time-scale ν −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 fluid, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

Linear stability of the Couette flow for the non-isentropic compressible fluid

“…Zeng et al [38] considered the linear stability of the three dimensional isentropic compressible Navier-Stokes equations on T × R × T. They proved the enhanced dissipation phenomenon and the lift-up phenomenon around the Couette flow (y, 0, 0) ⊤ . The motivation of the present paper is to generalize the results obtained by Antonelli et al [1], [2] to the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity.…”

“…The equations (1.1) then express respectively the conservation of mass, the balance of momentum, and the balance of energy under internal pressure, viscosity forces, and the conduction of thermal energy. A comprehensive understanding of the stability of compressible or incompressible shear flows is a fundamental problem in fluid mechanics and has been the subject of both theoretical and practical interest in astrophysics and engineering, see [1], [2], [6]- [9], [14]- [25], [27], [29]- [33], [38], [39] for the compressible fluid and [3]- [5], [10]- [13], [26], [28], [34]- [37] for incompressible fluid. The aim of the present paper is to study the long-time asymptotic behaviour of the linearized non-isentropic compressible Navier-Stokes equations around the Couette flow.…”

“…Due to the complicate form of ϑ sh (y), to study the long-time asymptotic behaviour of (1.1) around the stationary solution defined in (1.2) and (1.4) is a very difficult problem. To our best knowledge, there are few results in this direction, see [1], [2], [6], [7], [8], [15]- [25], [31], [33] .…”

“…They shown that the plane Couette flow is asymptotically stable for small perturbation provided that the slip length, Reynolds and Mach numbers satisfy some restricted relation. Recently, Antonelli et al [1] studied the linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain T × R. Later then, they in [2] also studied the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic inviscid or viscous compressible fluid. Moreover, in the inviscid case, they proved the inviscid damping for the solenoidal component of the velocity field and Lyapunov type instability for the density and the irrotational component of the velocity field.…”

Linear stability analysis of the Couette flow for the two dimensional non-isentropic compressible Euler equations

“…Inspired by [1] and [2], in the present paper, we are interested in the long-time asymptotic behaviour of the linearized two dimensional non-isentropic compressible Euler equations in a domain T × R. The governing equations (in non-dimensional variables) are ∂̺ ∂t + u • ∇̺ + ̺div u = 0, (1.1)…”