2021/01/05 by Paolo Antonelli, Michele Dolce, Antonelli, Paolo +3 · 1 citation
Engineering · Physics and Astronomy · Mathematics · #Fluid Dynamics and Turbulent Flows #Advanced Thermodynamics and Statistical Mechanics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2101.01696
In this paper, we study the linear stability properties of perturbations\naround the homogeneous Couette flow for a 2D isentropic compressible fluid in\nthe domain mathbbT\× \ℝ. In the inviscid case there is a\ngeneric Lyapunov type instability for the density and the irrotational\ncomponent of the velocity field. More precisely, we prove that their L2 norm\ngrows as t1/2 and this confirms previous observations in the physics\nliterature. Instead, the solenoidal component of the velocity field experience\ninviscid damping, meaning that it decays to zero even in the absence of\nviscosity. For a viscous compressible fluid, we show that the perturbations may\nhave a transient growth of order \ν-1/6 (with \ν-1 being\nproportional to the Reynolds number) on a time-scale \ν-1/3, after which\nit decays exponentially fast. This phenomenon is also called enhanced\ndissipation and our result appears to be the first to detect this mechanism for\na compressible fluid, where an exponential decay for the density is not a\npriori trivial given the absence of dissipation in the continuity equation.\n