Effect of bulk viscosity on Kelvin-Helmholtz instability

An energy functional leading to a resolvable variational problem for determining the critical Reynolds number of laminar-turbulent transition Recr is constructed within the framework of the nonlinear energy stability theory of compressible flows. Asymptotic estimates containing the characteristic dependence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname{Re} _{cr} \sim \sqrt {\alpha + 4/3} (\alpha = \eta _b /\eta )$$ \end{document} in the main order are obtained for the stability of various modes of Couette compressible gas flow. The asymptotics considered are long-wave approximations. This suggests that the obtained dependence describes the effect of bulk viscosity on the large-scale vortex structures characteristic of Kelvin-Helmholtz instability.