Nonlinear theory of Kelvin-Helmholtz instability (Q5905521)

The present paper deals with the instability of shear discontinuities in a flow of an unbounded fluid, in particular with the nonlinear evolution of this instability. The results of an analytic theory in the long- wavelength approximation for the case of identical fluids on the two sides of the shear are generalized to the case of different fluids. In the approximation of long-wavelength perturbation of irrotational flows of incompressible fluids, nonlinear equations are obtained which describe perturbations of the interface of fluids with a tangential discontinuity of the velocity. These equations belong to the class of quasi-Chaplygin (quasigas) equations with the azimuthal wave number \(m=1/2\), and describe a large class of nonlinear unstable media (more than 50). The latter are called quasi-Chaplygin (quasigas) media, the general theory of which developed elsewhere for \(m=-1/2\). By means of a hodograph transformation it can be shown that these equation \((m=1/2)\) can be reduced to the two- dimensional Laplace equation. The authors demonstrate that there exists a general relation between quasi-Chaplygin media with azimuthal numbers of opposite signs \(\pm m\). The basic equation of these media is the so- called Darboux equation. These results advance the general theory of quasi-Chaplygin media. Not only periodic perturbations but also local ones are considered.