Linear growth rate for Kelvin-Helmholtz instability appearing in a moving mixing layer

In this paper, we study a generalization of the classical Kelvin-Helmholtz instability, which appears when two fluids move with different parallel velocity on each side of an interface. Considering here that this interface is moving and that L(0) is mixing layer width, this generalization takes into account a family of a continuous density profile and a continuous parallel velocity profile of the form (rho*(z/L(0)), (u*(z/L(0)), 0, 0)) with rho* (X) -> rho(+/-) and u*(X) -> u(+/-) for X -> +/-infinity. Under suitable assumptions on density rho(0) and velocity u(0), we prove that the linear instability growth rate in this case converges, when L(0) -> 0, to the classical growth rate of Kelvin-Helmholtz. We are able to calculate the next term in the asymptotic expansion in L(0) when L(0) -> 0 of the growth rate, under additional assumptions on rho(0) and u(0), hence obtaining a new approximation of the growth rate. This result on the growth rate of instability has been established rigorously, and takes into account general situations, such as the presence of a mixing region. The numerical approach is used to prove the existence of the linear growth rate of such instability.