BENJAMIN-FEIR AND ECKHAUS INSTABILITIES WITH GALILEAN INVARIANCE - THE CASE OF INTERFACIAL WAVES IN VISCOUS SHEAR FLOWS

We consider the stability of interfacial waves in a two-layer viscous shear flow. The weakly nonlinear dynamics are governed by a set of two coupled amplitude equations (Renardy and Renardy, 1993): a complex Gi nzburg-Landau equation for the travelling wave with finite wavenumber kc, and a Burgers-type equation for the neutral mode with wavenumber k = 0. We study here the linear stability, against long wavelength dist urbances, of the travelling wave solutions of these equations. For the travelling wave with k = k,, the Lange & Newell criterion for the Ben jamin-Feir instability is modified by the coupling, and a new kind of instability may arise from the neutral mode k = 0. For travelling wave s with k not equal k(c), several wavenumber bands may be Eckhaus-unsta ble, with growth rate much larger than for the classical Eckhaus insta bility. A physical mechanism for this instability is proposed. Beyond the case of interfacial waves in viscous shear flows, the results appl y for physical systems with translational and galilean invariances, an d no-space-reflection symmetry. (C) Elsevier, Paris.