On the absence of dissipative instability of negative energy waves in hydrodynamic shear flows

Stability criterion for the surface gravity capillary waves in a flowing two-layered fluid system with viscous dissipation is investigated. It is seen that the dissipative instability of negative energy waves is absent,- contrary to what earlier authors have concluded. Their error is identified to arise from an erroneous choice of the dissipation law, in which the wave profile velocity is wrongly equated to the particle velocity. Our corrected dissipation law is also shown to restore Galilean invariance to the stability condition of the system.