Spatiotemporal instability of a shear-imposed viscous flow

We study the linear spatiotemporal instability of a two-dimensional gravity-driven viscous fluid flow where the fluid surface is subjected to an imposed shear stress. The fourth order Orr-Sommerfeld boundary value problem is derived and solved numerically up to moderate values of the Reynolds number. Numerical solution based on AUTO07p identifies four spatial branches, viz., I, II, III, and IV, where the spatial branches I, II, and IV lie in the upper half zone, while the spatial branch III lies in the lower half zone of the complex wavenumber plane. The spatial growth rate -k(i) corresponding to branch I becomes stronger as long as the imposed shear stress increases and ensures a destabilizing effect. Furthermore, the spatial branch I enters in the lower half zone of the complex wavenumber plane as soon as the temporal growth rate omega(i) decreases and may collide with other spatial branch lying in the lower half zone of the complex wavenumber plane. Moreover, a study of absolute and convective instabilities is carried out within the frameworks of saddle point technique and collision criterion. The saddle point technique provides only one unstable branch of the unstable wavepacket, while the collision criterion provides two unstable branches of the wavepacket. The unstable range of the wavepacket with ray velocity enhances in the presence of imposed shear stress. It is observed that the shear-imposed fluid flow is convectively unstable. In addition, the simplified second order two-equation model is developed for a shear-imposed flow in terms of the local fluid layer thickness and local flow rate, which in fact renders three spatial branches rather than four. However, the two-equation model recovers the physically relevant spatial branch I very well. Finally, nonlinear spatiotemporal simulation of the two-equation model displays a formation of the regular train of solitary waves downstream at low forcing frequency.