Effect of shear flow on the stability of domains in two-dimensional phase-separating binary fluids

We perform a linear stability analysis of extended domains in phase-separating fluids of equal viscosity in two dimensions. Using the coupled Cahn-Hilliard and Stokes equations, we derive analytically the stability eigenvalues for long wavelength fluctuations. In the quiescent state we find an unstable varicose mode that corresponds to an instability towards coarsening. This mode is stabilized when an external shear flow is imposed on the fluid. The effect of the shear is seen to be qualitatively similar to that found in experiments.