  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 which 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.

Statistical Mechanics;Soft Condensed Matter