Instability of co-flow in a Hele-Shaw cell with cross-flow varying thickness

We analyse the stability of interface between two immiscible fluids both flowing in horizontal direction a thin cell with vertically varying gap width. The dispersion relation for growth rate each mode is derived. approximately determined by sign simple expression, which incorporates density difference and effect surface tension along- cross-cell directions. latter arises from channel width: if non-wetting fluid thinner part channel, unstable as it will preferentially migrate into thicker part. may suppress or complement this effect. system always stable to sufficiently large wavenumbers owing along-flow component tension.