Instabilities in rapid directional solidification under weak flow

We examine a rapidly solidifying binary alloy under directional solidification with nonequilibrium interfacial thermodynamics viz. the segregation coefficient and the liquidus slope are speed dependent and attachment-kinetic effects are present. Both of these effects alone give rise to (steady) cellular instabilities, mode S, and a pulsatile instability, mode P. We examine how weak imposed boundary-layer flow of magnitude |V| affects these instabilities. For small |V|, mode S becomes a traveling and the flow stabilizes (destabilizes) the interface for small (large) surface energies. For small |V|, mode P has a critical wave number that shifts from zero to nonzero giving spatial structure. The flow promotes this instability and the frequencies of the complex conjugate pairs each increase (decrease) with flow for large (small) wave numbers. These results are obtained by regular perturbation theory in powers of V far from the point where the neutral curves cross, but requires a modified expansion in powers of V^{1/3} near the crossing. A uniform composite expansion is then obtained valid for all small |V|.