A linear stability analysis as well as nonlinear simulations are performed in order to analyse the coupling between the directional solidification of a binary alloy and the flow in its melt. An incompressible, potential flow model is assumed, whose validity is tested through comparisons with the accompanying experiments of Zhang & Maxworthy (2002) in a Hele-Shaw cell. The linear stability analysis predicts that a uniform flow parallel to the interface reduces the growth rates of directional solidification instabilities. In addition, the dominant wavelength is shifted to larger values by the flow, and a small propagation velocity in the downstream direction is observed. These findings are confirmed by the nonlinear simulations as well. While the overall stabilization is confirmed by the experiments, the predicted values of the dominant wavenumber and its growth rate are too high by factors of two and four, respectively. These differences are attributed to the existence of a velocity boundary layer in the melt, which strongly affects the lateral solute transport.