It is often assumed that the molten regions of planetary interiors can be modeled as a compacting viscous matrix permeated by a less viscous melt phase. In order to assess the true behaviour, it is important to test as many different aspects of these models as possible. In the calculations presented here, the matrix viscosity is made to decrease as the proportion of melt increases. New results in two dimensions show that if only gravity drives the flow, pockets of melt travel upwards as solitary waves, and the behaviour is similar to the well known constant viscosity case. However, when an external shear strain is applied, an instability occurs and melt flows into elongated pockets, or veins, perpendicular to the least compressive stress. Whilst veins much greater than a compaction length in width are suppressed, their minimum width is not clearly constrained. The growth rate of veins is directly proportional to the applied strain rate, so that their degree of development is proportional to the total strain. It is suggested that under mid‐ocean ridges, the total strain may be enough for such veins to be important.