The Saffman–Taylor instability: From the linear to the circular geometry
Résumé
The Saffman–Taylor fingers are studied in cells that have the form of sectors of a disk. The less viscous fluid can be injected at the apex (divergent flow) or at the periphery (convergent flow). As in the linear geometry, at large velocities, a unique finger tends to occupy a well determined fraction λ of the cell angular width. This fraction is a function of the angle of the cell, being larger than 0.5 in the divergent case and smaller in the convergent case. In both cases these fractions tend linearly toward λ=0.5 when the angle of the cell tends to zero. In support of recent theories, these results show how the selection is changed when the geometry induces an increase or a decrease of the curvature of the profiles. The formation of fingers in the circular geometry is revisited. In a divergent flow, the circular front appears to break into independent parts so that each finger grows as if it were contained in a sector shaped cell. The rate of occupancy of the cell by one of the fluids as a function of the distance to the center is then discussed. Finally, the existence of the mathematical counterpart to the well-known Saffman–Taylor finger solutions in a nonparallel cell is discussed in the Appendix.