Narrow fingers in the Saffman-Taylor instability

Saffman-Taylor fingers with a relative width much smaller than the classical limit $\lambda =0.5\mathrm{}$ are found when a small isolated bubble is located at their tip. These solutions are members of a family found by Saffman and Taylor neglecting superficial tension. Recent theories have shown that when capillary forces are taken into account an unphysical cusplike singularity would appear at the tip of all the fingers with $\lambda <0.5\mathrm{}$. Conversely, here the replacement of the tip by a small bubble makes these solutions possible. At large velocity these fingers show dendritic instability.