Analytic Description of the Viscous Fingering Interface in a Hele-Shaw Cell

The determination of the interface shape between two fluids during the process of viscous fingering (Saffman--Taylor instability) in a Hele-Shaw cell is addressed here. The parametric equations describing this interface are obtained in an explicit analytic form. With these results, a number of interface shapes are presented, including shapes with opposite sides in contact that are considered as the onset of droplet pinch-off.