  We study self-similar viscous fingering for the case of divergent flow within
a wedge-shaped Hele-Shaw cell. Previous authors have conjectured the existence
of a countably-infinite number of selected solutions, each distinguished by a
different value of the relative finger angle. Interestingly, the associated
solution branches have been posited to merge and disappear in pairs as the
surface tension decreases. We demonstrate how exponential asymptotics is used
to derive the selection mechanism. In addition, asymptotic predictions of the
finger-to-wedge angle are given for different sized wedges and surface-tension
values. The merging of solution branches is explained; this feature is
qualitatively different to the case of classic Saffman-Taylor viscous fingering
in a parallel channel configuration. The phenomena of branch merging in our
self-similar problem relates to tip splitting instabilities in time-dependent
flows in a circular geometry, where the viscous fingers destabilise and divide
in two.
