  The three-layer Saffman-Taylor problem introduces two coupled moving
interfaces separating the three fluids. A very recent weakly nonlinear analysis
of this problem in a radial Hele-Shaw cell setup has shown that the
morphologies of the emerging fingering patterns strongly depend on the initial
thickness of the intermediate layer connecting the two interfaces. Here we go
beyond the weakly nonlinear regime and explore full nonlinear interfacial
dynamics using a spectrally accurate boundary integral method. We quantify the
nonlinear instability of both interfaces as the relevant physical parameters
(e.g., viscosities and surface tensions) are varied and show that our nonlinear
computations are in good agreement with the experimental observations and the
weakly nonlinear analysis. Nonlinear simulations reveal that due to the
existence of a second interface, the classical highly branched morphologies are
replaced by less unstable structures in which finger tip-splitting and finger
competition phenomena are evidently restrained as the initial annulus'
thickness is reduced. In addition, these novels patterns develop fingers with a
series of low-amplitude bumps at their tips, associated with the enhanced
growth of high-frequency modes promoted by the increasing coupling strength of
interfaces.
