Weakly nonlinear analysis of the Saffman-Taylor problem in a radially spreading fluid annulus

We study viscous fingering formation in an immiscible three-layer radial Hele-Shaw flow, where two coupled interfaces spread radially. More specifically, we examine how the initial distance d between the inner and outer interfaces (i.e., annulus' thickness) influences the shape of the emerging fingering patterns. Our theoretical analysis extends the perturbation theory beyond linear stability to a second-order mode-coupling theory. An important feature of our second-order perturbative approach is the fact that through the coupling of the appropriate Fourier modes, one is able to extract key analytical information about the morphology of the interface at the onset of nonlinearities. Under the circumstances where the inner interface is unstable and the outer one is stable, our theoretical results indicate that as d decreases, the coupling between the interfaces becomes stronger and the nearly matched final shapes exhibit the formation of wide fingers with bifurcated tips. However, if d is reduced further, we observe an unexpected change in the morphology of the patterns, where the conventional finger-splitting morphologies are replaced by polygonal-like structures with narrow fingers. The opposite scenario where the inner interface is stable and the outer one is unstable reveals two interesting situations: the onset of formation of drops due to the rupture of the intermediate layer of fluid, and an overall stabilization of the outer interface by the presence of the fluid annulus.