Elastic fingering in three dimensions

Recent studies on quasi-two-dimensional (2D) fluid flows in Hele-Shaw cells revealed the emergence of the so-called elastic fingering phenomenon. This pattern-forming process takes place when a reaction occurs at the fluid-fluid interface, transforming it into an elastic gel-like boundary. The interplay of viscous and elastic forces leads to the development of pattern morphologies significantly different from those seen in the conventional, purely hydrodynamic viscous fingering problem. In this work, we investigate the occurrence of elastic fingering for radial fluid displacements in a 3D uniform porous medium. A perturbative third-order mode-coupling approach is employed to examine how the combined action of viscous and elastic effects influences the linear stability of the interface, and the weakly nonlinear pattern formation in such a 3D environment. In addition, a variational method is used to determine how to minimize the growth of interfacial perturbation amplitudes via a time-dependent injection rate scheme.