Development of tip-splitting and side-branching patterns in elastic fingering

Area of Science:

• Fluid Dynamics
• Soft Matter Physics
• Pattern Formation

Background:

• Viscous fingering in Hele-Shaw cells describes fluid-fluid interfaces driven by viscosity differences.
• Elastic fingering introduces membrane elasticity to the interface, leading to complex pattern evolution.
• Previous studies relied on complex numerical simulations to understand these phenomena.

Purpose of the Study:

• To analytically investigate the onset of pattern formation in elastic fingering.
• To develop a predictive model for interfacial patterns without extensive simulations.
• To elucidate the role of elastic properties in dictating fingering morphology.

Main Methods:

• A perturbative mode-coupling scheme was applied at the lowest nonlinear order.
• The analysis focused on the interaction of three Fourier modes: n, 2n, and 3n.
• A morphology diagram was constructed using two dimensionless parameters: rigidity fraction (C) and elastic-viscous ratio (Γ).

Main Results:

• The study identified key mechanisms for finger tip-splitting and side-branching.
• A parameter space map (morphology diagram) was generated, predicting pattern types.
• Visualization revealed an elastic weakening mechanism promoting finger growth in low-rigidity areas.

Conclusions:

• The perturbative approach provides significant insights into elastic fingering dynamics.
• The developed model simplifies the prediction of interfacial patterns in Hele-Shaw cells.
• Elastic properties play a crucial role in controlling the morphology of fluid instabilities.