Control of radial fingering patterns: a weakly nonlinear approach

Area of Science:

• Fluid dynamics
• Pattern formation
• Nonlinear dynamics

Background:

• Radial Hele-Shaw cells with constant injection rates typically produce highly branched patterns with frequent finger tip-splitting.
• Lifting Hele-Shaw flow, characterized by a linearly time-increasing gap width, exhibits pattern morphology dominated by competing inward-moving fingers.

Purpose of the Study:

• To investigate methods for controlling pattern formation in radial Hele-Shaw flows.
• To analyze the mechanisms behind finger tip-splitting and finger competition.
• To develop a theoretical model for understanding these controlling mechanisms.

Main Methods:

• Mode-coupling theory was employed to analyze the fluid dynamics.
• An analytical theoretical model was developed to approach the problem.
• The model captures controlling mechanisms at weakly nonlinear stages.

Main Results:

• Finger tip-splitting in constant injection rate flows can be restrained by adjusting the injection rate.
• Finger competition in lifting Hele-Shaw flows can be restrained by adjusting the time-dependent gap width.
• The theoretical model successfully captures these controlling effects.

Conclusions:

• The study demonstrates analytical control over complex fluid patterns in Hele-Shaw cells.
• Injection rate and gap width are key parameters for manipulating pattern morphology.
• The findings offer insights into controlling instabilities in fluid systems.