Mode-coupling approach to near-cuspidal patterns in planar fluid flows.

Area of Science:

• Fluid dynamics
• Interface phenomena
• Non-Newtonian fluid mechanics

Background:

• Investigating the dynamics of interfaces between fluids is crucial for understanding various natural and industrial processes.
• Two-dimensional Stokes flow provides a simplified yet insightful model for studying low-Reynolds-number fluid behavior.

Purpose of the Study:

• To analyze the evolution of interfaces between two Newtonian fluids with differing viscosities under suction or injection.
• To explore interfacial pattern formation in the transitional regime between linear and nonlinear flow dynamics.

Main Methods:

• Utilizing a second-order mode-coupling theory to examine interfacial morphology.
• Comparing theoretical predictions with established results from conformal mapping techniques.

Main Results:

• Linear analysis shows injection-driven interfaces are stable, while suction-driven interfaces are unstable, with instability growth rate independent of viscosity contrast.
• Second-order analysis reveals viscosity contrast significantly influences interfacial morphology.
• The theory successfully reproduces near-cusp pattern-forming structures.

Conclusions:

• Viscosity contrast plays a critical role in determining interfacial pattern morphology beyond the linear regime.
• The developed mode-coupling theory is valid across all viscosity contrasts and offers insights into complex pattern formation.