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Area of Science:

  • Fluid Dynamics
  • Non-Newtonian Fluid Mechanics
  • Pattern Formation

Background:

  • Hele-Shaw flow is crucial for understanding fluid displacement in porous media.
  • Interfacial instabilities drive pattern formation, influencing flow behavior.
  • The role of non-Newtonian fluid rheology in these patterns is not fully understood.

Purpose of the Study:

  • To analytically investigate interfacial pattern formation in radial Hele-Shaw flow.
  • To determine how the power-law index of a displaced non-Newtonian fluid influences pattern morphology.
  • To develop a time-dependent injection strategy to control pattern development.

Main Methods:

  • Utilized a Darcy-law-like formulation for fluid displacement.
  • Employed perturbative analysis to study interface evolution.
  • Derived second-order mode-coupling equations to model perturbation amplitudes.

Main Results:

  • Shear-thinning displaced fluids lead to side-branching patterns.
  • Shear-thickening displaced fluids result in petal-like shapes with tip-splitting.
  • A time-dependent injection protocol successfully controlled finger proliferation.

Conclusions:

  • The rheology of the displaced non-Newtonian fluid dictates interfacial pattern morphology.
  • The proposed injection protocol offers control over pattern formation, generalizing to power-law fluids.
  • This research provides insights into controlling complex fluid interfaces in Hele-Shaw systems.