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Marginal stability and traveling fronts in two-phase mixtures
N G Cogan1, Matthew Donahue, Mark Whidden
1Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
Summary
This study introduces a minimal two-phase model for viscous materials. It accurately predicts traveling front velocity using marginal stability for high diffusion, but this fails at lower diffusion constants.
Area of Science:
- Multiphase flow dynamics
- Biophysical modeling
- Nonlinear systems analysis
Background:
- Complex material mixtures exhibit diverse behaviors like phase separation and traveling waves.
- Existing multiphase models vary significantly, necessitating simplified approaches for fundamental understanding.
- Minimal models are crucial for analyzing bifurcations and instabilities in complex systems.
Purpose of the Study:
- To develop and analyze a simplified two-phase system model.
- To investigate the dynamics of a traveling front separating distinct phases.
- To determine the velocity of this traveling front under varying conditions.
Main Methods:
- Formulation of a minimal two-phase system with viscous forces and osmotic response.
- Inclusion of a drag term for inter-phase interaction.
- Analysis of traveling front solutions and their stability.
Main Results:
- A traveling front emerges, separating a uniform unstable phase from a patterned phase-separated solution.
- Marginal stability provides a simple and accurate prediction for front velocity when diffusion is high.
- Linear prediction of front velocity fails for smaller diffusion constants, indicating a 'pushed' front regime.
Conclusions:
- The minimal model successfully captures essential dynamics of phase separation and front propagation.
- Marginal stability is a valid approach for predicting traveling front velocities in specific parameter regimes.
- The study highlights limitations of linear predictions in nonlinear systems with varying diffusion.
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