A New Approach to the Rayleigh-Taylor Instability

Area of Science:

• Fluid dynamics
• Partial differential equations
• Mathematical physics

Background:

• The inhomogeneous incompressible Euler equations model fluid interfaces under gravity.
• Rayleigh-Taylor instability occurs when a heavier fluid is above a lighter fluid, leading to turbulent mixing.
• Understanding weak solutions is crucial for modeling complex fluid behaviors.

Purpose of the Study:

• To formulate a general criterion for the existence of infinitely many weak solutions for the Euler equations.
• To analyze turbulent mixing in two-fluid systems with different densities.
• To investigate the impact of relaxing constitutive laws on fluid behavior.

Main Methods:

• Modeling the Euler equations as a differential inclusion.
• Considering the relaxation of constitutive laws.
• Developing a criterion for existence of weak solutions.
• Analyzing the Rayleigh-Taylor instability configuration.

Main Results:

• A general criterion for infinitely many weak solutions is established.
• The criterion is verified for the Rayleigh-Taylor instability with a heavier fluid initially above a lighter fluid.
• Specific examples show mixing zones growing quadratically in time for ultra-high Atwood numbers.

Conclusions:

• The study provides a theoretical framework for understanding turbulent mixing in stratified fluids.
• The findings are particularly relevant for scenarios exhibiting strong density differences (ultra-high Atwood numbers).
• The relaxation of constitutive laws offers a pathway to model complex, turbulent phenomena.