Simple model for linear and nonlinear mixing at unstable fluid interfaces in spherical geometry

Area of Science:

• Fluid Dynamics
• Plasma Physics
• Astrophysical Fluid Flows

Background:

• Previous models successfully predicted mixing at planar fluid interfaces under variable acceleration.
• Extending these models to spherical geometries is crucial for understanding complex fluid phenomena.

Purpose of the Study:

• To develop an analogous model for predicting fluid mixing in spherical shells with time-dependent interface radii.
• To provide a framework for analyzing nonlinear mixing in spherical systems.

Main Methods:

• A heuristic kinetic energy expression, adapted from linear perturbation theory, was employed.
• Dynamically renormalized effective wavelengths were used to capture nonlinear effects.
• Lagrange's equations were utilized to derive an equation of motion for the mixing layer.

Main Results:

• The derived evolution equation correctly reduces to Plesset's equation for small perturbations.
• The model accurately reproduces results from the planar interface model in the limit of large radii.
• Conservation properties of the model were established and a numerical scheme was proposed.

Conclusions:

• The new model offers a robust method for analyzing fluid mixing in spherical shells.
• It successfully extends the understanding of mixing dynamics from planar to spherical geometries.
• The proposed numerical scheme ensures the preservation of crucial physical properties.