Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics

Area of Science:

• Fluid dynamics
• Acoustics
• Nonlinear dynamics

Background:

• Understanding bubble dynamics is crucial in various fields, including acoustics and fluid mechanics.
• Previous models often simplified bubble interactions or excluded external acoustic influences.

Purpose of the Study:

• To develop and compare two distinct mathematical models for interacting bubble dynamics.
• To incorporate the effects of external acoustic sources on bubble behavior.
• To assess the numerical stability of different modeling approaches.

Main Methods:

• Formulation of a coupled system of second-order differential equations using Lagrangian mechanics.
• Development of a first-order system of differential equations based on Hamiltonian mechanics.
• Inclusion of bubble pulsation, translation, and external acoustic field effects in both models.

Main Results:

• Both models successfully describe the pulsation and translation of multiple interacting spherical bubbles.
• The Hamiltonian model demonstrates superior numerical stability for a large number of interacting bubbles.
• External acoustic sources influence bubble dynamics through acoustic pressure, its gradient, and particle velocity.

Conclusions:

• The Hamiltonian mechanics approach provides a more stable framework for simulating complex interacting bubble systems.
• The derived models offer a comprehensive description of bubble dynamics under external acoustic forcing.
• The findings highlight the importance of including particle velocity in models of bubble acoustics.