Nonspherical oscillations of an encapsulated microbubble with interface energy under the acoustic field

Spherical instability in acoustically driven encapsulated microbubbles (EBs) suspended in a fluid can trigger nonspherical oscillations within them. We apply the interface energy model [N. Dash and G. Tamadapu, J. Fluid Mech. 932, A26 (2022b)] to investigate nonspherical oscillations of smaller radius microbubbles encapsulated with a viscoelastic shell membrane under acoustic field. Using the Lagrangian energy formulation, coupled governing equations for spherical and nonspherical modes are derived, incorporating interface energy effects, shell elasticity, and viscosity. Numerical simulations of governing equations revealed that the parametrically forced even mode excites even modes, while the odd modes excite both even and odd modes. The model demonstrates that finite amplitude nonspherical oscillations are identifiable in smaller radius EBs only when the interface parameters are introduced into the model; otherwise, they are not. Realizing that nonlinear mode coupling is responsible for saturation of instability resulting in stable nonspherical oscillations, we perform a steady-state and stability analysis using the slow-time equations obtained from Krylov-Bogoliubov perturbation method. Analytical expressions for modal amplitudes and stability thresholds are derived in terms of interface and material parameters. The stability curves are invaluable in determining the precise range of excitation pressure and frequency values required for the EB to exhibit finite amplitude nonspherical oscillations.