Dr Nehal Dash - Encapsulated bubbles

Mechanics of Encapsulated Microbubbles: Continuum-Based Mathematical Models

During my doctoral research, I have worked extensively on the nonlinear mechanics and dynamics of gas-filled encapsulated (viscoelastic shelled) microbubble (used as contrast agents in ultrasound imaging and vehicles in targeted drug delivery applications) with interface energy formulation to capture the essential interface mechanics using the Steigmann-Ogden surface energy theory. This model precisely mimicked the physical experimental settings, provided a better fit with the existing experimental data, and explained certain critical aspects of encapsulated microbubbles, which remained hitherto unexplained. Further, this model is extended to study the nonspherical axisymmetric deformation of microbubbles, and carry out a conditional stability analysis to determine the finite amplitude shape mode oscillations using the perturbation method.

A detailed overview of the work and the results stemming from it are listed below.

Radial dynamics of an encapsulated microbubble with interface energy

In this work a mathematical model based on interface energy is proposed within the framework of surface continuum mechanics to study the dynamics of encapsulated bubbles. The interface model naturally induces a residual stress field into the bulk of the bubble, with possible expansion or shrinkage from a stress-free configuration to a natural equilibrium configuration. The significant influence of interface area strain and the coupled effect of stretch and curvature is observed in the numerical simulations based on constrained optimization. Due to the bending rigidity related to additional terms, the dynamic interface tension can become negative, but not due to the interface area strain. The coupled effect of interface strain and curvature term observed is new and plays a dominant role in the dominant compression behaviour of encapsulated bubbles observed in the experiments. The present model is validated by fitting the experimental data of 1.7μm, 1.4μm and 1μm radii bubbles by calculating the optimized parameters. This work also highlights the role of interface parameters and natural configuration gas pressure in estimating the size-independent viscoelastic material properties of encapsulated bubbles with interesting future developments.

For more details, please refer to:
Dash, N. and Tamadapu, G., Journal of Fluid Mechanics (2022).

Describing the dynamics of a nonlinear viscoelastic shelled microbubble with an interface energy model

The present work introduces an interesting revamp to the recently proposed interface energy model [N. Dash and G. Tamadapu, J. Fluid Mech. 932, A26 (2022)] for gas-filled encapsulated bubbles (EBs) suspended in a viscous fluid. Here, the elastic and viscous parts of the viscoelastic shell material are described by the Gent hyperelastic material model and a polymer solute following upper-convected Maxwell (UCM) constitutive relations, respectively. Using the aforementioned framework, the integrodifferential type governing equation has been derived, and the physical features of the radial dynamics of the EB model are studied in detail using numerical simulations. The nonlinear behavior and the underlying implications of the newly introduced interface energy model for EBs are also investigated. It was observed that the interface parameters arising from the interface energy formulation and the Gent material model collectively introduce a stiffening effect into the EB model and the extension limit parameter at its lower values affects the radial dynamics of the bubble. Analysis has been carried out at different relaxation time scales, where the viscoelastic shell material resembles a fluid-like or solid-like behavior. The UCM-type viscous part of the viscoelastic shell material introduces strong nonlinear effects into the bubble model and significantly influences the EB’s behavior. For the present model, a detailed study has been conducted to capture the dynamic behavior of the bubble through the time series curves, phase space analysis, and the nonlinear frequency response of the bubble.

For more details, please refer to:
Dash, N. and Tamadapu, G., Journal of Applied Physics (2022).

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.

For more details, please refer to:
Dash, N. and Tamadapu, G. The Journal of the Acoustical Society of America (2024).

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