Tensorial modeling of an oscillating and cavitating microshell used as a contrast agent

1. Tensorial modeling of an oscillating and cavitatingmicroshell used as a contrast agent

2. Objectives • Formulate an equation for the shell with tensorial analysis using the Mooney Rivlinhyperelastic model. • Determine the parametric relations • Solve the equation to predict the behaviour of the system

3. Mathematical model • Using the Cauchy Stress equation together with the Navier-Stokes equations with their conditions and taking into account spherical symmetry for a thin microshell.

4. Mathematicalmodel • The transient Cauchy Eq. With the stresses

5. Mathematicalmodel • For a Mooney Rivlin material we have the elastic potential.

6. Cauchy’s Eq. can be integrated as:

7. Mathematicalmodel • At the same time we have the R-P Eq.

8. Mathematicalmodel • The stresses at both inside as a gas and outside of the shell as a liquid, must stand equilibrium.

9. Mathematicalmodel • With both equations and the balance equations we have:

10. Mathematicalmodel • Introducing the nondimensional variables

11. Mathematicalmodel • We can rewrite the dimensionless equation as

12. Mathematicalmodel • For the last equation the initial conditions are • And the dimensionless parameters are

13. Results • For typical experimental physical values

14. Results

15. Results

16. Results

17. Results

18. Conclusions • We obtained a simple model for a Money Rivlin shell • The thin shell approach led to a very close interval in the parameters, which showed two modes of collapse. • The violent collapse • The ever growing collapse, we suppose an elastic response from the shell deformation

19. Conclusions • The main parameters P and bA showed to be the main drivers of the collapse however the elastic parameters can shorten or prolong the collapse • The linearized equation shows this competence

20. Conclusions • Further studies on the frequency and stability of the equation should be done