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

Objectives

• Formulate an equation for the shell with tensorial analysis using the Mooney Rivlin hyperelastic model.

• Determine the parametric relations• Solve the equation to predict the behaviour of

the system

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.

Mathematical model

• The transient Cauchy Eq. With the stresses

Mathematical model

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

Cauchy’s Eq. can be integrated as:

Mathematical model

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

Mathematical model

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

Mathematical model

• With both equations and the balance equations we have:

Mathematical model

• Introducing the nondimensional variables

Mathematical model

• We can rewrite the dimensionless equation as

Mathematical model

• For the last equation the initial conditions are

• And the dimensionless parameters are

Results

• For typical experimental physical values

Results

Results

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

=0.6, =0.4=6, =4

Results

Results

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

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

Conclusions

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