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