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The Rayleigh-Taylor instability with foams

Antoine Bret, Audrey DeVault, Skylar Dannhoff, Maria Gatu Johnson, Chikang Li, Johan Frenje

TL;DR

This work analyzes the Rayleigh–Taylor instability in the presence of a dry, 2D foam, motivated by ICF applications. Using a non-standard RTI formalism that incorporates foam elasticity, the authors derive how the foam's elastic phase reduces growth rates and introduces a wavenumber threshold k_m, with γ'^2 = A k g (1 − k/k_m) and k_m = g(ρ_up − ρ)/E. In the plastic phase, RTI growth resumes at the fluid rate once the plateau stress σ_el^* is reached, while the fracture phase lies outside the linear regime. The results show that homogenized foam models overestimate RTI growth by ignoring elasticity, especially for small k, and provide a framework to connect foam microstructure to RTI behavior in both ICF and related fields.

Abstract

We analyse the behaviour of the Rayleigh-Taylor instability (RTI) in the presence of a foam. Such a problem may be relevant, for example, to some inertial confinement fusion (ICF) scenarios such as foams within the capsule or lining the inner hohlraum wall. The foam displays 3 different phases: by order of increasing stress, it is first elastic, then plastic, and then fractures. Only the elastic and plastic phases can be subject to a linear analysis of the instability. The growth rate is analytically computed in these 2 phases, in terms of the micro-structure of the foam. In the first, elastic, phase, the RTI can be stabilized for some wavelengths. In this elastic phase, a homogenous foam model overestimates the growth because it ignores the elastic nature of the foam. Although this result is derived for a simplified foam model, it is likely valid for most of them. Besides the ICF context considered here, our results could be relevant for many fields of science.

The Rayleigh-Taylor instability with foams

TL;DR

This work analyzes the Rayleigh–Taylor instability in the presence of a dry, 2D foam, motivated by ICF applications. Using a non-standard RTI formalism that incorporates foam elasticity, the authors derive how the foam's elastic phase reduces growth rates and introduces a wavenumber threshold k_m, with γ'^2 = A k g (1 − k/k_m) and k_m = g(ρ_up − ρ)/E. In the plastic phase, RTI growth resumes at the fluid rate once the plateau stress σ_el^* is reached, while the fracture phase lies outside the linear regime. The results show that homogenized foam models overestimate RTI growth by ignoring elasticity, especially for small k, and provide a framework to connect foam microstructure to RTI behavior in both ICF and related fields.

Abstract

We analyse the behaviour of the Rayleigh-Taylor instability (RTI) in the presence of a foam. Such a problem may be relevant, for example, to some inertial confinement fusion (ICF) scenarios such as foams within the capsule or lining the inner hohlraum wall. The foam displays 3 different phases: by order of increasing stress, it is first elastic, then plastic, and then fractures. Only the elastic and plastic phases can be subject to a linear analysis of the instability. The growth rate is analytically computed in these 2 phases, in terms of the micro-structure of the foam. In the first, elastic, phase, the RTI can be stabilized for some wavelengths. In this elastic phase, a homogenous foam model overestimates the growth because it ignores the elastic nature of the foam. Although this result is derived for a simplified foam model, it is likely valid for most of them. Besides the ICF context considered here, our results could be relevant for many fields of science.

Paper Structure

This paper contains 11 sections, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model of foam
  3. Elastic phase
  4. Plastic phase
  5. Fracture phase
  6. RTI formalism
  7. RTI with a foam
  8. Elastic phase
  9. Plastic phase
  10. Conclusion
  11. Acknowledgments

Figures (4)

  • Figure 1: Model of cell of a foam in 2D. Beams of a material of density $\rho_s$ connected to each other according to the displayed geometry. The whole structure is obtained replicating this unit in every direction. From Gibson1982_2D.
  • Figure 2: Typical stress-strain curve of a foam. See Eq. (\ref{['eq:EpsDens']}) for $\varepsilon_D$. Adapted from Gibson1982_2D.
  • Figure 3: Setup considered for the RTI. The foam average density is $\rho$. It is placed below a fluid of density $\rho_{up} > \rho$.
  • Figure 4: Ratio of the growth rate $\gamma '$ of the foam-RTI, to the growth rate $\gamma$ of the averaged medium-RTI in the elastic phase of the foam. From Eq. (\ref{['eq:gammaprime']}).