Finite-thickness effects on the Rayleigh-Taylor instability in accelerated elastic solids

A physical model has been developed for the linear Rayleigh-Taylor instability of a finite-thickness elastic slab laying on top of a semi-infinite ideal fluid. The model includes the nonideal effects of elasticity as boundary conditions at the top and bottom interfaces of the slab and also takes into account the finite transit time of the elastic waves across the slab thickness. For Atwood number $A_T=1$, the asymptotic growth rate is found to be in excellent agreement with the exact solution [Plohr and Sharp, Z. Angew. Math. Mech. 49, 786 (1998)], and a physical explanation is given for the reduction of the stabilizing effectiveness of the elasticity for the thinner slabs. The feedthrough factor is also calculated.

Figure 1

Schematic of the two-interface system formed by the elastic slab on top of an ideal fluid.

Figure 2

Asymptotic dimensionless growth rate $\sigma =\gamma /\sqrt{k_{0}g}$ as a function of the dimensionless wave number $\kappa =k/k_0$ for several values of $a=k_{0}h$ and $A_T=1$. Dots for $a\le 1$ are calculated with the theory of Ref. [49], and for $a\gg 1$ Ref. [50] has been used.

Figure 3

Dimensionless cutoff wave number ${\kappa }_c=k_c/k_0$ as a function of the dimensionless slab thickness $a=k_{0}h$ for $A_T=1$. The solid line is the result of the present model, the dotted line is given in Refs. [3, 4], and the dash-dotted line is obtained from the theory of Ref. [49].

Figure 4

Asymptotic dimensionless growth rate $\sigma =\gamma /\sqrt{k_{0}g}$ as a function of the dimensionless wave number $\kappa =k/k_0$ for several values of $a=k_{0}h$ and $A_T=0.5$.

Figure 5

Asymptotic dimensionless growth rate $\sigma =\gamma /\sqrt{k_{0}g}$ as a function of the dimensionless wave number $\kappa =k/k_0$ for several values of $A_T$ and $a=k_{0}h=0.1$.

Figure 6

Feedthrough factor $F={\xi }_b/\xi$ as a function of the dimensionless wave number $\kappa =k/k_0$ for several values of $a=k_{0}h$ and $A_T=1$.

Figure 7

Dimensionless cutoff wave number ${\kappa }_c=k_c/k_0$ as a function of the dimensionless slab thickness $a=k_{0}h$ for several values of $A_T$. Solid lines are given by Eq. (30), and dotted lines are the results of the exact theory outlined in Appendix pp2.