Rayleigh-Taylor instability in accelerated elastic-solid slabs

We develop the linear theory for the asymptotic growth of the incompressible Rayleigh-Taylor instability of an accelerated solid slab of density ${\rho }_2$, shear modulus $G$, and thickness $h$, placed over a semi-infinite ideal fluid of density ${\rho }_1<{\rho }_2$. It extends previous results for Atwood number $A_T=1$ [B. J. Plohr and D. H. Sharp, Z. Angew. Math. Phys. 49, 786 (1998)] to arbitrary values of $A_T$ and unveil the singular feature of an instability threshold below which the slab is stable for any perturbation wavelength. As a consequence, an accelerated elastic-solid slab is stable if ${\rho }_{2}gh/G\le 2(1-A_T)/A_T$.

Figure 1

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

Figure 2

(a) Slab thickness $\alpha =k_{0}h$ as a function of the cutoff wave number $w=k_{c}h$ for several values of the Atwood number $A_T$. (b) Normalized cutoff wave number $(1+A_T){\kappa }_c/A_T$ (${\kappa }_c=k_c/k_0=w/\alpha$) as a function of the dimensionless slab thickness $\alpha =k_{0}h$ for several values of $A_T$.

Figure 3

Asymptotic growth rate $\sigma =\gamma /\sqrt{k_{0}g}$ as a function of the wave number $\kappa =k/k_0$ for several values of $\alpha =k_{0}h$ and for (a) $A_T=1$, (b) $A_T=0.8$, and (c) $A_T=0.4$.

Figure 4

Growth rate $\sigma =\gamma /\sqrt{k_{0}g}$ as a function of the wave number $\kappa =k/k_0$ for several values of $A_T$ and $\alpha =k_{0}h=4$.