Linear analysis of Rayleigh-Taylor instability in viscoelastic materials

Rayleigh-Taylor instability (RTI) has become a powerful tool for determining the mechanical properties of materials under extreme conditions. In this paper, we first present the exact and approximate linear dispersion relations for RTI in viscoelastic materials based on the Maxwell and Kelvin-Voigt models. The approximate dispersion relation produces good predictions of growth rates in comparison with the exact one. The motion of the interface in Maxwell flow is mainly controlled by viscosity and elasticity dominates this behavior in Kelvin-Voigt flow. Since elasticity plays a distinct role from viscosity, cutoff wavelengths arise only in Kelvin-Voigt flow. The variation of the maximum growth rates and their corresponding wave numbers are also carefully studied. For both types of materials, viscosity suppresses the growth of instability, while elasticity speeds it up. This is at odds with the well-known understanding that elasticity suppresses hydrodynamic instabilities. The dependence of the maximum growth rate on slab thickness is also investigated for RTI in both types of flow, since the metal slab as a pusher has been extensively employed in high-energy-density physics. The model presented here allows study of more realistic situations by considering convergent effects and shock wave interactions, for the traditional potential flow theory is not suitable. To summary, it is able to provide guidances for future experimental designs for studies of materials under high strain and high strain rate conditions, as well as allow us to study RTI theoretically in more complicated conditions.

Figure 1

Schematic configuration of RTI in viscoelastic materials.

Figure 2

(a) $\gamma$ as a function of $K$ for RTI in Maxwell materials with varying $m$. (b) The $({\gamma }_{\max },K_{\max })$ plane for $0.01<m<20$.

Figure 3

$\gamma$ as a function of $K$ in Maxwell materials with varying parameters: (a) $A_T$, (b) $m$, (c) $S$, and (d) $T$.

Figure 4

(a) $\gamma$ as a function of $K$ for RTI in Kelvin-Voigt materials with varying $m$. (b) The $({\gamma }_{\max },K_{\max })$ plane for $0.01<m<20$.

Figure 5

$\gamma$ as a function of $K$ in Kelvin-Voigt materials with varying parameters: (a) $A_T$, (b) $m$, (c) $S$, and (d) $T$.

Figure 6

$\alpha$ as a function of $H$ in Maxwell materials with varying $R_m$.

Figure 7

$\alpha$ as a function of $H$ in Kelvin-Voigt materials with varying $R_m$.