Unified decomposition method to study Rayleigh-Taylor instability in liquids and solids

In the previous studies of Rayleigh-Taylor instability, different methods were used to consider the effects of elasticity, viscosity, and magnetic fields. In this paper, a unified method, which was first used for fluids, is validated for different physical models, where the unstable mode is decomposed into an irrotational part and a rotational part, and the latter one includes the effects of nonconservative forces and constitutive relations. Previous results of solid and liquid with or without the effects of magnetic fields and finite thickness can be easily recovered after applying the corresponding interface boundary conditions. In addition, new approximate but analytical solutions of the growth rates for a semi-infinite solid-solid interface and solid-fluid interface are obtained with substantially improved accuracy in comparison with the previous ones.

Figure 1

Configuration of the physical model.

Figure 2

The dimensionless growth rate $\sigma =n{(G_2/{\rho }_2{g}^2)}^{1/2}$ as a function of the dimensionless wave number $K=k{G}_2/\left({\rho }_{2}g\right)$ for a solid-solid interface. (a) The ratio $R=G_1/G_2$ is $0.5,2,$ and 5 from top to bottom with $A_T=0.8$, and (b) $A_T$ is $0.9,0.5,$ and 0.2 from top to bottom with $R=1$, respectively.

Figure 3

The dimensionless growth rate $\sigma =n{(G/{\rho }_2{g}^2)}^{1/2}$ as a function of the dimensionless wave number $K=kG/\left({\rho }_{2}g\right)$ for a liquid-solid interface. (a) The dimensionless viscosity ${\left[\mu g{({\rho }_2/G^3)}^{1/2}\right]}^{1/2}$ is 0.5, 2, and 5 from top to bottom with $A_T=0.8$ and (b) $A_T$ is $0.9,0.5,$ and 0.2 from top to bottom with $m=1$, respectively.