Effects of viscosity and elasticity on the Richtmyer-Meshkov instability

The impulsively accelerated Richtmyer-Meshkov instability (RMI) of a fluid-solid interface is theoretically studied with a decomposition method, and the least stable mode is shown to oscillate but decay due to the combined effects of elasticity and viscosity. The dispersion relation of RMI in viscous fluids is obtained as well, and the predicted interface amplitude agrees with numerical simulations better than the previous linear and weakly nonlinear theories. As time tends to infinity, the asymptotic amplitude of the disturbed interface is found to be inversely proportional to the wave number in viscous fluids but to be independent of the wave number for the interface between elastic solids.

Figure 1

The scheme of the perturbed interface.

Figure 2

The temporal evolution of the total $y\text{−}\mathrm{direction}$ velocity amplitude $V$ normalized with its initial value $V_0$ for the least stable mode as $A_T=1/3$ and $r=\mu k/\sqrt{G{\rho }_2}=0,0.2,0.5,0.8,1.2$ and 2 from up to bottom, respectively.

Figure 3

The evolutions of the interface perturbation amplitudes obtained by simulations and stability theories. In panel (a) ${\eta }_0=1\mathrm{mm}$, $\lambda =13\mathrm{cm}$, ${\rho }_1=0.166\text{mg}/{\text{cm}}^3$, ${\mu }_1={\mu }_2=0.1\mathrm{Pa}\mathrm{s}$, ${\rho }_2=4{\rho }_1$, and $\mathrm{\Delta }V=18.9\text{cm}/\text{ms}$. In panel (b) ${\eta }_0=30\mu \text{m}$, $\lambda =2.5\text{cm}$, ${\rho }_1=1\text{g}/{\text{cm}}^3$, ${\mu }_1=2{\mu }_2$, ${\mu }_2=0.32\text{Pa}\text{s}$, ${\rho }_2=4\text{g}/{\text{cm}}^3$, and $\mathrm{\Delta }V=0.1\text{cm}/\text{ms}$.