Linear Rayleigh-Taylor instability in an accelerated Newtonian fluid with finite width

The linear theory of Rayleigh-Taylor instability is developed for the case of a viscous fluid layer accelerated by a semi-infinite viscous fluid, considering that the top interface is a free surface. Effects of the surface tensions at both interfaces are taken into account. When viscous effects dominate on surface tensions, an interplay of two mechanisms determines opposite behaviors of the instability growth rate with the thickness of the heavy layer for an Atwood number $A_T=1$ and for sufficiently small values of $A_T$. In the former case, viscosity is a less effective stabilizing mechanism for the thinnest layers. However, the finite thickness of the heavy layer enhances its viscous effects that, in general, prevail on the viscous effects of the semi-infinite medium.

Figure 1

Schematic of the two-interfaces system formed by a viscous layer on the top of a semi-infinite viscous fluid.

Figure 2

Dimensionless growth rate $\sigma$ as a function of the dimensionless perturbation wave number $\kappa$ for several values of the dimensionless thickness $\alpha =k_{0}h$ of the heavy fluid layer and different Atwood numbers $A_T$. (a) $A_T=1$; (b) $A_T=0.8$; (c) $A_T=0.4$.

Figure 3

Dimensionless growth rate $\sigma$ as a function of the dimensionless perturbation wave number $\kappa$ for several values of the Atwood number $A_T$ and different dimensionless thicknesses $\alpha =k_{0}h$. (a) $\alpha =1$; (b) $\alpha =0.4$.

Figure 4

Dimensionless growth rate $\sigma$ as a function of the dimensionless perturbation wave number $\kappa$ when both fluids are viscous (${\nu }_1={\nu }_2$), and the lighter fluid is irrotational. (a) Comparison with exact rotational theory (Chandrasekhar) for the case of two semi-infinite fluids for several Atwood numbers $A_T$; (b) Comparison with the case of an ideal light fluid (${\nu }_1=0$) for several thicknesses $\alpha =k_{0}h$ of the heavy layer.

Figure 5

Dimensionless growth rate $\sigma$ as a function of the dimensionless perturbation wave number $\kappa$ for several values of the dimensionless thickness $\alpha =k_{0}h$ of the heavy fluid layer, and of the dimensionless surface tensions $S_a$ and $S_b$. Comparisons with the cases of two ideal fluids (${\nu }_1={\nu }_2=0$), and of a light ideal fluid (${\nu }_1=0$, ${\nu }_2\ne 0$). (a) $A_T=0.4$, $S_a=15$, $S_b=150$; (b) $A_T=0.4$, $S_a=5$, $S_b=45$; (c) $A_T=0.25$, $S_a=0.2$, $S_b=0.4$.

Figure 6

Comparison of the present theory for two viscous fluids with surface tensions (full lines), with the experimental results (dots) of Ref. [54]. (a) $\alpha =7.897$ ($h=1$ mm); (b) $\alpha =15.794$ ($h=2$ mm). Results for the cases of two ideal fluids (dotted lines), and two semi-infinite fluids are also shown ($\alpha \gg 1$).