Coupling effects and thin-shell corrections for surface instabilities of cylindrical fluid shells

We show that when linear azimuthal perturbations on the surfaces of a fluid shell are regrouped according to ${\alpha }^m$, they can be divided into Bell model terms, coupling terms, and the newly identified thin-shell correction terms, where $\alpha$ is the ratio of $R_{\text{out}}$ to $R_{\text{in}}$, and $m$ is the mode number of a given unstable mode on the surfaces. It is also revealed that ${\alpha }^m$ is a convenient index variable of coupling effects, with which we show that the evolution of instability is composed of three stages, i.e., strongly coupled stage, transition stage, and uncoupled stage. Roughly, when ${\alpha }^m<6$, the fluid shell is in the strongly coupled stage, where both coupling effects and the newly identified thin-shell corrections play important roles. Strong feed through is expected to be observed. The uncoupled stage is reached at ${\alpha }^m\sim 36$, where Bell's model of independent surface holds. In between is the transition stage, where mode competitions on the two surfaces are expected to be observed. These results afford an intuitive picture which is easy to use in guiding the design of experiments. They may also help to quickly grasp major features of instability experiments of this kind.

Figure 1

Schematic illustration of a contracting cylindrical shell considered in this work. $R_{\text{in}}$ and $R_{\text{out}}$ here are inner and outer radii of undisturbed surfaces.

Figure 2

Profiles of a fluid shell in accelerated (a, c, e) and coasting (b, d, f) contractions at $\mathrm{Cr}=1$ (black dots), 1.3 (blue dashed lines), and 1.7 (red solid lines), for mode number $m=3$ (a, b), 4 (c, d), and 8 (e, f). The corresponding $\alpha$ are 1.11, 1.21, and 1.49. Their initial amplitude perturbations on outer surfaces have the same value ${\eta }_{\mathrm{out}}\left(0\right)=0.01$.

Figure 3

Profiles of a fluid shell in accelerated (a, c, e) and coasting (b, d, f) collapse at $\text{Cr}=1$ (black dots), 1.3 (blue dashed lines), and 1.7 (red solid lines), for mode number $m=3$ (a, b), 4 (c, d), and 8 (e, f). The corresponding $\alpha$ are 1.11, 1.21, and 1.49. Their initial velocity perturbations on outer surfaces have the same value ${\dot{\eta }}_{\text{out}}\left(0\right)=0.04\overline{v}$.

Figure 4

Instantaneous growth rates ${\sigma }_1$ (a), ${\sigma }_2$ (b) and the instantaneous coupling factors ${\mathrm{\Omega }}_1$ (c), ${\mathrm{\Omega }}_2$ (d) for unstable mode $m=4$ in an accelerated contraction. The green vertical dashed dots represent ${\alpha }^m=6$, and the pink vertical dashed dots represent ${\alpha }^m=36$.

Figure 5

Instantaneous growth rates ${\sigma }_1$ (a), ${\sigma }_2$ (b) and the instantaneous coupling factors ${\mathrm{\Omega }}_1$ (c), ${\mathrm{\Omega }}_2$ (d) for $m=4$ in a coasting contraction. The green vertical dashed dot line represent ${\alpha }^m=6$, and the pink vertical dashed dot line represent ${\alpha }^m=36$.

Figure 6

Instantaneous growth rate ${\sigma }_1$ (a) and the instantaneous coupling factor ${\mathrm{\Omega }}_1$ (b) for mode $m=4$ in an accelerated explosion. The green vertical dashed dot line represent ${\alpha }^m=6$, and the pink vertical dashed dot line represent ${\alpha }^m=36$.

Figure 7

Ratios ${\gamma }_{\text{in}}$ and ${\gamma }_{\text{out}}$ for $m=$ 2, 4, and 7, which represent the influence of thin-shell corrections. Short vertical segments shows the positions that ${\alpha }^m=6$, at which $\gamma$ drops to about 0.1.