Rayleigh-Taylor instability under a spherical substrate

We investigate the Rayleigh-Taylor instability of a thin viscous film coating the inside of a spherical substrate. The aim of this work is to find and characterize the instability pattern in this spherical geometry. In contrast to the Rayleigh-Taylor instability under a planar substrate, where the interface is asymptotically unstable with respect to infinitesimal perturbations, the drainage induced by the component of gravity tangent to a curved substrate stabilizes the liquid interface, making the system linearly asymptotically stable. By performing a linear optimal transient growth analysis we show that the double curvature of a spherical substrate yields a critical Bond number, prescribing the ratio between gravitational and capillary forces, before an initial growth of perturbations is possible two-times larger than for a circular cylindrical substrate. This linear transient growth analysis however does not yield any selection principle for an optimal azimuthal wave number and we have to resort to a fully nonlinear analysis. By numerically solving the nonlinear lubrication equation we find that the most amplified azimuthal wave number increases with the Bond number. Nonlinear interactions are responsible for the transfer of energy to higher-order harmonics. The larger the Bond number and the farther away from the apex of the sphere, the richer the wave-number spectrum.

Figure 1

Sketch of the problem's geometry.

Figure 2

Temporal evolution of the linear amplitude $A$ for different Bond numbers $Bo=10$, 25, 50, 75, 100, 125, 150, 175, and 200, with the initial optimal wave number $k_0$. Asterisks denote the largest amplitudes.

Figure 3

(a) Optimal time $T_{A_{\text{max}}}$ (blue solid line), with its approximation (black dashed line) and wave number $k_{\text{max}}$ (inset) giving (b) the largest amplitude $A_{\text{max}}$ as a function of the Bond number.

Figure 4

Comparison between the linear evolution of the film thickness given by Eq. (7) (dashed line) and the one of the fundamental (solid line) obtained by the nonlinear evolution of the initial condition (13) with $\text{Bo}=25$, $m_0=1$, and $\varepsilon =A(T=1)={10}^{-2}$ at the times $t=0$, 1, 2, 3, and 4. The film thickness is along (a) the polar direction at $\theta =\pi /30$ for $\varphi \in [0,2\pi ]$ and (b) the azimuthal direction for $\theta \in [0,\pi /20]$ and $\varphi =0$. The agreement remains good also for later times.

Figure 5

Nonlinear evolution of the film thickness for the initial condition given by Eq. (13) for $\text{Bo}=150$, $m_0=6$, and $\varepsilon ={10}^{-2}$ at the times $t=1$, 2, 3, 4, 6, 8, 10, 12, and 14. The film thickness is indicated by the colorbar, which changes scaled nonmonotonically with time. The evolution with a constant colorbar is shown in Fig. 22 in the Appendix pp4. Particular spatial locations are indicated by a white dashed line; the numbers are used also in the following figures. The droplet generations $G_{m_0}$, $G_{2m_0}$, $G_{2m_0}$, and $G_0$ are also highlighted. See movie 1 in [24] for the entire evolution.

Figure 6

Azimuthal disturbance norm $\mathcal{H}$ for the evolution shown in Fig. 5, namely, for $\text{Bo}=150$, $m_0=6$, and $\varepsilon ={10}^{-2}$ with a linear optimal initial condition. The numbered points correspond to the locations specified in Fig. 5 and the red star highlights the instant of largest amplitude. The amplitude ridges of the droplets generations $G_{m_0}^{1,2,3}$ are also indicated. The discontinuous appearance of the amplitude is due to the limited temporal sampling. The film thickness disturbance amplitude at the pole $\varepsilon h(\theta =0)$ is shown in the inset on the left (black solid line), together with the growth predicted by the linear theory (red dashed line).

Figure 7

Amplitude of the azimuthal modes (a) $m=0$, (b) $m=m_0$, (c) $m=2m_0$, and (d) $m=4m_0$ for $\text{Bo}=150$, $m_0=6$, and $\varepsilon ={10}^{-2}$ with a linear optimal initial condition. The numbered points correspond to the locations specified in Fig. 5.

Figure 8

Azimuthal mode $m$ with the largest amplitude for $\text{Bo}=150$, $m_0=6$, and $\varepsilon ={10}^{-2}$ with a linear optimal initial condition. The colorbar indicates the $m$ values. Numbered points correspond to the locations in Fig. 5 and numbers on a white background correspond to the $m$ value. Here ${\theta }_1$ and ${\theta }_2$ indicate the polar angles at $t=20$ where the largest mode switches to the higher harmonics.

Figure 9

Schematic representation of (a) the inverted-$\mathsf{U}$ pattern and (b) the late-time pattern, with the corresponding locations in Fig. 8 (left) and the wave numbers (right).

Figure 10

Evolution of the energy norm ${\mathcal{E}}_m$ of the different modes over the upper hemisphere for $\text{Bo}=150$, $m_0=6$, and $\varepsilon ={10}^{-2}$ with a linear optimal initial condition. The colorbar indicates the $m$ values, also highlighted on the plot for the dominant modes.

Figure 11

Effect of the azimuthal wave number $m_0$ and of the Bond number Bo for the initial condition given by Eq. (13). Here $\varepsilon ={10}^{-2}$; $m_0=1$, 6, and 12; $\text{Bo}=100$, 150, and 175; and $t=4$.

Figure 12

Effect of the azimuthal wave number on the azimuthal disturbance norm $\mathcal{H}$ for $\text{Bo}=150$, $\varepsilon ={10}^{-2}$, and $m_0=1$, 6, and 12 with a linear optimal initial condition. The red star highlights the instant of largest amplitude.

Figure 13

Effect of the azimuthal wave number on the azimuthal mode $m$ with the largest amplitude for $\text{Bo}=150$, $\varepsilon ={10}^{-2}$, and $m_0=1$, 6, and 12 with a linear optimal initial condition. The colorbar indicates the $m$ values.

Figure 14

Effect of the azimuthal wave number on the evolution of the energy norm ${\mathcal{E}}_m$ for $\text{Bo}=150$, $\varepsilon ={10}^{-2}$, and $m_0=1$, 6, and 12 with a linear optimal initial condition. The colorbar indicates the $m$ values.

Figure 15

(a) Effect of the Bond number on the angles ${\theta }_1$ and ${\theta }_2$ for the transition of the azimuthal mode with the largest amplitude from $m_0$ to $2m_0$ and from $2m_0$ to $4m_0$, respectively, at $t=20$ (see Fig. 8). (b) Maximal azimuthal energy ${\mathcal{H}}_{\text{max}}={max}_{(t,\theta )}\mathcal{H}$ as a function of the initial azimuthal wave number $m_0$ for $\varepsilon ={10}^{-2}$; $\text{Bo}=100$, 125, 150, and 175 with a linear optimal initial condition. Arrows for $m_0=2$, 3, and 4 indicate that the largest azimuthal energy for $\text{Bo}=125$, 150, and 175 is larger than 5, but cannot be computed as a thick droplet forms at the north pole and the simulations have to be interrupted. Here $\varepsilon ={10}^{-2}$, $m_0=6$, and $\text{Bo}=100,110,120,...,160$ with a linear optimal initial condition. Dashed lines are added for visualization purposes.

Figure 16

(a) Spatiotemporal diagram of the film-thickness disturbance $\varepsilon h=\overline{H}-H$ at $\varphi =0$ together with the isocontours of the free-surface velocity $U=0.1$, 0.4, 1.5, 2.3, and 3 for the asymptotic solution (19) (turquoise dashed line) and for Eq. (18) with the numerical solution of the draining film (yellow solid line), respectively, for $\text{Bo}=100$. (b) Spatiotemporal diagram of the film-thickness disturbance $\varepsilon h=\overline{H}-H$ at $\varphi =0$ with the sliding-droplet trajectories obtained by the integration of the shock speed predicted by Eq. (20) (yellow circles) for the three droplet generations at $\varphi =0$ for $\text{Bo}=150$. For both panels, $m_0=6$ and $\varepsilon ={10}^{-2}$ with a linear optimal initial condition.

Figure 17

Effect of the Bond number for a random noise initial condition with maximal amplitude $\varepsilon ={10}^{-2}$; $\text{Bo}=100$, 125, 150, and 175; and $t=4$, 8, and 12. See movie 2 in [24] for the entire evolution for $\text{Bo}=175$.

Figure 18

Azimuthal disturbance norm $\mathcal{H}$ for a random noise initial condition with maximal amplitude $\varepsilon ={10}^{-2}$ and $\text{Bo}=100$, 125, and 175. The red star highlights the instant of largest amplitude.

Figure 19

Azimuthal mode $m$ with the largest amplitude for a random noise initial condition with maximal amplitude $\varepsilon ={10}^{-2}$ and $\text{Bo}=100$, 125, and 175. The colorbar indicates the $m$ values.

Figure 20

Evolution of the energy norm ${\mathcal{E}}_m$ for a random noise initial condition with maximal amplitude $\varepsilon ={10}^{-2}$ and $\text{Bo}=100$, 125, and 175. The colorbar indicates the $m$ values.

Figure 21

Validation of the numerical scheme against the theoretical predictions: (a) drainage at the pole [Eq. (B1)] and (b) drainage over the polar angle for $t=20$ [Eq. (B3)]. Here $\text{Bo}=25$.

Figure 22

Same as Fig. 5 with a constant colorbar for the film thickness. Nonlinear evolution of the film thickness for the initial condition given by Eq. (13) for $\text{Bo}=150$, $m_0=6$, and $\varepsilon ={10}^{-2}$ at the times $t=1$, 2, 3, 4, 6, 8, 10, 12, and 14. The droplet generations $G_{m_0}$, $G_{2m_0}$, $G_{2m_0}$, and $G_0$ are also highlighted.