Pinch-off dynamics to elucidate animal lapping

Some carnivorous mammals (e.g., cats and dogs) lap water with their tongues to drink water at high frequencies. Such a fast moving tongue creates a liquid column out of a bath which is bitten by the mouth for drinking. Presumably, the animals bite just before the pinch-off time of the water column to maximize the water intake. Otherwise, the water column falls back to the bath before being bitten. Such a pinch-off phenomenon in the liquid column can be described as the acceleration-induced (i.e., unsteady) inertia balances with the capillary force. The classical Rayleigh-Plateau instability explains the competition of the steady inertia with the capillarity, but not with the acceleration-induced inertia. In this study, we modify the Rayleigh-Plateau instability in the presence of the fluid acceleration, and show that the most unstable wavenumber and growth rate increase with acceleration. The pinch-off time is theoretically predicted as the $-1/3$ power of the Bond number (i.e, a ratio of the acceleration-induced inertia to capillarity). Finally, measured pinch-off times from previous physical experiments and dog and cat jaw-closing times are shown to be in good agreement with our theoretical pinch-off time. Therefore, our study shows that animals presumably modulate their lapping and jaw-closing time to bite down on the water column before the pinch-off to maximize the water intake.

Figure 1

(a) Photos show the formation of a water column while a dog drinks (image credit to J. Jocha, S. Gart, and S. Jung). (b) Water columns are formed while a killer whale jumps out of the water (image credit to S. Jung). (c) Schematic of a liquid column underneath an object exiting the liquid surface. (d) Averaged tongue velocity of two dogs (27 kg and 9 kg in Ref. [5]) and one cat (7 kg in Ref. [4]). Time zero is when the tongue starts to move out of a liquid bath. A linear increase in the velocity of the dog's tongue indicates a constant acceleration ($2g–4g$) during the withdrawal phase, whereas a cat drinking exhibits nonuniform acceleration as the tongue accelerates in the beginning and then decelerates in the end.

Figure 2

(a) Dispersion relation of the classical Rayleigh-Plateau instability with $\gamma =0.072$ N/m, $\rho =1000$ kg/${\mathrm{m}}^3$, and $R_0={10}^{-2}$ m. The growth rate is maximized at ${\left(k{R}_0\right)}^{*}=0.708$ regardless of fluid properties. (b) Dispersion relation of the modified Rayleigh-Plateau instability with acceleration. Here, both the growth rate and nondimensional wavenumber of the most unstable mode increase with acceleration.

Figure 3

Fluid properties are $\gamma =0.072$ N/m, $\rho =1000$ kg/${\mathrm{m}}^3$, and $R_0={10}^{-2}$ m. (a) Imaginary part of the most unstable nondimensional wavenumbers ($\mathrm{Im}\left[{\left(k{R}_0\right)}^{*}\right]$) scales as the 1/3 power of acceleration: $\mathrm{Im}\left[{\left(k{R}_0\right)}^{*}\right]\propto a^{1/3}$. Three solutions are obtained from Eq. (27), one red line and two blue lines. (b) The most unstable growth rate ($\mathrm{Im}\left[{\omega }^{*}\right]$) scales as the 2/3 power of acceleration: $\mathrm{Im}\left[{\left(\omega \right)}^{*}\right]\propto a^{2/3}$.

Figure 4

(a) Dimensionless pinch-off time (${\tilde{t}}_{\text{pinch-off}}$) versus the acceleration-based Bond number $\left[\mathrm{Bo}=\rho a{\left(2R_0\right)}^2/\gamma \right]$. The purple line is from our present theory, ${\tilde{t}}_{\text{pinch-off}}=2/\sqrt{3}{\mathrm{Bo}}^{-1/3}$, whereas the yellow line indicates the previous theory, ${\tilde{t}}_{\text{pinch-off}}=\sqrt{8\pi /9}{\mathrm{Bo}}^{-1/2}$. It shows that our new theoretical pinch-off time works quite well with both physical experiments and dog and cat lapping. (b) Jaw-closing timescale normalized by our predicted pinch-off time versus animal weight. As shown here, most animals close their jaws before the pinch-off time ($t_{\text{jaw-closing}}<t_{\text{pinch-off}}$).