Stick-slip-to-stick transition of liquid oscillations in a U-shaped tube

The nonlinear decay of oscillations of a liquid column in a U-shaped tube is investigated within the theoretical framework of the projection method formalized by Bongarzone et al. [Chaos 31, 123124 (2021)]. Starting from the full hydrodynamic system supplemented by a phenomenological contact line model, this physics-inspired method uses successive linear eigenmode projections to simulate the relaxation dynamics of liquid oscillations in the presence of sliding triple lines. Each projection is shown to eventually induce a rapid loss of total energy in the liquid motion, thus contributing to its nonlinear damping. A thorough quantitative comparison with experiments by Dollet et al. [Phys. Rev. Lett. 124, 104502 (2020)] demonstrates that, in contradistinction with their simplistic one-degree-of-freedom model, the present approach not only describes well the transient stick-slip dynamics, but also correctly captures the global stick-slip to stick transition, as well as the residual exponentially decaying bulk motion following the arrest of the contact line, which has been so far overlooked by existing theoretical analyses but is clearly attested experimentally. This study offers a further contribution to rationalizing the impact of contact angle hysteresis and its associated solidlike friction on the decay of liquid oscillations in the presence of sliding triple lines.

Figure 1

(a) Contact angle dynamics in a vertically vibrating droplet. For these oscillatory flows, experiments from (b) Ref. [23] suggest as suitable phenomenological contact angle laws the (c) nonlinear Dussan model [38, 45]. (d) Transition between stick and stick-slip motions in a water sessile drop deposited on a vertically vibrating substrate characterized by a finite contact angle hysteresis ($\mathrm{\Delta }\approx 10-15$ degrees) [46]. Lower curves are contact angle variations vs time; the dashed line represents ${\theta }_s$. Higher curves are the contact line position around the starting position before vibrations. The six curves for different nondimensional acceleration amplitudes $f/g$ are joined in the same plot for comparison. The driving frequency is $1/T=9\text{Hz}$. Panels (b) and (d) are modified versions of figures given in Refs. [23] and [46], respectively.

Figure 2

(a) Contact angle dynamics in sloshing waves (snapshots over a period) [47]. For this oscillatory flow, experiments from Ref. [48] suggest as suitable phenomenological contact angle laws the (b) Hocking linear law [22] supplemented with hysteresis. (c) Experimental contact angle dependence on the capillary number as measured in Ref. [48] during the natural relaxation dynamics of water oscillations in a cylindrical container initially perturbed. (d) Associated damping rate vs the amplitude of the angle, $\phi$, measured at the container axis as indicated in (a). The vertical dashed line indicates the value for which the contact line irreversibly pins. Panels (a) and (c)–(d) are modified versions of figures reported in Refs. [47] and [48], respectively.

Figure 3

(a) Dimensional interface height $\overline{h}\left(t\right)\text{mm}$ vs dimensional time $\overline{t}\left(s\right)$ for water and ethanol in the hydrophobic tube and for liquid column length $l=14.6\text{mm}$. (b) Rescaled interface height, $h$, vs time $\overline{t}\left(s\right)$, for water in the hydrophobic tube with a fixed liquid column length $l=14.6\text{mm}$ and at different initial elevation $h_{\mathrm{in}}$. The solid curves correspond to the predictions from Eq. (5) with an oscillation period $T=2\pi /{\omega }_0=2\pi \sqrt{l/2g}$ and with $\sigma =0.06$ as a free-fitting parameter common for all experiments. (c) Phenomenological law used in the present work to model the apparent dynamic contact angle, $\theta$, vs the nondimensional contact line speed, ${\text{Ca}}^{\prime }=\text{Ca}\partial \eta /\partial t$, with $\text{Ca}=\nu \rho \sqrt{gl/2}/\gamma , \nu$ the kinematic liquid viscosity, $\rho$ the liquid density, and $\gamma$ the liquid-air surface tension. Panels (a) and (b) are modified versions of figures given in Ref. [53].

Figure 4

Sketch of the U-tube configuration. (a) Full 3D geometry. (b) 2D view of the centerline plane. The tube radius is assumed constant and denoted by $a$. The length of the liquid column is $l$. $h$ indicates the height difference of the liquid column between the left and right straight channels. $g$ is the gravity acceleration. The advancing and receding dynamic contact angles are, respectively, ${\theta }_a$ and ${\theta }_r$, whereas the static contact angle is labeled as ${\theta }_s$ and it is in general $\ne {90}^{\circ }$. (c) If the tube curvature is neglected, the 3D geometry can be reduced to an axisymmetric configuration, by considering only half of the liquid column, of length $l/2$, and by imposing antisymmetry conditions at the bottom boundary so as to restore the effect of the gravity term on the missing straight channel.

Figure 5

Shape of the dimensional static meniscus, ${\overline{\eta }}_0$, computed numerically for ${\theta }_s=({\theta }_a+{\theta }_r)/2=(93+68)/2=80.5^{\circ }$.

Figure 6

Eigenvalue spectrum associated with the two contact line boundary conditions, i.e., pinned (green markers) and free (blue markers), computed numerically by solving the generalized eigenvalue problem (19). For the case of a free contact line condition, the calculation here reported has been performed by imposing a value of $\alpha =0$. Both spectra are computed for a liquid column length $l=14.6\text{cm}$. Fluid properties: water, $\rho =1000\text{kg/m}{}^3, \gamma =0.0725\text{N/m}$, and $\nu =1\times {10}^{-6}\text{m}{}^2\text{/s}$.

Figure 7

(a) Eigen-interface associated with the U-tube free mode computed in Fig. 6. The free surface dynamics in the free-phase consists of an upward-downward oscillation of a flat interface. (b) Eigen-interface associated with U-tube pinned mode computed in Fig. 6. Instead, the surface dynamics in the pinned-phase consists of an interface oscillating with a bell-like shape whose edges are anchored at the wall.

Figure 8

Dimensional oscillation period, $T$, and damping coefficient, $\overline{\sigma }$, vs the water column length and associated with a pinned contact line dynamics of the fundamental U-tube mode. Green diamonds: values computed fully numerical eigenvalue calculation. White circles: values measured experimentally as reported in Ref. [53].

Figure 9

(a) Dimensional oscillation period, $T$, and (b) damping coefficient, $\overline{\sigma }$, vs the water column length, $l\left(\text{cm}\right)$ and associated with a free contact line dynamics of the fundamental U-tube mode. Blue diamonds: values computed fully numerical eigenvalue calculation by accounting for the variable slip length model discussed in Eq. (16) with $\alpha =0$. White circles in (a): values measured experimentally as reported in Ref. [53]. The experimental range investigated in Ref. [53] is indicated by the gray arrow in (b). Within this range, the damping coefficient is nearly constant with the tube length.

Figure 10

Same as in Fig. 9 (here in $\sigma \text{−}log$ scale), but with the light blue crosses indicating the values computed by also accounting for extra contact line dissipation produced by Hocking's law [22, 54] with $\alpha =200\text{rad}$). Within this range, the damping coefficient is nearly constant with the tube length, $l$, even for $\alpha =200\text{rad}$. The average value in this range is $\sigma \approx 0.06$, which matches the one used in Fig. 3 and obtained from the best fit of the experiments.

Figure 11

(a) Axisymmetric meniscus modes associated with the free-phase and (b) with the pinned-phase. In (a), the slope at the wall is 1, whereas the contact line elevation is $F_0$. In (b), the slope is $1/F_0$, whereas the contact line elevation is 1. (c) Real part of the eigen-interface associated with the free and (d) pinned U-tube modes, with the corresponding eigenvalues, ${\lambda }_{f_0}=-{\sigma }_{f_0}+\text{i}{\omega }_{f_0}$ and ${\lambda }_{p_0}=-{\sigma }_{p_0}+\text{i}{\omega }_{p_0}$ reported on top. The free mode is normalized such that the contact line elevation is 1, while the pinned mode is normalized such that the slope at the wall is 1. For completeness, in (c), we have also reported the interface shape when $\alpha =0$ (thin blue line) as shown in Fig. 7. [(e)–(i)] Real part of the eigen-interface associated with the five least damped free and [(j)–(n)] pinned capillary-gravity waves. The same normalization as in (c) and (d) is employed.

Figure 12

Workflow of the projection algorithm [from (a) to (c)].

Figure 13

Contact line elevation vs time for different initial conditions. Dashed line: 1dof model. Red solid lines: predictions from the projection model. Markers: experiments by Dollet et al. [53]. We note that in performing the calculation, we have considered an effective tube length of $16.2\text{cm}$, where an excess length of $l^{\prime }=1.6\text{cm}$ is introduced to take into account the fact that the cross section along the curved part of the tube is not constant due to the fabrication process. See Ref. [53] for further details.

Figure 14

Finite time of arrest vs the imposed initial elevation. Black solid line: analytical prediction from the 1dof model proposed by Dollet et al. [53]. White triangles: experimental measurements by Dollet et al. Colored circles: projection method. The black dashed line only serves a guide for the eye.

Figure 15

Centerline free surface elevation, $r=0$ and $z=0$ (in log-scale), vs time for different initial elevations, $h_{\mathrm{in}}$. The gray solid lines show the actual signal produced by the projection method, and the colored solid lines indicate the amplitude envelope only. The colored dashed lines correspond to the analytical prediction given by the 1dof model employed by Dollet et al. [53]. An almost abrupt change in the trend of these signals is well visible. This is a clear sign of the final transition to a pinned contact line dynamics following the contact line arrest.

Figure 16

(a) Dimensional damping rate and (b) frequency modulation vs time at different initial conditions. The damping rate, $DR\left(t\right)$ is computed as the logarithmic decrement of the amplitude of the centerline free surface elevation, shown in Fig. 15. The frequency is computed from the same signal by evaluating the period from peak to peak, with the resulting value that is then roughly assigned to the midpoint of the corresponding time interval [colored filled circles in (a) and (b)]. The colored solid lines represent the best fit (smoothing splines) of these time signals, whereas the colored dashed lines correspond to the analytical prediction given by the single-degree-of-freedom model employed by Dollet et al. [53].

Figure 17

(a) Three-dimensional natural U-tube mode for a pinned contact line. The full domain has been resolved, but only a quarter of it is shown here for visualization purposes. (b) Axial velocity profile plotted at different sections along the tube, as indicated by the colored arrows. The liquid column length in (a) and (b) has been set to $l=14.6\text{cm}$. (c) Dimensional oscillation period, $T=2\pi /\omega$, associated with the pinned contact line dynamics and as a function of the liquid column length, $l$. (d) Same as in (c), but for the dimensional damping coefficient. In (c) and (d), empty circles correspond to the present 3D calculation, black crosses are from our axisymmetric model, while filled black diamonds are experimental measurements from Ref. [53]. Only one measurement has been reported for the oscillation period.

Figure 18

(a) Nondimensional, $\sigma$, and (b) dimensional, $\overline{\sigma }=\sigma \sqrt{2g/l}$, damping coefficient versus the water column length, $l\left(\text{cm}\right)$ and associated with a free contact line dynamics of the fundamental U-tube mode for $\alpha =0\text{rad}$. Blue diamonds: values computed fully numerical eigenvalue calculation by accounting for the variable slip length model (16). The red solid lines correspond to the analytical estimate of the damping coefficient as estimated in this Appendix according to Eq. (B12). The vertical black dashed lines in (a) and (b) indicate the length of the U-turn region, $\pi R\approx 7\text{cm}$. Below this length, the liquid column is all contained in the U-turn region. In proximity and, particularly, below this limit value (as indicated by the gray-shaded regions), neglecting the curvature of the tube is no longer a justifiable assumption.

Figure 19

Different slip-length models employed in this Appendix: (a), (b) exponential, (c), (d) Gaussian, (e), (f) linear, (g), (h) bell, and (i), (j) inverse. The abscissa in panels (a), (c), (e), (g), and (i) is in linear scale, whereas the logarithmic scale is used in panels (b), (d), (f), (h), and (j). Black solid lines show the various continuous functions; while markers represent the Chebyshev collocation points actually resolved. For the specific calculations outlined in this Appendix, 40 radial and 200 axial points were used.

Figure 20

(a) Dimensional damping coefficient associated with the fundamental U-tube mode, $\overline{\sigma }=\sigma \sqrt{2g/l}$, numerically computed for a fixed water column length, $l=14\text{cm}$ and for a free contact line dynamics $\alpha =0\text{rad}$, but changing the depth-dependent slip-length model and its main free parameter $\delta$, i.e., the extension of the slip region. Different markers correspond to different models, namely, (circles) exponential, (squares) Gaussian, (triangles) linear, (diamonds) bell, and (crosses) inverse. The black dashed line indicates the analytical estimate according to Eq. (B12) for $l=14\text{cm}$. The abscissa value $\sqrt{2\nu /{\omega }_0}$, with ${\omega }_0^2=2g/l$, denotes the thickness of the Stokes boundary layer. (b) Corresponding dimensional oscillation period. The black dotted line corresponds to the theoretical value $2\pi /{\omega }_0=2\pi \sqrt{l/2g}$ and serves to highlight the viscous correction introduced by the sidewall boundary layer, whose numerical description according to the variable slip-length model adopted. The viscous frequency correction to the theoretical value (horizontal dotted line) is also indicated by the black vertical double-headed arrow.