Theory for undercompressive shocks in tears of wine

We revisit the tears of wine problem for thin films in water-ethanol mixtures and present a model for the climbing dynamics. The formulation includes a Marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. The prior literature did not address the wine tears but rather the behavior of the film at earlier stages and the behavior of the meniscus. In the lubrication limit we obtain an equation that is already well known for rising films in the presence of thermal gradients. Such models can exhibit nonclassical shocks that are undercompressive. We present basic theory that allows one to identify the signature of an undercompressive wave. We observe both compressive and undercompressive waves in new experiments, and we argue that, in the case of a preswirled glass, the famous “wine tears” emerge from a reverse undercompressive shock originating at the meniscus.

Figure 1

(Left) Schematic illustration of a conical-shaped cocktail glass of inclination angle $\alpha$ and (right) the corresponding one-dimensional thin wine film traveling up an inclined flat glass surface. The film height $h^{*}$ is exaggerated for clarity of the illustration.

Figure 2

Schematics for the characteristics of (left) a compressive shock that satisfies (13) and (middle) an undercompressive shock that satisfies (14) shown in a moving reference frame in which the shock is stationary. (Right) Flux function $f\left(h\right)$ in (9) with a compressive connection from $h_{\infty ,C}=0.4$ to $b=0.1$ and an undercompressive connection from $h_{\infty ,UC}=0.6$ to $b=0.1$ where $h_{\infty }$ and $b$ are the left and right boundary conditions of Eq. (7).

Figure 3

(Left) Numerical simulation of (7) with parameters extracted from experiment (A) [2]. The simulation illustrates a compressive-undercompressive shock pair in the long time with $b=0.0317$ and $h_{\infty }=0.43$. (Right) Consecutive time steps $\sim 0.1\left(\text{s}\right)$ starting from the meniscus initial condition.

Figure 4

(Left) Shock types of the hydrodynamic model (7) discussed in the present study. (Right) A shock bifurcation diagram parametrized by $(h_{\infty },D)$ pairs obtained by numerically solving the ODE (18a) subject to the boundary condition (18b) for $b=0.0353$.

Figure 5

Numerical simulations of Vuiellemuier et al. experiments (BI), (BII) for long times. (Left) experiment (BI) exhibiting a less distinct compressive-undercompressive double shock; cf. Fig. 3. (Right) experiment (BII) exhibiting undercompressive shock.

Figure 6

A comparison of shock types affected by the precursor thickness $b$, showing that (left) $b=0.0353$ results in a compressive-undercompressive shock and (right) $b=0.1585$ (corresponding to a hypothetical dimensional thickness $b^{*}=10\mu \mathrm{m}$) leads to the formation of a compressive wave. The other parameters in the two simulations are identical to those in Fig. 3 and correspond to measurements from experiment A.

Figure 7

The evolution of film height of Ref. [6] (wine setting) with $b=0.028$ (corresponding to a hypothetical dimensional thickness $b^{*}=0.5\mu \mathrm{m}$) showing the formation of a compressive wave. The meniscus dynamics with $D=0.0338$ yields a left state thickness $h_{\infty }=0.036.$

Figure 8

Spontaneous climb images of ethanol-water mixture with ethanol concentration $C=0.7$ on a dry watch glass (no preswirl) of diameter 75 mm and angle ranging between $9^{\circ }<\alpha <{20}^{\circ }$.

Figure 9

Long-time shock profiles of (22) for $A=0$ (no curvature effects) and $A=2.83, x_0=5$ (with curvature effects) at $t=700$. Other system parameters are $b=0.0317, D=0.353$ corresponding to experiment (A).

Figure 10

Top view images of tears of wine experiment at $t=0,5,10,20$ s in a stemless martini glass (conical substrate) using $18\%$ alcohol by volume port wine. Swirling the wine around the glass creates a reverse front that forms out of the meniscus, advances up the glass, and destabilizes into wine tears.

Figure 11

Tears of wine experiment. (Left) Top view and (right) side view and projection of a stemless martini glass with inclination angle $\alpha ={65}^{\circ }$, using $18\%$ port wine. Swirling the wine around the glass creates a front that forms out of the meniscus. The draining film advances up the glass and destabilizes into wine tears.

Figure 12

(Left) The formation of a reverse-undercompressive (RUC) shock with initial condition (23) and boundary conditions (24). (Right) A closeup view of the boxed region on the left, at dimensionless time $t=800$, showing a good agreement with the rarefaction wave (dot-dashed line) $H\left(\eta \right)$ in (12) for $\eta =x/t$. The rarefaction separating from the RUC shock is a signature of a nonclassical shock; it is shown by the flat state $h_l$ between the rarefaction and the RUC shock.

Figure 13

(Left) A comparison of shock solutions at $t=450$ for initial data (23) with varying $h_{\infty }$ showing reverse-undercompressive (RUC) shocks for $h_{\infty }=0.6,0.8$, and a single rarefaction wave for $h_{\infty }=0.4$. The critical thicknesses $h_{\infty }$ and $h_l$ are marked for the $h_{\infty }=0.8$ profile. Note that the rarefaction part of the solution is independent of $h_{\infty }$. (Right) The flux diagram with two undercompressive connections for the RUC shocks with $h_{\infty }=0.6,0.8$.

Figure 14

(Left) Advancing waves starting from initial condition (17) at times $t=0,100,200,300$ governed by (22) with meniscus boundary conditions (16), showing a comparison of fronts influenced by curvature effects $(A=68.9,x_0=25)$ and without curvature effects $(A=0)$. (Right) A closeup view of the boxed region on the left at time $t=300$ with the curve for $A=68.9$ shifted by $\mathrm{\Delta }x=5.3$. The other settings are identical to those in Fig. 12.