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Freezing a rivulet

Abstract : We investigate experimentally the formation of the particular ice structure obtained when a capillary trickle of water flows on a cold substrate. We show that after a few minutes the water ends up flowing on a tiny ice wall whose shape is permanent. We characterize and understand quantitatively the formation dynamics and the final thickness of this ice structure. In particular, we identify two growth regimes. First, a 1D solidification diffusive regime, where ice is building independently of the flowing water. And second, once the ice is thick enough, the heat flux in the water comes into play, breaking the 1D symmetry of the problem, and the ice ends up thickening linearly downward. This linear pattern is explained by considering the confinement of the thermal boundary layer in the water by the free surface. The partial freezing of Niagara falls and the cancellation of thousands of flights during the cold snap of winter 2019 are a few examples of the disturbances caused by extreme weather events. Indeed, the accretion of ice on superstructures such as planes [1-3], power-lines [4], bridge cables [5] or wind turbines [6] can have dramatic consequences. Nowadays, the main strategy to prevent most of these undesirable effects is to develop anti-icing surfaces [7, 8], but new paradigms could emerge from a better understanding of the freezing dynamics in complex configurations. When water flows on a cold surface for example, the resulting ice structure is reminiscent from the manifold interaction between the heat transport and the flow [9]. The presence of a free-surface is also determinant in these problems, resulting in the apparition of a tip on frozen sessile drops [10], or in the explosion of droplets cooled from the outside-in [11] in static conditions. Freezing of capillary flows, widely encountered in the previous examples, can consequently reveal a very rich behavior [12-15] as in the formation of icicles [16] or ice structures following drop impact on cold surfaces [17]. In this Letter, we investigate experimentally the freezing of a capillary water river, the so-called rivulet [18-20] (see Fig. 1), flowing over a cold substrate. We show for the first time that the growing ice structure reaches a static shape after few minutes. The water then flows on a tiny ice wall that thickens downward, an observation we quantitatively explain considering the confinement of a thermal boundary layer. These results bring new understanding of the ice crust formation in the presence of streaming water and improve the prediction of its shape. The experiment consists in flowing distilled water dyed with fluorescein at 0.5 g.L −1 along a cold aluminum block of 10 cm long, with an inclination of α = 30 • to the horizontal. The temperature of the injected water T in ranges from 8 to 35 • C, see Fig. 1(a). The water is injected through a needle (inner diameter 1.6 mm) at a flow-rate Q = 20 mL.min −1 , such that there is no meander at room temperature [19]. A straight water rivulet is then formed [18], with a typical width of 2 mm, a thickness of h w = 800 µm, and a characteristic velocity of the buoyant flow u 0 ≈ 10 cm.s −1. As the Reynolds number of the flow is sufficiently small (Re = u 0 h w /ν = 80), the flow is laminar and mass conservation implies that the liquid layer thickness h w is constant [18]. The temperature of the aluminum substrate T s is set by plunging the block in liquid nitrogen for a given amount of time so that it ranges from −9 to −44 • C. T s is measured during the experiment and remains constant (±1 • C). Experiments performed with substrate temperatures below −44 • C consistently lead to the fracture [21] or the self-peeling [22] of the ice and are not considered here. Upon contact with the cold substrate, the water freezes and an ice layer grows while the water continues to flow on top, as shown on the sequence of snapshots of inset in Fig. 1(a) and in the Sup. Mat. movie. During that process, the fluorescein concentrates between the ice dendrites, causing self-quenching and fluorescence dimming in the ice [23]. This allows us to clearly distinguish between the ice and the water layers under UV light. The ice layer thickness h i (x, t) is then measured using a Nikon D800 camera recording from the side at 30 fps. The setup is placed in a humidity control box to avoid frost formation (H r ≈ 5 − 10%). Figure 1(b) presents the ice layer profile along the direction of the flow (x = 0 at the needle) at different times for T in =10 • C and T s =-36 • C. The analysis is restricted to the middle of the plate (x ∈ [1,8] cm) to avoid input and output influences. At early times, the ice layer grows homogeneously along the plane and the successive profiles are parallel to the substrate. After that, the ice layer continues to grow but not uniformly: its thickness increases along the plane. Finally, the ice layer stops growing and the system reaches a permanent regime consisting of a static ice structure, of thickness h max , on top of which a water layer is flowing. The final shape of the ice can be well described by a line of slope β as illustrated by the dashed lines in Fig. 1(b): h max (x) = h i,0 + βx, with β varying in our experiments between 0 and 4 •. The
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Submitted on : Tuesday, November 17, 2020 - 11:45:01 AM
Last modification on : Tuesday, December 8, 2020 - 3:35:49 AM
Long-term archiving on: : Thursday, February 18, 2021 - 7:12:53 PM

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Antoine Monier, Axel Huerre, Christophe Josserand, Thomas Séon. Freezing a rivulet. Physical Review Fluids, American Physical Society, 2020, 5 (6), ⟨10.1103/PhysRevFluids.5.062301⟩. ⟨hal-02882552⟩

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