Stretching water between two grooves

TL;DR

The paper addresses the challenge of forming and stabilising continuous thin water films on untreated vertical surfaces. It introduces a purely geometrical solution—a pair of laser-engraved vertical grooves—that edge-pin the film and sustain a film thicker than typical capillary-thin films, with height exceeding $100$ capillary lengths and thickness $e$ tunable via the input flow rate $Q_{ m in}$ and groove geometry. The authors develop a coupled experimental–theoretical framework: PTV measurements reveal a parabolic velocity profile and, together with a lubrication-based model, establish a thickness scaling $e \sim (Q_f/s)^{1/3}$ after groove saturation, while a gravity–capillarity balance predicts the retraction height $H_{ m max} \sim (s/Q_f)^{1/3}$ and a robust cycle that yields regular droplet release. Taken together, the work demonstrates a simple, scalable method for passive liquid control—stabilising, shaping, and triggering droplets from thin films using just groove geometry, with potential applications in condensation management, surface cleaning, and 2D millifluidic systems.

Abstract

Controlling water motion on surfaces is critical for applications ranging from thermal management, passive water harvesting, to self-cleaning coatings. Yet stabilising continuous water films, desirable for their high surface coverage and drainage capacity, remains challenging with pure water, due to its high surface tension. Existing strategies rely on extreme wettability achieved by coating or fine-scale patterning, which are costly, fragile, or complex to scale. A robust, purely geometric solution is still lacking. We demonstrate that a pair of laser-engraved grooves on a moderately hydrophilic vertical substrate can laterally anchor water and stretch a continuous thin film, without any other surface treatment. Once stabilised, the film extends vertically over 100 capillary lengths (> 30cm), with thickness tunable via both flow rate and groove geometry. At the groove extremities, the end of anchoring triggers a cyclic instability, characterised by film rupture, retraction, and droplet release. The thickness of the film and retraction height obey predictive models, while droplet mass varies systematically with spacing and surface tension. This groove-based method offers a straightforward and scalable approach to creating, sustaining, and controlling thin water films. It opens new directions for passive liquid control in condensation, surface cleaning, and 2D millifluidic systems.

Figures (8)

• Figure 1: Experimental setup and groove geometry. (a) Front and side views of the acrylic substrate (TroGlass Clear, $3\,\mathrm{mm}$ thick) engraved with two vertical grooves spaced by a distance $s$. The grooves, $115\,\mathrm{mm}$ long, terminate a few centimetres above the bottom edge to trigger film rupture. A syringe pump delivers water between the grooves through a $0.8\,\mathrm{mm}$ nozzle at a flow rate $Q_{\mathrm{in}}$. The side view illustrates how the thin film of thickness $e$ connects the syringe tip to the droplet near the groove termination. (b) Groove geometry visualised using a Keyence VHX optical microscope. The three defining parameters are groove depth $d$, width $w$, and centre-to-centre spacing $s$. The example shown corresponds to a substrate with $s = 1.75\,\mathrm{mm}$ and $d/w = 0.54$. Arrows illustrate the partitioning of the injected flow $Q_{\mathrm{in}}$ into a film contribution $Q_{\mathrm{f}}$ and groove contribution $Q_{\mathrm{g}}$. (c) Side-view image showing a retracting droplet at the lower end of the film. We measure the front contact angle $\theta_{\mathrm{F}}$ throughout the periodic instability. In contrast, we assume the rear angle to be zero, given the continuous and smooth connection with the film.
• Figure 2: Sequence of flow regimes observed on vertical substrates as a function of flow rate $Q_{\mathrm{in}}$. Each image displays the fluid morphology for increasing $Q_{\mathrm{in}}$, with a checkered background that reveals film thickness via refraction: compressed patterns indicate higher curvature and thicker liquid layers. Schematics above each image illustrate the fluid cross-section at the location marked by the colored line in the corresponding photo. Smooth surfaces are shaded purple, and grooved cross-sections are shaded orange. At low flow rate on a smooth substrate, droplets form and slide once their weight overcomes contact angle hysteresis (a). At high flow rates, these droplets merge into a rivulet that eventually meanders due to inertial effects (f). On grooved substrates, new regimes emerge. When grooves are dry, they confine droplet width but do not transport liquid (b). Once prewetted, the same $Q_{\mathrm{in}}$ yields Groove Flow, where a puddle at the syringe tip feeds the grooves and droplets detach from the bottom (c). At a higher flow rate, a stabilised film stretches between the grooves and periodically ruptures at their termination, initiating the Film Flow regime (d). At very high flow, this film becomes a rivulet, anchored by the grooves and held in a straight path (e).
• Figure 3: Cyclic dynamics of film rupture and droplet release in the stabilised film regime. Top and bottom rows show synchronised front and side views of the same experiment: groove spacing of $s = 2.25\,\mathrm{mm}$, an aspect ratio $d/w = 1.80$, and an input flow rate of $Q_{\mathrm{in}} = 3.33\,\mathrm{mm^3/s}$. The time elapsed between each pair of images is indicated. (Top) A checkered pattern is placed between the backlight and the substrate to visualise film curvature via refraction: strong pattern deformation signals thicker fluid layers. Yet the film between the droplet and syringe is nearly imperceptible; the pattern remains barely distorted, highlighting how thin the film is. As the droplet slides and crosses the groove termini, at $t=0\,\rm{s}$, lateral confinement disappears and the film ruptures. The film retracts upward, forming at its base a new puddle that grows as the film pulls back. Retraction and early puddle growth occur within $0.64\,\mathrm{s}$. The puddle then pins at a stable retraction height $H_{\mathrm{max}}$, where capillary forces balance its weight, and continues to inflate for another $2.5\,\mathrm{s}$ (until $t = 3.07\,\mathrm{s}$). Once the puddle becomes heavy enough, it depins and slides downward over approximately $5\,\mathrm{s}$ (reaching $t = 7.94\,\mathrm{s}$). When the rear of the droplet reaches the end of the groove, the film ruptures again, restarting the cycle. (Bottom) Side view of the same dynamics, emphasising puddle inflation and the evolution of the front contact angle $\theta_{\mathrm{F}}$, which increases steadily until the droplet detaches.
• Figure 4: Film velocity measurements using particle tracking velocimetry (PTV). (a) Superposition of 2000 consecutive frames showing $20\,\mu\mathrm{m}$ tracer particles flowing through a thin film confined between two grooves spaced by $s=2.25\,\mathrm{mm}$ with $Q_{\rm{in}} = 6.67\,\rm{mm^3/s}$. Tracers cluster near the centre, indicating faster flow in this region. (b) Reconstructed horizontal velocity profiles across the film, exhibiting a parabolic shape consistent with confined Stokes flow. The apparent “filled” profiles reflect the 3D nature of the flow: particles occupy different heights, resulting in multiple velocities being detected at each horizontal position. (c) Maximum velocity $V_{\mathrm{max}}$, averaged from the 10 fastest particles, plotted as a function of $Q_{\mathrm{in}}$ for various groove spacings $s$ and geometries $d/w$. Velocity increases with $Q_{\mathrm{in}}$ and reveals two trends: (i) at fixed groove geometry (same shade), smaller spacings (blue) yield higher $V_{\mathrm{max}}$ due to thicker films; (ii) at fixed spacing (same color), deeper grooves (darker tones) reduce $V_{\mathrm{max}}$, consistent with flow being diverted into the grooves.
• Figure 5: Film thickness measurements and model validation. (a) Comparison between experimental film thickness $e_{\mathrm{PTV}}$, inferred from particle tracking velocimetry (Equation \ref{['eq:ePTV']}), and the theoretical prediction $e_{\mathrm{flow}}$ based on a uniform Stokes flow model (Equation \ref{['eq:eflow']}). A systematic offset (slope $\xi = 4/5$) is attributed to the finite width of the film, which reduces the effective flow area. (b) Collapse of all measured film thicknesses $e_{\mathrm{PTV}}$ as a function of scaled film flow rate $Q_f/s$. This collapse confirms the model central assumption: once groove flow saturates, the remaining input is redirected entirely into the intergroove film (Equation \ref{['eq: saturation']}). The dashed line represents the predicted model (Equation \ref{['eq: thickness']}), which is validated across various groove geometries and spacings. (c) Schematic showing how a corner enables the triple line to accommodate a range of contact angles $\Delta\theta$, enhancing pinning and film stability. (d) Vertical cross-section of the outer groove edge. Despite thinning at low $Q_{\mathrm{in}}$, the film remains pinned as the contact angle stays within the hysteresis range (grey dashed lines). The red circle highlights a region of steep velocity gradients between film and groove flow, where small perturbations may trigger dewetting.