Constant Contact Angle Research Articles

Effects of Heat Treatment on The Wettability and Color Properties of Betung Bamboo (Dendrocalamus Asper) Strand

Betung bamboo (Dendrocalamus Asper) has recently attracted interest as a sustainable reinforcement fiber in (polymer) composite materials. Utilization of bamboo has an advantages and disadvantages. It has a short growth cycle, lightweight, and good mechanical properties, but with poor wettability, dimensional instability, and low resistance againts biodeterioration agent. Some efforts have been devoted to alter and enhance its properties, especially wettability properties. Heat treatment under various temperature can enhance wettability properties due to chemical changes and migration of extractives, but also can induce color changes. Therefore, this study aims to investigate wettability properties and color changes of heat-treated Betung bamboo strand. The 1/3 middle of Betung bamboo was used as main material and converted into strand. Heat treatment temperatures applied were 140oC and 160oC for 3 hours. The wettability of Phenol Formaldehyde (PF) adhesive on heat treated Betung bamboo strand was measured using a sessile drop contact angle method. Constant contact angle and K-value were used as parameter of wettability. Color change of heat-treated Betung bamboo strand was measured by portable color difference meter model CDX 105 and characterized by CIE Lab. Results showed that the increasing temperature of heat treatment affected wettability and color properties. Contact angle tended to increase with an increase of temperature which is indicated an enhancement of wettability by making wood more hydrophobic. Increasing of temperature made Betung bamboo strand to be darker than that of control sample. Total color change (AE) values were 21.06 and 29.23 at 140 and 160 oC, respectively.

Droplet evaporation characteristics on heated superhydrophobic surface subjected to airflow

In a novel frost suppression method for air source heat pumps, the frost-free operation of the system is realized by vaporizing the condensed droplets on superhydrophobic heat exchanger fins before freezing. Due to the control requirements of this method, a multiphase flow model based on VOF method for droplet evaporation is proposed. The evaporation characteristics and survival time of a sessile water droplet on heated superhydrophobic surface subjected to airflow are investigated through the model. The convection and evaporative cooling effect are considered in the simulation, experiments showed that the prediction error for droplet survival time is within 8%. The effects of boundary layer thickness, substrate temperature, surface wettability, air temperature and humidity on the heat and mass transfer process of droplet evaporation are studied with this model. The survival time of water droplet with diameter below 100 μm evaporating in different evaporation modes is also calculated. The results showed that under the conditions of common parameters in winter, the survival time of a droplet evaporating in fully developed flow is in the range of 4∼20 s. The survival time of droplets evaporating in constant contact radius mode is 15.4% shorter than that evaporating in constant contact angle mode.

Parametric and algebraic study of an isolated pore shape in solid after unidirectional solidification

This study applies algebraic polynomials to investigate the effects of independent dimensionless parameters on the shape of an isolated pore resulting from a bubble completely entrapped by a solidification front. The pore shape in solid is contemporarily challenging in controlling directional properties of natural or functional materials in MEMS, packaging, manufacturing and medical and micro-or nano-technologies. In this work, polynomials with different degrees are provided by extending previous work, accounting for conservation of total solute content in the pore and boundary layer on the bubble cap, on which solute transport in different directions, balance of pressures and physico-chemical equilibrium are satisfied. Dimensionless independent parameters include product of Bond number with initial liquid concentration, solidification rate, mass transfer coefficient, Henry’s law constant, atmospheric pressure, and initial contact angle. By comparison with numerical solutions of thermal fluid and concentration equations, it is found that Cases 1 and 2 referred to different mass transfer coefficients and directions of solute transport from the pore to surrounding liquid in the early stage are relevant, rather than available Cases 3 and 4 with different modeling of solute transfer rates. Polynomials with 4 and 5 degrees to predict shapes of the isolated pore are in good agreement with numerical solution of simultaneous first-order unsteady ordinary equations governing solute gas pressure and geometry of the pore and experimental data.

Evaporation of a sessile droplet on flat surfaces: An axisymmetric lattice Boltzmann model with consideration of contact angle hysteresis

Droplet evaporation is a widely encountered heat and mass transfer phenomenon. Due to its 3D in nature, it is challenging to simulate the complex 3D droplet evaporation process. Here, we propose an axisymmetric lattice Boltzmann (LB) model to simulate the dynamics of contact line motion during sessile droplet evaporation on a flat heated surface, taking contact angle hysteresis into consideration. We demonstrate that this LB model can numerically simulate the evaporation processes of sessile droplet evaporation processes undergoing constant contact angle (CCA), constant contact radius (CCR), and mixed modes sequentially. The classical D2-law for droplet surface mass transfer is confirmed numerically by using the case of a floating droplet evaporation, thereby proving that the axisymmetric LB model is capable of 3D simulation. Our simulated results on temporal variations of contact radius, contact angle are found in good agreement with two different sets of literature experimental data. Specifically, we compute the local mass transfer rate on the droplet surface from the simulated velocity field to show that the evaporation rate near the triple-phase contact region is much higher than that at the droplet apex region, with the peak heat flux occurring near the location of the dynamic contact line. The contact angle fluctuates with stick-slip motion are found during the CCA stage of the evaporation process at higher substrate temperatures. Additionally, we analyse the non-quasi-equilibrium evaporation mode and contact line slip velocity on substrates surface at higher temperatures.

Measurement of Contact Angles at Carbon Fiber-Water-Air Triple Phase Boundaries inside Gas Diffusion Media of Polymer Electrolyte Membrane Fuel Cells from Xray Computed Tomography

Gas diffusion layers (GDLs) are porous fiber layers that are used for various electrochemical applications, including fuel cells, electrolyzers, CO2 reduction reaction electrolyzers, and redox flow batteries. Depending on its application, a GDL’s wettability is controlled by the addition of polytetrafluoroethylene (PTFE). The PTFE addition is highly stochastic, resulting in inhomogeneous distribution. Furthermore, the fibers, carbonaceous binder, and PTFE all have different surface roughness and surface properties that dictate GDL wettability. To determine the contact angle of the GDL, goniometer measurements of external contact angle are used, where a droplet of water is placed on the top of the GDL and a contact angle is projected onto a detector. However, this form of measurement has limitations that need to be accounted for as the GDL is highly porous and its surface and internal contact angles might be different due to inhomogeneous binder and PTFE distribution. For the first time, we have applied a method to calculate the internal contact angle distribution of water inside various commercial GDLs. Lattice Boltzmann water transport simulations using these contact angle distributions yielded results much closer to experimental values than simulations that used goniometer-measured constant contact angles. XPS was used to correlate surface oxide groups to measured internal contact angles.AlRatrout et al. developed an open-source C++ code to compute local contact angles at triple phase boundary points (comprised of a solid and two fluids) from micro Xray computed tomography (XCT) images of porous media [1]. We applied it to segmented micro XCT images of water filled commercial GDL samples and computed local contact angles at air-GDL-water triple phase contacts points (Figure 1A). We obtained a state of mixed wettability inside all GDL samples, with contact angles ranging from 0° to 180° inside the pores (Figure 1B). Several works have calculated average contact angles similar to our results using other methods [2, 3]. Hydrophilicity is due to the presence of surface oxide groups on GDL fibers while hydrophobicity is due to the chemical nature of carbon and the roughness of the GDL fibers. Lattice Boltzmann water transport simulations were performed with the spatially distributed, mapped contact angles and constant contact angles measured with a goniometer (Figure 1C-F) [4]. When our mapped, mixed-wettability contact angle distributions were used, the simulation revealed several regions of high liquid saturation that the constant contact angle simulations did not capture. These results agreed with experimental observations. Thus, for robust water transport simulations, we suggest that distributed contact angles should be used instead of goniometer-measured constant contact angles.

Hydrothermal waves in sessile droplets evaporating at a constant contact angle mode

Although various Marangoni instability patterns have been observed in evaporating sessile droplets, most are limited to a small contact angle. As a result, the droplets resemble a flat liquid layer, where the vertical temperature gradient dominates, and the hydrothermal waves (HTWs) predominantly driven by the tangential temperature gradient do not occur. In this paper, we focus on the possibility of HTWs in droplets with moderate contact angles using numerical simulation. For this purpose, a three-dimensional numerical model is proposed, in which the deformation of the droplet surface induced by evaporation can be directly calculated from the local evaporation flux. We predict fan-like HTWs with three blades in the droplet. The waves are located at the center region of the droplet and are uniformly distributed throughout the circumference; they simultaneously propagate along the anticlockwise and radial directions. With evaporation, the blades become shorter, whereas the fan-like shape and wave number remain without significant variation. The behavior of the fluid flow inside the droplet and its oscillatory characteristics were analyzed carefully. The numerical results were verified by experiments, and confirm that HTWs would only occur under moderate contact angles. Additionally, a higher substrate temperature or a larger initial radius would promote the appearance of the irregular fan-like HTWs with two blades, whereas a sufficient small initial radius will inhibit the appearance of HTWs. The effects of the substrate temperature, the initial radius and the contact angle on the critical radius and the corresponding critical normal Marangoni number (Mav,c) for the disappearance of the HTWs were also studied.

Droplet Evaporation on a Hydrophobic Photothermal Conversion Substrate

Use of the photothermal effect to control droplet evaporation shows great promise in various applications. Droplet evaporation initiated by direct photothermal conversion between light and fluid is limited by the interaction between the wavelength and fluid type. Here, we report laser-induced sessile droplet evaporation on a hydrophobic photothermal conversion substrate formed by embedding Ti₂O₃ particles into polydimethylsiloxane. Our results indicated that the photothermal conversion substrate could efficiently convert light energy into heat to enable droplet evaporation on it. Unlike droplet evaporation actuated by direct photothermal conversion, in which higher interface temperature was located at the laser projection region, the droplet evaporation on the photothermal conversion substrate showed higher temperature near the triple-phase contact line because of heat transfer from the substrate to the droplet and small heat transfer resistance resulting from small height at the edge. Due to the increased substrate temperature, the droplet evaporation on the photothermal conversion substrate could exhibit a relatively stable and long-term constant contact angle mode even under the localized heating effect. Moreover, by controlling the laser beam position, droplet evaporation could exhibit a directional long-term constant contact angle mode. The obtained results will be helpful for the design and operation of droplet microfluidics based on the photothermal conversion substrate.

Evaporation characteristics and heat transfer enhancement of sessile droplets under non-uniform electric field

Electrohydrodynamic (EHD) enhancement of water droplet evaporation by non-uniform electric field and the subsequent dynamic characteristics were experimentally investigated in this paper. A pin-to-plate electric field was produced by a needle connected to a high-voltage power supply and the grounded heated surface coated by thin Teflon film with a contact angle of about 100°. The droplet was generated onto the hot surface by a micro-syringe with a constant initial volume of 4 μl, and a high-speed camera was used to capture the droplet morphology. Effects of electric field and surface temperature on the contact angle, droplet height, evaporation stage, volume variation, and evaporation rate were studied. The results showed that both the increase in external electric field strength and surface temperature enhance droplet evaporation with a maximum enhancement ratio of 6.8 times in the present work. In addition, a special extra constant contact angle (CCA) stage was found in droplet evaporation under electric field, while only constant contact radium (CCR) and Mixed stages existed in neutral (no voltage) evaporation. In particular, the evaporation enhancing mechanism of no-uniform electric field considering corona wind blowing, surface tension weakening and molecular orientational alignment was also presented.

Evaporation of Sessile Droplets of Polyelectrolyte/Surfactant Mixtures on Silicon Wafers

The wetting and evaporation behavior of droplets of aqueous solutions of mixtures of poly(diallyldimethylammonium chloride) solution, PDADMAC, with two different anionic surfactants, sodium laureth sulfate, SLES, and sodium N-lauroyl N-methyl taurate, SLMT, were studied in terms of the changes of the contact angle θ and contact length L of sessile droplets of the mixtures on silicon wafers at a temperature of 25 °C and different relative humidities in the range of 30–90%. The advancing contact angle θa was found to depend on the surfactant concentration, independent of the relative humidity, with the mixtures containing SLES presenting improved wetting behaviors. Furthermore, a constant droplet contact angle was not observed during evaporation due to pinning of the droplet at the coffee-ring that was formed. The kinetics for the first evaporation stage of the mixture were independent of the relative humidity, with the evaporation behavior being well described in terms of the universal law for evaporation.

Forced oscillation dynamics of surface nanobubbles.

Surface nanobubbles have potential applications in the manipulation of nanoscale and biological materials, waste-water treatment, and surface cleaning. These spherically capped bubbles of gas can exist in stable diffusive equilibrium on chemically patterned or rough hydrophobic surfaces, under supersaturated conditions. Previous studies have investigated their long-term response to pressure variations, which is governed by the surrounding liquid's local supersaturation; however, not much is known about their short-term response to rapid pressure changes, i.e., their cavitation dynamics. Here, we present molecular dynamics simulations of a surface nanobubble subjected to an external oscillating pressure field. The surface nanobubble is found to oscillate with a pinned contact line, while still retaining a mostly spherical cap shape. The amplitude-frequency response is typical of an underdamped system, with a peak amplitude near the estimated natural frequency, despite the strong viscous effects at the nanoscale. This peak is enhanced by the surface nanobubble's high internal gas pressure, a result of the Laplace pressure. We find that accurately capturing the gas pressure, bubble volume, and pinned growth mode is important for estimating the natural frequency, and we propose a simple model for the surface nanobubble frequency response, with comparisons made to other common models for a spherical bubble, a constant contact angle surface bubble, and a bubble entrapped within a cylindrical micropore. This work reveals the initial stages of growth of cavitation nanobubbles on surfaces, common in heterogeneous nucleation, where classical models based on spherical bubble growth break down.

Impact and lifecycle of superfluid helium drops on a solid surface

Superfluid helium droplets impacting on a solid surface behave much differently than any other fluid. After a short period of initial spreading that is similar to classical fluids, superfluid helium drops quickly shrink and disappear as the superfluid drains out through a thin adsorbed layer of helium on the surface. The lifetime and contact angle of these drops is strongly temperature-dependent, and colder drops (with high superfluid fractions) maintain a constant contact angle throughout the contraction. Above ${T}_{l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}b\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}a}$, helium spreads slowly like other classical fluids.

Evaporation and Electrowetting of Sessile Droplets on Slippery Liquid-Like Surfaces and Slippery Liquid-Infused Porous Surfaces (SLIPS).

Sessile droplet evaporation underpins a wide range of applications from inkjet printing to coating. However, drying times can be variable and contact-line pinning often leads to undesirable effects, such as ring stain formation. Here, we show voltage programmable control of contact angles during evaporation on two pinning-free surfaces. We use an electrowetting-on-dielectric approach and Slippery Liquid-Infused Porous (SLIP) and Slippery Omniphobic Covalently Attached Liquid-Like (SOCAL) surfaces to achieve a constant contact angle mode of evaporation. We report evaporation sequences and droplet lifetimes across a broad range of contact angles from 105°–67°. The values of the contact angles during evaporation are consistent with expectations from electrowetting and the Young-Lippman equation. The droplet contact areas reduce linearly in time, and this provides estimates of diffusion coefficients close to the expected literature value. We further find that the total time of evaporation over the broad contact angle range studied is only weakly dependent on the value of the contact angle. We conclude that on these types of slippery surfaces, droplet lifetimes can be predicted and controlled by the droplet’s volume and physical properties (density, diffusion coefficient, and vapor concentration difference to the vapor phase) largely independent of the precise value of contact angle. These results are relevant to applications, such as printing, spraying, coating, and other processes, where controlling droplet evaporation and drying is important.

Dynamic wetting failure in curtain coating by the Volume-of-Fluid method

In this paper, we investigate dynamic wetting in the curtain coating configuration. The two-phase Navier–Stokes equations are solved by a Volume-of-Fluid method on an adaptive Cartesian mesh. We introduce the Navier boundary condition to regularize the solution at the triple point and remove the implicit numerical slip induced by the cell-centered interface advection. We use a constant contact angle to describe the dynamic contact line. The resolution of the governing equations allows us to predict the substrate velocity at which wetting failure occurs. The model predictions are compared with prior computations of Liu et al. [C.Y. Liu, E. Vandre, M. Carvalho, S. Kumar, J. Fluid Mech. 808, 290 (2016); C.Y. Liu, M. Carvalho, S. Kumar, Chem. Eng. Sci. 195, 74 (2019)] and experimental observations of Blake et al. [T. Blake, M. Bracke, Y. Shikhmurzaev, Phys. Fluids 11, 1995 (1999)] and Marston et al. [J. Marston, V. Hawkins, S. Decent, M. Simmons, Exp. Fluids 46, 549 (2009)].

Numerical study of droplet evaporation on heated flat and micro-pillared hydrophobic surfaces by using the lattice Boltzmann method

The evaporation processes of sessile droplets on both heated flat and structured hydrophobic surfaces are numerically studied by using a thermal multi-phase lattice Boltzmann model. The contact-line dynamics and heat transfer regarding the evaporation of droplets are simulated and the effects of the heat conductivity, wettability, and roughness of the substrate on the droplet evaporation are investigated. The results indicate that the different evaporation modes of the droplets on the rough surfaces are attributed to the microscopic “stick–slip” motion of the contact line. The constant contact angle (CCA) mode of droplet evaporation on the rough hydrophobic surface can be regarded as a macroscopic average of the microscopic “stick–slip” processes with a short stick period and small change in apparent contact angle. The weaker stick effect of the surface with a higher hydrophobicity and smaller solid fraction makes the droplet evaporation more likely to occur in the CCA manner.

Evaporation dynamics of water droplets on superhydrophobic nanograss surfaces

Evaporation of a sessile droplet on superhydrophobic surfaces has various applications due to their utmost water-repellent and self-cleaning wetting properties. We experimentally investigate the evaporation dynamics of water droplets and the time evolution of the contact angle and drop dimensions on superhydrophobic, nanograss substrates of an extremely-low solid packing fraction (ϕ ≈ 0.005). The experimental data shows that all the droplets deposited initially form a gas-trapping, Cassie-Baxter state. Small droplets subsequently evaporate with a constant contact angle mode, followed by a mixed mode at the end of the droplet lifetime. On the contrary, for relatively large droplets, two distinct evaporation modes are found. Some of the larger evaporating droplets were initially in a constant contact angle mode and underwent a mixed mode, while others began with a mixed mode with slowly decreasing base diameter and contact angle. Intriguingly, stick-slip motions of the contact line for large droplets are observed using superhydrophobic nanograss surfaces. Such slip or jumping motion could be related to the excess free energy available in the system when the receding contact angle is reached, resulting in the contact line movement. Finally, the experimental data of contact angle dependent evaporative mass flux are found to nearly collapse onto one universal curve for different droplet sizes and initial contact angles, in agreement with an evaporative cooling model.

Magnetic suppression of the coffee ring effect

This paper reports the evaporation and drying characteristics of ferrofluid droplets on a PDMS (Polydimethylsiloxane) substrate under the influence of magnetic field. The effect of magnetic field strength and asymmetry of the external magnetic field with respect to the axis of the droplet is studied. Goniometric measurement is used for visualization of the evaporating droplets. Confocal microscopy is used for imaging the drying pattern and particle motion inside the droplet. The magnetic field strength is controlled by varying the distance between the magnet and the substrate. COMSOL Multiphysics software is used to report the magnetic field and magnetic force distribution. The uniform deposition of drying pattern is observed under the influence of magnetic field. The effect of magnetic field on evaporation dynamics is attributed to the modification in drying pattern. The evaporation of the droplet takes place in constant contact radius (CCR) mode, constant contact angle (CCA) mode, mixed mode, and late pinning mode. Both sliding of the droplet and receding of the contact line is observed during evaporation when the center of magnet is not aligned with the center of the droplet. The suppression in duration of the CCR and CCA modes of evaporation and increase in duration of mixed mode evaporation is observed at higher magnetic field strength. The initial contact angle decreases and receding contact angle increases with increasing strength of magnetic field. This is attributed to lower concentration of nanoparticle in the contact line region resulting in lower frictional resistance to the contact line motion. Hump formation takes place at the apex region of the droplet, which depends on the strength of the magnetic field and local concentration of nanoparticle. The hump formation is followed by splitting of the droplet.

Uncertainty of rivulet flows on heterogeneous surfaces – A numerical study

This paper presents a three-dimensional numerical simulation of water flow on an inclined plate using the software package OpenFOAM. We specifically investigate the influence of a heterogeneous surface on the flow pattern and the degree of wetting. By analyzing 400 simulation results a recently developed method for applying random contact angle inhomogeneities to surfaces is investigated. The simulations were carried out using the empirical contact angle model, which enables stable rivulet development. Part of the analysis is conducted with the help of the statistical software Dakota, which provides information about the probability distribution of the amount of wetted area when varying the two important influencing factors: the amplitude of the contact angle disturbance or the size of contiguous hydrophobic or hydrophilic areas. The results show that a variation of the two input parameters as desired leads to the introduction of a random component and thus to a change in the flow behavior. However, the changes remain within a narrow range around the mean value and produce only a few outliers, which occur mainly at large contact angle disturbance amplitudes and thus indicate the determining influence of the amplitude. The method has proven useful for performing several runs of the same simulation configuration for collecting statistics. Furthermore the inclination of the plate and the contact angle using a constant contact angle model are varied. Here, a clear dominance of the influence of the inclination also manifests over a wide contact angle range.

The evaporation of surfactant-laden droplets: A comparison between contact line models

HypothesisThere are two different sharp-interface models for moving contact lines: slip models and precursor film models. While both models predict a mostly constant contact angle during the evaporation of pure droplets, it is expected that they behave differently when surfactants are present, because of the inherent dissimilarities in their respective interface definitions. SimulationsBoth contact line models are numerically implemented using lubrication theory to analyze evaporating droplets. A convection-diffusion equation is implemented for insoluble surfactants. For pure droplets the models are compared with experiments performed by Nguyen et al. (2012). FindingsThe two contact line models show results comparable to the experiments with pure droplets. If insoluble surfactants are present, the slip model increasingly shows pinning-like behavior as the initial surfactant concentration is increased. This ‘quasi-pinning’ is found to be consistent with experimental results in literature. The precursor film model, in contrast, shows no significant change when surfactants are added. This lack of change is a result of surfactant flowing from the droplet into the precursor film and vice versa. While suggesting potential solutions to this unphysical behavior, it is concluded that in the context of surfactants, slip models are preferable over precursor film models given the current state of the art.

Interfacial Strategies for Smart Slippery Surfaces

The problem of contact line pinning on surfaces is pervasive and contributes to problems from ring stains to ice formation. Here we provide a single conceptual framework for interfacial strategies encompassing five strategies for modifying the solid-liquid interface to remove pinning and increase droplet mobility. Three biomimetic strategies are included, (i) reducing the liquid-solid interfacial area inspired by the Lotus effect, (ii) converting the liquid-solid contact to a solid-solid contact by the formation of a liquid marble inspired by how galling aphids remove honeydew, and (iii) converting the liquid-solid interface to a liquid-lubricant contact by the use of a lubricant impregnated surface inspired by the Nepenthes Pitcher plant. Two further strategies are, (iv) converting the liquid-solid contact to a liquid-vapor contact by using the Leidenfrost effect, and (v) converting the contact to a liquid-liquid-like contact using slippery omniphobic covalent attachment of a liquid-like coating (SOCAL). Using these approaches, we explain how surfaces can be designed to have smart functionality whilst retaining the mobility of contact lines and droplets. Furthermore, we show how droplets can evaporate at constant contact angle, be positioned using a Cheerios effect, transported by boundary reconfiguration in an energy invariant manner, and drive the rotation of solid components in a Leidenfrost heat engine. Our conceptual framework enables the rationale design of surfaces which are slippery to liquids and is relevant to a diverse range of applications.

Rivulet flow down a slippery substrate

A detailed analysis of small-scale locally unidirectional gravity-driven rivulet flow with prescribed volume flux down an inclined slippery substrate for a rivulet with either constant width (i.e., pinned contact lines) or constant contact angle is undertaken. In particular, we determine the effect of varying the Navier slip length λ (i.e., the strength of the slip at the solid–fluid interface) on the rivulet. The present analysis shows that the shape and size of the rivulet and the velocity within it depend strongly on the value of λ. Increasing the value of λ reduces the viscous resistance at the substrate and, hence, leads to a larger velocity within the rivulet, and so the prescribed flux is achieved with a smaller rivulet. In particular, in the limit of strong slip, λ → ∞, for a rivulet of a perfectly wetting fluid and a rivulet with constant width, the velocity becomes large and plug-like like O(λ1/2) ≫ 1, and the rivulet becomes shallow like O(λ−1/2) ≪ 1, while for a rivulet with positive constant contact angle, the velocity becomes large and plug-like like O(λ2/3) ≫ 1, and the rivulet becomes narrow like O(λ−1/3) ≪ 1 and shallow like O(λ−1/3) ≪ 1.