1. Introduction
Bead-on-fibre patterns form in liquids flowing down a fibre due to hydrodynamic instabilities, such as the Kapitza (Ter Haar Reference Ter Haar2016) and Plateau–Rayleigh (Plateau Reference Plateau1873; Rayleigh Reference Rayleigh1878) instabilities. Such flows have received increased attention over the past decade, in part due to nature's nifty use of bead-on-fibre structures for water harvesting and transport (Zheng et al. Reference Zheng, Bai, Huang, Tian, Nie, Zhao, Zhai and Jiang2010; Ju et al. Reference Ju, Bai, Zheng, Zhao, Fang and Jiang2012; Xue et al. Reference Xue, Wang, Shi, Sun and Zheng2014; Guo & Tang Reference Guo and Tang2015), and our attempts to imitate it may help to curb the anticipated global water crisis (Shi et al. Reference Shi, Anderson, Tulkoff, Kennedy and Boreyko2018; Sadeghpour et al. Reference Sadeghpour, Zeng, Ji, Dehdari Ebrahimi, Bertozzi and Ju2019; Zeng, Sadeghpour & Ju Reference Zeng, Sadeghpour and Ju2019; Moncuquet et al. Reference Moncuquet, Mitranescu, Marchand, Ramananarivo and Duprat2022; Jin et al. Reference Jin, Zheng, Zhao, Ma, Wang, Qi and Kong2023). These high surface area flows are suitable for numerous heat and mass transfer applications, including heat exchange (Zeng et al. Reference Zeng, Sadeghpour, Warrier and Ju2017), particle capture (Sadeghpour et al. Reference Sadeghpour, Oroumiyeh, Zhu, Ko, Ji, Bertozzi and Ju2021), aerosol capture (Labbé & Duprat Reference Labbé and Duprat2019) and gas absorption (Chinju, Uchiyama & Mori Reference Chinju, Uchiyama and Mori2000; Migita, Soga & Mori Reference Migita, Soga and Mori2005). In practice, optimizing mass and heat transfer requires parallelization through large vertical fibre arrays whose size is often constrained by the system geometry (cf. figure 1a). Recent investigations have explored the role of fibre spacing in determining the critical fibre density where isolated bead-on-fibre patterns are formed (Wagstaff et al. Reference Wagstaff, Gubba, Truscott, Algashgari and Roberts2023) and how this compares to a structured packing with the same dimensions (Zeng et al. Reference Zeng, Sadeghpour, Warrier and Ju2017), in the context of wet scrubbers. When the fibres are closely spaced, the beads can interact with adjacent fibres forming flowing liquid bridges, as shown in figure 1(a). In this study, we present an experimental investigation of liquid bridges flowing between vertical fibres, aiming to contribute to the existing literature by elucidating the role of a base flow on bridge shape and dynamics.
Liquids between fibres. (a) Schematic of flow down a fibre array showing bead-on-fibre patterns when the fibre spacing is large, and interactions between adjacent fibres when the fibre spacing is small, resulting in a mix of traditional bead-on-fibre patterns and bridge patterns. Inset images show associated bead-on-fibre and bridge patterns with a 3 mm scale bar. (b) A small liquid volume suspended between two parallel fibres of radius 
$r$ can take on a rounded drop-like shape that transitions to a thin column shape as the fibre gap approaches 
$w \approx 2 \sqrt {2}\,r$ (Protiere, Duprat & Stone Reference Protiere, Duprat and Stone2013). The dimensions of the fibre radius 
$r$ and gap 
$w$ determine the length of the liquid profile 
$\ell$ for a given volume 
$V$.
Early studies on liquid–fibre interactions were motivated by textile applications (Minor et al. Reference Minor, Schwartz, Wulkow and Buckles1959; Kissa Reference Kissa1981; Kawase et al. Reference Kawase, Sekoguchi, Fuj and Minagawa1986; Chen et al. Reference Chen, Kornev, Kamath and Neimark2001; Patnaik et al. Reference Patnaik, Rengasamy, Kothari and Ghosh2006). In this context, Princen (Reference Princen1970) provided a detailed mathematical treatment for the equilibrium configuration of liquid confined between parallel fibres. Princen observed the existence of two states: (1) a rounded drop profile; and (2) a column profile that is convex outwards, extends further along the fibres, and is well predicted by his theoretical formulation. These two states are shown in figure 1(b). A number of recent investigations have focused on the transition between these two equilibrium states. For example, the wettability of the two parallel fibres was considered by Lukáš et al. (Reference Lukáš, Chaloupek, Košt’áková, Pan and Martinkova2006), who determined the transition criteria for complete wetting, partial wetting and super-hydrophobic surfaces, and reported hysteretic behaviour, which was later explored in detail by Wang & Schiller (Reference Wang and Schiller2021) using lattice Boltzmann simulations.
Protiere et al. (Reference Protiere, Duprat and Stone2013) determined empirically the transition point between the drop and column profile, and showed that it depends critically upon the volume 
$V$, fibre gap 
$w$, and fibre radius 
$r$. They showed that when the volume of liquid is small (
$V/r^{3} <700$), the transition point is 
$w=2\sqrt {2}\,r$. However, both states can exist for large volumes. Additionally, they derived a scaling law for the bridge length by considering the relevant variables 
$\ell =f(w,V,r)$ and obtaining three dimensionless groups 
$\ell / r = f(w/r, V/r^{3})$ governing the problem. They recognized the intuitive dependence of 
$\ell$ on 
$V$, and defined a composite parameter 
$(\ell / r)/(V / r^{3}) = \ell r^{2}/V$, similar to that defined previously by Duprat et al. (Reference Duprat, Protiere, Beebe and Stone2012), which explained their experimental data. Their experimental results showed 
$\ell r^{2}/V \sim w/r$. Additional works have addressed more realistic scenarios observable in nature, namely elastic and non-parallel fibres. Duprat et al. (Reference Duprat, Protiere, Beebe and Stone2012) explored the shape transition and elastocapillary adhesion of drops between flexible fibres, showing that the drop could spread completely, partially or not at all, and that this distinction depends upon the drop volume and fibre geometry and wetting properties. Furthermore, varying the tension of flexible parallel fibres can ‘zip’ them together or cause the breakup of the liquid drop (Duprat & Protiere Reference Duprat and Protiere2015). Fibres with dissimilar radii (Sauret et al. Reference Sauret, Boulogne, Cébron, Dressaire and Stone2015a) and random orientation (Sauret et al. Reference Sauret, Boulogne, Soh, Dressaire and Stone2015b) have also been considered and compared with the theoretical and experimental results for equivalent and parallel fibres (Princen Reference Princen1970; Protiere et al. Reference Protiere, Duprat and Stone2013).
The results above provide physical insight into the static equilibrium shape of a liquid between parallel fibres. The literature on the dynamics of liquids between parallel fibres is comparatively sparse and mostly limited to the extensional dynamics of liquid bridges, which are useful for characterizing viscoelastic fluid properties through thinning and breakup (Sridhar et al. Reference Sridhar, Tirtaatmadja, Nguyen and Gupta1991; Entov & Hinch Reference Entov and Hinch1997), and have wide-ranging industrial and biological applications, as reviewed by Montanero & Ponce-Torres (Reference Montanero and Ponce-Torres2020). Furthermore, the capillary and viscous forces associated with bridges in extension have been explored (Pitois, Moucheront & Chateau Reference Pitois, Moucheront and Chateau2000), and viscosity has been shown to play a critical role in bridge rupture (Mazzone, Tardos & Pfeffer Reference Mazzone, Tardos and Pfeffer1987). To our knowledge, the addition of a base flow to the drop between fibres system has yet to be explored. The shear dynamics of bridges is relevant to numerous heat and mass transfer applications mentioned earlier, as well as more traditional processes such as fibre coating (Quéré Reference Quéré1999), where liquid bridging fibres leads to partially coated and uneven surfaces. For example, Kurtyigit et al. (Reference Kurtyigit, Guo, Singer, DeMauro and Drazer2023) showed the potential for static bridges in a cross-flow to capture particles. However, a base flow, which is essential for transporting the particle-laden bridges, has not been considered. The dynamics of beads on fibres is affected by base flow stability, resulting in bead patterns and interactions (Kliakhandler, Davis & Bankoff Reference Kliakhandler, Davis and Bankoff2001; Duprat et al. Reference Duprat, Ruyer-Quil, Kalliadasis and Giorgiutti-Dauphiné2007), as well as film-mediated and aerodynamic interactions that can spatially orient beads along a fibre (Duprat et al. Reference Duprat, Giorgiutti-Dauphiné, Tseluiko, Saprykin and Kalliadasis2009) and across multiple parallel fibres (Wilson et al. Reference Wilson, Pahlavan, Erinin, Duprat, Deike and Stone2023), respectively. Here, we present an experimental study of flowing bridge patterns driven by gravity that introduce inertial and viscous forces that were unimportant in the previous work on the equilibrium shape between fibres (Protiere et al. Reference Protiere, Duprat and Stone2013).
We begin this paper by describing the experimental apparatus and protocol that we used to create flowing liquid bridges between parallel fibres, as well as the image processing techniques that define the dynamic liquid bridge properties, as discussed in § 2. In § 3, we describe the formation of bridge-between-fibre patterns, and identify the conditions under which steady patterns emerge. Our experimental results are presented in § 4 and highlight how different experimental parameters impact the bridge pattern, as well as the shape and speed of individual bridges. We also show how the bridge geometry can be controlled using dynamic fibres. We offer concluding remarks in § 5, and provide recommendations for future research directions and potential applications of thin film flow between fibres.
2. Experiment
Flowing liquid bridges were created using the experimental set-up in figure 2. A linear actuator driven by an Arduino-controlled stepper motor was used to dynamically control the spacing 
$w$ between two vertical fibres. A microstep driver and vibration dampeners were used so the fibres could be displaced smoothly for speeds between 
$0.01\,{\rm mm}\,{\rm s}^{-1}$ and 
$4\,{\rm mm}\,{\rm s}^{-1}$. The fibres were nylon monofilament with radius 
$0.15\,{\rm mm} \leq r \leq 0.95\,{\rm mm}$, tightened with inline tensioners to eliminate deformation-induced spreading (Duprat & Protiere Reference Duprat and Protiere2015) and fibre entanglement (Shi et al. Reference Shi, De Koninck, Hart, Kowalski, Fugaro, van der Sloot, Ott, Kennedy and Boreyko2020). An NE-1000 syringe pump was used to apply a flow rate 
$Q$ through a stainless steel nozzle of diameter 
$r_{n}$ onto the side of a fibre that could interact with the neighbouring fibre while destabilizing. The resulting bridge patterns were captured using a Phantom VEO-410L high-speed camera at a position 400 mm below the nozzle – a sufficient distance for capturing steady-state dynamics. An LED panel was used as a backlight for all experiments.
Experimental set-up and image analysis. (a) A syringe pump delivers a controlled flow to two vertical fibres, spaced at distance 
$w$ apart, and adjusted using a custom linear actuator. The inset illustrates a typical bridge pattern with associated experimental variables. (b) A bridge pattern with midline shown as the black dashed line is used to create (c) a spatiotemporal diagram that demonstrates both uniform spacing and steady dynamics. (d) A binarized section (with liquid white) of the spatiotemporal diagram (black box in c) shows the variable measurements (
$\ell,\lambda,f,v_b$).
Glycerol–water mixtures and silicone oil were used as the working fluids, which provided a viscosity range 
$\mu = 9.5\unicode{x2013}699\,{\rm mPa}\cdot{\rm s}$, surface tension range 
$\sigma = $ 
$21.1\unicode{x2013}62.7\,{\rm mN}\,{\rm m}^{-1}$, and density range 
$\rho =969\unicode{x2013}1250\,{\rm kg}\,{\rm m}^{-3}$. The viscosity was measured using a cone–plate shear rheometer (Anton Paar MCR 302), and the surface tension and density were measured using an Attension Sigma 702 force tensiometer with a Wilhelmy plate and density probe, respectively. Glycerol–water mixtures were used as our primary working fluids because of their tunable viscosity and high surface tension, which was critical to produce large asymmetric beads that could interact with a neighbouring fibre at large 
$w$ (Gabbard & Bostwick Reference Gabbard and Bostwick2021a). The contact angle between the glycerol–water mixtures and nylon fibres was 
$\phi = 65^{\circ } \pm 9.3^{\circ }$, measured optically using the largest fibre 
$r=0.95\,{\rm mm}$. To capture wettability effects, we also tested silicone oils with symmetric bead profiles for all 
$r$, limiting the range of interfibre spacing to 
$w \leq 1.5\,{\rm mm}$ (Wagstaff et al. Reference Wagstaff, Gubba, Truscott, Algashgari and Roberts2023). The relevant dimensionless parameters include the Reynolds number 
$Re \equiv \rho v_{b} w / \mu \sim 10^{-3}\unicode{x2013}10^{-1}$ and Weber number 
$We \equiv \rho v_{b}^{2} w / \sigma \sim 10^{-4}\unicode{x2013}10^{1}$, indicating that viscous and capillary forces both dominate inertial forces for the majority of tests. The role of inertia may also be compared to gravity by comparing 
$F_{g}/w^{2} = \rho g V / w^{2}$ with the dynamic pressure 
$p_d \sim \rho v_{b}^{2}$, which shows a negligible role of inertia when 
$v_{b} w / \sqrt {g V} \leq 1$. This holds true for all experiments. The dominant role of viscosity and capillarity over inertia leads us to also define the capillary number 
$Ca \equiv v_{b} \mu / \sigma \sim 10^{-2}\unicode{x2013}10^{-1}$. Finally, we define the Bond number 
$Bo \equiv \rho g w^{2} / \sigma \sim 10^{-1}\unicode{x2013}10^{0}$. Herein, the Bond number plays a critical role in this gravity-driven capillary phenomena where the maximum 
$w$ that we could test for each liquid corresponds to 
$Bo \approx 1$, or 
$\ell _{c} \approx w$, where 
$\ell _{c}=\sqrt {\sigma / \rho g}$ is the capillary length.
Each experiment began by attaching and tensioning two dry fibres to the linear actuator. The fibres were slowly brought together until they touched and then separated by a user-defined distance 
$w$. The nozzle was oriented perpendicular to the fibres and positioned carefully to touch a portion of one of the fibres but not the other. A flow rate 
$Q$ was then applied by the syringe pump for several minutes to achieve a steady pattern, which was then imaged at 1000 f.p.s. The constant flow resulted in periodic bridge formation, which ensured that each bridge flowed at a constant velocity on the thin film deposited by the previous bridge. These steps were repeated for increasing 
$Q$ until the flowing bridges began to interact at a critical flow rate 
$Q=Q_{c}$. All bridge patterns were qualitatively similar and exhibited a periodic structure and uniform motion within a flow rate range 
$Q_{b} \leq Q \leq Q_{c}$, which will be discussed in § 3.
We used MATLAB and ImageJ for image processing to extract the bridge properties defined in figure 2(a). A frame from a typical experiment is shown in figure 2(b). The dashed vertical line corresponds to the midpoint between the fibres. We construct the spatiotemporal diagram shown in figure 2(c) by tracking the pixel column associated with the midpoint with time. The orange diagonal lines correspond to the liquid bridge, where the lines’ negative slope indicates downward motion. Additionally, the constant slope and the even spacing between diagonal lines indicate that the bridges move at a constant velocity and spacing – permitting easy data acquisition from the spatiotemporal diagram. A small region of the spatiotemporal diagram is binarized and displayed in figure 2(d) with the measurements extracted during image processing overlaid. These values are taken for the full spatiotemporal diagram, and their mean and standard deviation are used for the final value and error of each experiment. The average bridge volume is 
$V=Q/f$.
3. Phenomenology: forming the bridge pattern
Depositing a drop of non-volatile liquid with known volume between two fibres is straightforward using a pipette. However, creating dynamic liquid bridges of known volume is more challenging: a dynamic bridge sliding down two fibres will lose mass as it coats the fibres, resulting in a non-constant size and speed. We circumvent this complexity by forming a series of flowing bridges that (i) have nearly identical volumes due to their periodic genesis, and (ii) do not change volume since they flow on the film deposited by the previous bridge. These periodic bridges result from capillary instability of the flow along a single fibre. The initial bead-on-fibre patterns grow in the transient region near the nozzle, and interact with the neighbouring nozzle periodically, thereby restructuring the bead-on-fibre pattern into a bridge-between-fibres pattern, as shown in movie 1 of the supplementary material, available at https://doi.org/10.1017/jfm.2024.794. We note that this method faithfully reproduces the patterns that may arise in fibre arrays with high fibre density – a scenario especially relevant for patterns that are not symmetric about the fibre (Gabbard & Bostwick Reference Gabbard and Bostwick2021a, Reference Gabbard and Bostwick2023a; Eghbali et al. Reference Eghbali, Keiser, Boujo and Gallaire2022; Cazaubiel & Carlson Reference Cazaubiel and Carlson2023).
Figure 3(a) plots bridge frequency 
$f$ (red, left-hand axis) and bridge volume 
$V$ (blue, right-hand axis) against flow rate 
$Q$. As flow rate 
$Q$ increases, bridge frequency 
$f$ increases monotonically until a critical flow rate 
$Q_c$, beyond which the bridges interact, resulting in a stark increase in frequency variability. These interactions result from mass differences in the bridges that originate from a convectively unstable base flow in the initial coating on a single fibre, which causes growing beads to interact before forming bridges of different mass. Consequently, all data points associated with flow rates exceeding 
$Q_{c}$ have been excluded from this study. Movie 2 of the supplementary material shows a typical coalescence event. For low flow rates 
$Q < Q_{b}$, the film deposited by a bridge destabilizes into a bead-on-fibre pattern before the subsequent bridge arrives. The destabilization is due to the Plateau–Rayleigh instability and has a typical time scale 
$\tau _{PR} = 3 \mu r^{4} / \sigma h_{0}^{3}$, where 
$h_{0}$ is the uniform coating of the base flow (Gallaire & Brun Reference Gallaire and Brun2017). Thus film breakup occurs when 
$\tau _{bridge} > \tau _{PR}$ or 
$f < \sigma h_{0}^{3} / 3 \mu R^{4}$. The overlaid experimental images show increased bridge density with 
$Q$ and interacting bridges above 
$Q_c$. Despite the monotonic relationship between 
$f$ and 
$Q$, the estimated bridge volume 
$V=Q/f$ exhibits non-monotonic behaviour as 
$V$ achieves a local maximum before 
$Q_c$. This implies that a decrease in bridge size can be a precursor to the transition to convectively unstable flow, and is a sign of nonlinearity. This non-intuitive behaviour aligns with previous observations of thin film flow down a fibre (Gabbard & Bostwick Reference Gabbard and Bostwick2021b). We note that while the bridge volume 
$V$ depends upon the flow rate 
$Q$, its effect is minor. To achieve a larger range of 
$V$, the nozzle radius 
$r_n$ is varied between 0.6 and 1.45 mm, resulting in a large range of bridge volumes, 
$V=6.4 \unicode{x2013}53 \, {\rm mm}^3$.
(a) Bridge frequency 
$f$ (red, left-hand axis) and bridge volume 
$V$ (blue, right-hand axis) plotted against flow rate 
$Q$, with inset images showing the bridge patterns. The shaded areas represent the data collection range: to the left, thin films between bridges destabilize due to the Plateau–Rayleigh instability; to the right, the shaded area and the inset's rightmost bridge pattern denote convectively unstable base flows. (b) At low flow rates, the bridge frequency is low, and the film between subsequent beads can break up. (c) When the frequency is sufficiently large to avoid film breakup, a steady bridge pattern forms and flows down the fibres on a uniform film. The highlighted areas indicate the meniscus linking the bulk fluid to the thin film on the fibres. For reference, the fibre radius is 
$r=0.25$ mm.
Figure 3(b) shows the structure of a bridge pattern with film breakup 
$Q < Q_{b}$. The beads seen below the downward flowing bridge cause two undesired oscillations in the bridge properties: (i) the bridge length increases as it passes over beads, then decreases over the dry region; and (ii) since the beads on neighbouring fibres can be at different heights, the bridge can oscillate in the image plane. For 
$Q_{b} \leq Q \leq Q_{c}$, the bridges flow smoothly on a uniform film coating the fibres. Figure 3(c) shows the structure of the bridges that we explored. The two highlighted areas show close-up images of the meniscus connecting the front and back of the bridge with the thin film for glycerol–water mixtures, showing that no contact line is present – a feature that we will leverage to develop a simple model of the bridge dynamics. A comparison of each flow type and a close-up of the menisci connecting the bridge to the thin film coating the fibre are included in movies 3 and 4 of the supplementary material.
4. Results
We performed 1112 experiments over an extensive range of system geometry, including fibre radius 
$r$ and interfibre spacing 
$w$, and liquid properties, including surface tension 
$\sigma$ and viscosity 
$\mu$. We quantify the bridge patterns through the bridge geometry 
$\ell$, and bridge dynamics 
$v_b, \lambda$. We begin by showing the experimental bridge patterns and how the bridge length 
$\ell$ depends on the system parameters, comparing our findings with those for static bridges. We then show how the bridge velocity 
$v_b$ depends upon the experimental parameters, including the viscosity over an order of magnitude. Finally, we set the fibres in motion and examine 
$\ell$ and 
$v_b$ as they converge or diverge.
To begin, we show some typical qualitative trends observed in our data. Figure 4(a) shows how the liquid bridge shape changes with increasing fibre spacing 
$w$. Here, the bridge length 
$\ell$ decreases nonlinearly with increasing 
$w$, which is consistent with the behaviour expected for a static drop formed between two fibres (Protiere et al. Reference Protiere, Duprat and Stone2013). A significant reduction in 
$\ell$ is particularly evident between the first two images, as this corresponds to the transition from a column-like shape to a drop-like shape for small 
$w$. As 
$w$ increases further, the changes in 
$\ell$ become less pronounced until the bridges detach from one of the fibres, causing them to flow asymmetrically down a single fibre thereafter. Figure 4(b) shows how the liquid bridge shape changes with viscosity 
$\mu$. Here, we note that viscosity has a negligible effect on 
$\ell$, but does play a significant role in the dynamics, as we will discuss. For reference, we include the average bridge volume 
$V$ in figures 4(a,b), as this implicitly affects the length 
$\ell$.
Bridge geometry. (a,b) Typical bridge profile for (a) increasing fibre gap 
$w$ with fixed viscosity 
$\mu$, and (b) increasing viscosity 
$\mu$ with fixed fibre gap 
$w$, both with 
$r_{f}=0.25$ mm. (c) Bridge length 
$\ell$ and (d) velocity 
$v_b$ against fibre gap 
$w$ for bridge volume 
$11.5 \leq V \leq 22\,{\rm mm}^3$, where the marker colour corresponds to the volume given in the colour bar. The fibre radius is 
$r = 0.25$ mm, and liquid viscosity is 
$\mu = 90\,{\rm mPa}\cdot{\rm s}$. Bridge length decreases nonlinearly as 
$w$ increases, while bridge velocity increases with 
$w$ and increases with 
$V$.
Figure 4(c) plots bridge length 
$\ell$ against fibre spacing 
$w$ for all experiments with 
$r=0.25$ mm and 
$\mu = 90\,{\rm mPa}\cdot{\rm s}$. The colour bar correlates the marker colour with the estimated volume 
$V$ of the bridge. Two key trends are observed: (i) 
$\ell$ decreases nonlinearly as 
$w$ increases; and (ii) 
$\ell$ increases with volume, as shown by the transition from blue to red data markers as 
$\ell$ increases for constant 
$w$. The velocity 
$v_b$ from the same set of experiments is shown in figure 4(d) against fibre spacing 
$w$. The velocity increases with 
$w$ and shows a slight positive trend with 
$V$. These trends are typical of all experiments.
4.1. Bridge shape
The shape of the bridge between parallel fibres can take on a range of complex profiles with associated length 
$\ell$, width 
$w$, and thickness 
$h$. The primary shape descriptor is 
$\ell$, which is readily measurable and unbounded. We note that the bridge width is essentially determined by the user-defined fibre gap 
$w$ and radius 
$r$, and any variations in thickness 
$h$ are relatively small compared with changes in 
$\ell$. As mentioned previously for the static case (Protiere et al. Reference Protiere, Duprat and Stone2013), the bridge length varies significantly between the rounded drop state (small 
$\ell$) and thin column state (large 
$\ell$), with a sensitive transition region.
Since the bridge length 
$\ell$ depends on volume 
$V$, we express 
$\ell$ as a function of fibre gap 
$w$ and volume 
$V$, and apply dimensional analysis with 
$V^{1/3}$ the characteristic length. We find that when 
$r$ and 
$\sigma$ are constant, 
$\ell / V^{1/3} \sim w / V^{1/3}$. Figure 5(a) plots 
$\ell / V^{1/3}$ against 
$w / V^{1/3}$ for a range of viscosities 
$\mu$ and fixed fibre radius 
$r_f=0.25$ mm, showing a rapid decrease in 
$\ell$ with increasing 
$w$ that approaches a horizontal asymptote as 
$w / V^{1/3} \rightarrow 1$. Note the vertical asymptote for small 
$w$, below which 
$\ell$ is too large to form bridge patterns and instead results in infinite or interacting columns. The data suggest 
$\ell =f(w)$. If we consider a bridge with constant volume 
$V=w \ell h$ and assume that changes in 
$w$ alter 
$\ell$ such that 
$V/h=\ell w$ is constant, then an inverse relationship is expected. The dashed line in figure 5(a) shows the data fit to a curve of the form 
$\ell \sim 1/w$. The data fit the trend except at low 
$w$, where the transition from a column to drop occurs, and a swift change in bridge thickness is expected. Additionally, these results verify that 
$\mu$ plays a negligible role in the shape geometry.
(a) Dimensionless bridge length 
$\ell /V^{1/3}$ plotted against dimensionless fibre gap 
$w/V^{1/3}$ as it depends upon the viscosity 
$\mu$, for 
$r=0.25$ mm. The dashed line fits the data to the scaling 
$\ell \sim 1/w$, expected for bridges with equal thickness. (b) Dimensionless bridge length 
$\ell r^2/V$ plotted against aspect ratio 
$w/r$ for all data. The data collapse along a single trend that closely follows the Princen (Reference Princen1970) prediction for low aspect ratio (dashed line), and approaches a small, nearly constant value as aspect ratio increases. The inset shows the data plotted on a logarithmic scale and overlaid with a power-law fit (solid line) with power-law index 
$n \approx 7/4$ (
${R}^{2}=0.9$).
If we expand our analysis to include 
$r$, such that 
$\ell =f(w,V,r)$, then we arrive at the scaling law for static bridges between fibres given by Protiere et al. (Reference Protiere, Duprat and Stone2013) and discussed in § 1. We test this scaling in figure 5(b), which plots 
$\ell r^{2}/V$ against 
$w / r$ for all experiments. The data markers indicate the fibre radius 
$r$. Our data follow a single trend and realize many of the same conclusions shown in figure 5(a). The data closely mirror those for static bridges, and show that this geometric scaling accurately captures the shape of flowing liquid bridges between fibres. The dashed line for small aspect ratio 
$w/r$ corresponds to the analytical expression for a static liquid column between parallel fibres derived by Princen (Reference Princen1970) and compared favourably with static bridges by Protiere et al. (Reference Protiere, Duprat and Stone2013) for contact angle 
$\phi =0^{\circ }$ (see (A1)–(A3)). The derivation can be found in Appendix A for reference. This predicted line is calculated for a contact angle 
$\phi = 0^{\circ }$, a convenient assumption validated by the thin film of the same fluid that the bridge flows along (cf. figure 3c). The inset re-plots the data on a logarithmic scale and fits the data to a power-law trend across several decades of dimensionless bridge length. The power-law index is 
$n \approx 7/4$, with 
${R}^{2}=0.9$.
4.2. Bridge dynamics
The dynamics of flowing bridges is quantified by their velocity 
$v_{b}$, which is constant for patterns with 
$Q_b< Q< Q_{c}$. We begin with glycerol–water mixtures with fixed fibre size 
$r=0.25$ mm, which suggests that 
$v_{b}=f(w,V,g)$, resulting in two dimensionless groups: 
$v_{b}/V^{1/6}g^{1/2}$ with 
$v_{g}=V^{1/6} g^{1/2}$ being the gravitational velocity scale, and 
$w/V^{1/3}$, where 
$V^{1/3}$ is the characteristic length. Figure 6(a) plots the dimensionless velocity 
$v_{b}/V^{1/3}g$ against the dimensionless fibre spacing 
$w/V^{1/3}$, as it depends upon viscosity 
$\mu$, which reveals a power-law trend for each data set that shifts upwards with decreasing 
$\mu$. This result suggests that 
$v_{b}/v_{g}=A(\mu )\,f(w/V^{1/3})$. We assume that 
$A(\mu )$ takes the form 
$1/ \mu$, and re-plot the data in the inset. The collapse validates our assumption and provides a useful scaling when 
$r$ is neglected.
Bridge dynamics. (a) Dimensionless bridge velocity 
$v_{b}/V^{1/6}g^{1/2}$ against dimensionless fibre gap 
$w/V^{1/3}$ as it depends upon viscosity 
$\mu$ for all data with fibre radius 
$r=0.25$ mm. The inset shows a power-law scaling (solid line) collapse of the data, with coefficient 
$A$ as a function of 
$\mu$, proportional to 
$1/ \mu$, where 
$A$ is the coefficient multiplying the independent variable. (b) Dimensional bridge velocity 
$v_{b}$ against impedance 
${Z}=\mu \ell / V w$ as it depends upon the fibre radius 
$r$. (c) Bridge velocity against predicted velocity 
$v_p= \rho g w^{2} / 8 \mu$ for all experiments. The data trend linearly with the predicted velocity (
${R}^{2}=0.96$), and reveal a dissipation factor 
$\zeta = 1/0.89 \approx 1.12$.
Figure 4(d) revealed that 
$v_b$ has a positive linear trend with 
$w$, and figure 6 shows that it is inversely proportional to 
$\mu$. These observations form a basis for determining the viscous resistance or impedance 
$Z$ as the viscous pressure per fluid volume velocity. Thus we need to estimate the cross-sectional area of the bridge, 
$A_{b}$. Letting 
$A_{b}= V / \ell$, we determine 
$Z= \mu \ell / V w$. In figure 6(b), we plot 
$v_b$ against 
$Z$, and note that our data collapse to a single trend, suggesting that the impedance is defined correctly for this system, and is the primary determinant of the bridge velocity. Although this is true across all fibre sizes and glycerol–water mixtures, this expression is not generalized, as it involves the comparison of dissimilar dimensional quantities. Despite this, it provides insight into how viscous dissipation influences the bridge velocity, and gives credence to our assumption that 
$A_{b} = V / \ell$.
Viscosity 
$\mu$ and bridge volume 
$V$ play a critical role in determining the bridge velocity 
$v_b$, as shown in figure 6(b). This observation suggests that the velocity can be determined by balancing the viscous dissipation and gravity, particularly in situations where 
$\ell$ is small and 
$v_b$ is large. Neglecting capillary effects, the energy rates associated with potential gravitational energy and viscous dissipation in the bulk are
\begin{gather} \varPhi_{g} \approx \rho g V v_{b}, \end{gather}
\begin{gather}\varPhi_{\mu} \approx \zeta 2 \mu \left(\frac{\partial v}{\partial r}\right)^{2} V. \end{gather}
Here, 
$\zeta$ is an empirical coefficient associated with the dissipation, and 
$\partial v / \partial r$ is the shear rate, 
$\partial v / \partial r \approx 2 v_{b} / w$. The rate balance becomes
\begin{equation} \rho g V v_{b} = \zeta\,\frac{8 \mu V v_{b}^{2}}{w^{2}}, \end{equation}which gives a predictive scaling for the bridge velocity:
\begin{equation} v_{b} \sim \frac{1}{\zeta}\,\frac{\rho g w^{2}}{8 \mu}. \end{equation} Figure 6(c) plots the bridge velocity 
$v_b$ against the scaling defined in (4.3), showing a collapse of all experimental data onto the curve 
$v_{b} = 0.89 (\rho g w^{2} / 8 \mu )$, with a high coefficient of determination (
${R}^{2}=0.96$), and suggesting an empirical dissipation factor 
$\zeta \approx 1.12$. We note that this model is accurate across practical ranges of 
$w$ and 
$r$, as well as a large range of fluid properties (
$\sigma, \mu, \rho$).
4.3. Bridges flowing between dynamic fibres
We can actively control the bridge length and velocity by adjusting the separation distance 
$w=w(t)$. We explore this by forming a steady pattern of liquid bridges and then actuating one fibre, moving it towards or away from the other fibre at a velocity 
$v_{f}$. Figure 7(a) illustrates the two cases, where the fibres were brought together (
$v_{f} < 0$) or separated (
$v_{f} > 0$) at constant speed. The midline between the fibres was tracked over time and then converted to binary spatiotemporal diagrams for data extraction, as shown in figure 7(b). Here, the upper diagram shows the fibres converging, resulting in elongating bridges (white), and the lower diagram shows the fibres diverging until the bridges detach from one of the fibres. In order to determine the volume 
$V$ inside the bridges, a steady pattern was first formed at a fixed 
$w$ and used to determine the frequency, thereby yielding an estimate for 
$V$. At time 
$t=0\,{\rm s}$, the fibre was set in motion, and the changes in bridge shape and speed were tracked.
Geometry of a liquid bridge between dynamic fibres. (a) Schematic illustrating the two scenarios, with the fibres (1) being brought together at a constant velocity 
$v_{f} < 0$ (left to right), or (2) being separated at a constant velocity 
$v_{f} > 0$ (right to left). (b) Spatiotemporal diagram of the bridge pattern midline as the fibres are brought together (top) and separated (bottom) at 
$0.5\,{\rm mm}\,{\rm s}^{-1}$. The bridge pattern is established while the fibres are static, resulting in uniform white lines, and then evolves with time once the fibres begin actuating at 
$t = 0\,{\rm s}$, as marked by the vertical red line. (c) Dimensionless bridge length 
$\ell r^{2} / V$ against time 
$t$ for fibres of size 
$r=0.25$ mm separating at velocities 
$0.1 \leq v_{f} \leq 2\, {\rm mm}\,{\rm s}^{-1}$. The inset shows the dimensionless bridge length against aspect ratio 
$w / r$. (d) Dimensionless bridge length 
$\ell r^2/V$ against aspect ratio 
$w/r$ as the fibres were brought together (blue lines) or separated (red lines). Data from experiments with increasing fibre radius 
$r$ are differentiated by progressively darker shades. Grey markers represent static-fibre test results, while the dashed line indicates the theoretical length (Princen Reference Princen1970). Data from dynamic-fibre tests are consistent with static-fibre observations and theoretical predictions at low aspect ratios.
In § 4.1, we found that the appropriate dimensionless form of the bridge length is 
$\ell r^{2} / V$. In figure 7(c), we plot the dimensionless bridge length against time 
$t$ for fibre separation rates 
$v_{f}$ ranging from 
$0.1\,{\rm mm}\,{\rm s}^{-1}$ to 
$2\,{\rm mm}\,{\rm s}^{-1}$. Since each test began with 
$w=1$ mm, their initial lengths are similar; however, they evolve differently based on the 
$v_{f}$ values. For higher 
$v_{f}$, the bridge length decreases faster and reaches its minimum value before detaching from a fibre more quickly, as indicated by the farthest right point along each data set. We note that a qualitative similarity exists between the trends and collapse upon re-plotting the data against aspect ratio 
$w/r$, as shown in the inset. Thus for the range of 
$v_{f}$ tested, the velocity plays a minor role on the length of the bridge at a given 
$w$, and we can expect our previous scaling to hold. Figure 7(d) plots the dimensionless bridge length against aspect ratio 
$w/r$ for all experiments with dynamic fibres. The blue and red data show the dimensionless length during fibre actuation for decreasing and increasing spacing, respectively. Three fibre radii were tested, and the coloured data are darkened to show increasing 
$r$. Additionally, three viscosities 
$\mu$ were used, ranging from 
$9.5\,{\rm mPa}\cdot{\rm s}$ to 
$699\,{\rm mPa}\cdot{\rm s}$. The static-fibre data (grey markers) and theoretical curve from Princen (Reference Princen1970) from figure 5(b) are included, and show that the dynamic-fibre results directly overlay the static-fibre results. Thus we conclude that this scaling is appropriate even when the fibre spacing evolves, and holds for separation rates up to 
$v_{f}=2\, {\rm mm}\,{\rm s}^{-1}$.
Figure 8 plots the bridge velocity 
$v_{b}$ against time 
$t$ for the eight different separation rates 
$v_f$ ranging from 
$0.1\,{\rm mm}\,{\rm s}^{-1}$ to 
$2\,{\rm mm}\,{\rm s}^{-1}$ for fibre radius 
$r = 0.25$ mm. As 
$v_f$ increases, the velocity increases at a higher rate, leading to bridge detachment at smaller 
$t$. We again note the similarity among tests, and re-plot our data in terms of the fibre spacing 
$w$. Figure 8(b) shows 
$v_b$ against the aspect ratio 
$w/r$ for the tests shown in figure 8(a). Again, we see that the data collapse along a single trend, irrespective of 
$v_f$. The trend is approximately linear, matching the empirically determined dependence of 
$v_b$ on 
$w$. We therefore conclude that 
$v_f$ plays a minor role in setting 
$\ell$ and 
$v_b$, rendering our conclusions from the static-fibre tests applicable for fibres converging or diverging at rate 
$2\,{\rm mm}\,{\rm s}^{-1}$ or below. This demonstration of active control over shape and speed represents a novelty among liquid–fibre systems, and can be realized using actuators on individual fibres or using non-contact light-actuated materials (Bao et al. Reference Bao, Wang, Song, Zhang, Li, Zhang, Lan and Yang2023).
Bridge velocity between dynamic fibres. (a) Bridge velocity 
$v_b$ against time 
$t$ for fibres of size 
$r=0.25$ mm separating at 
$0.1 \,{\rm mm}\,{\rm s}^{-1} \leq v_{f} \leq 2\, {\rm mm}\,{\rm s}^{-1}$. (b) Bridge velocity 
$v_b$ against aspect ratio 
$w / r$ for the eight separation speeds detailed in (a). The trend of 
$v_{b}$ with respect to 
$w / r$ is invariant with respect to separation rate.
5. Concluding remarks
In this study, we conducted an extensive experimental investigation of the shape and dynamics of flowing liquid bridges. We described a method for producing flowing bridge patterns, and elucidated the flow rate range associated with homogeneous bridge patterns. We discussed some of the general trends observed, namely that the bridge length 
$\ell$ is independent of viscosity but sensitive to changes in volume 
$V$ and fibre spacing 
$w$. A dimensionless bridge length is defined following the work of Protiere et al. (Reference Protiere, Duprat and Stone2013), and scales with the aspect ratio 
$w/r$ for all our experiments. We derived a simple model for the bridge velocity by balancing the driving gravitational force with viscous dissipation, which showed good agreement with our experimental data. Finally, we showed that our results are robust even when the fibres are mobile, such that scalings for the bridge length and bridge velocity are unaffected whenever the separation rate is 
$v<2\, {\rm mm}\,{\rm s}^{-1}$.
Our results provide valuable practical insights into recent advances in heat and mass transfer applications that utilize bead-on-fibre flows in fibre arrays. In these systems, unintentional flowing bridges can emerge due to optimizing fibre density in a constrained geometry. However, flowing bridges can also be desirable to optimize liquid–fibre systems. For instance, they can exhibit diverse shapes and speeds compared to conventional bead-on-fibre patterns, and mobile fibres can provide active control over bridge properties, as shown in figures 7 and 8. Furthermore, the use of liquid between actuated fibres has already shown the potential to transport and mix drops (Khattak et al. Reference Khattak, Shanzeela, Raphael and Dalnoki-Veress2024). Our findings can further advance liquid control and transport between fibres in innovative and bio-inspired systems (Meng et al. Reference Meng, Wang, Liu and Jiang2014; Wang et al. Reference Wang, Bian, Meng, Liu and Jiang2017). Furthermore, flowing liquid between fibres is common in fog collectors, as observed in the laboratory (Shi et al. Reference Shi, Anderson, Tulkoff, Kennedy and Boreyko2018) and in the field (Moncuquet et al. Reference Moncuquet, Mitranescu, Marchand, Ramananarivo and Duprat2022). Thus novel geometries for water harvesting (Bintein et al. Reference Bintein, Cornu, Weyer, De Coster, Vandewalle and Terwagne2023) and materials capable of simultaneously harvesting and cleaning water (Ghosh Reference Ghosh2023) offer impactful opportunities for further research. Our findings offer valuable physical insights into flowing liquid bridges, with the potential to advance the current state of the art in these areas.
The study of flow down a fibre has revealed rich physics over the past two decades. Likewise, we expect the study of flow between fibres to provide both practical and physical intrigue. Here, we studied flowing bridges, but the transition from static to dynamic bridges exhibits interesting behaviour yet to be explored. For example, figure 9 shows a liquid bridge held statically between two closely spaced vertical fibres that is put into motion as the fibres are separated at a constant rate. Interestingly, the drop ‘floats’: its primary motion is inwards, pulling the bottom of the bridge up the fibre, until a critical time 
$\tau _{delay}$ when the bridge succumbs to gravity and falls. This phenomenon shares several qualitative similarities with the classic ‘falling slinky’ problem, where a stretched spring also appears to float for some short time 
$t< \tau _{delay}$ before falling (Cross & Wheatland Reference Cross and Wheatland2012). Thus the intriguing behaviour of falling liquid bridges between fibres offers a fluidic twist on this popular physics demonstration, as shown in movie 5 of the supplementary material. Alternatively, at the high end of possible 
$w$ values, many aspects of bridge detachment remain unexplored. For example, the role of viscous and capillary forces on the detachment of flowing bridges is worth pursuing, especially since previous works have shown that viscosity plays a critical role in bridge rupture (Mazzone et al. Reference Mazzone, Tardos and Pfeffer1987). These forces may also be considered for determining the physics underpinning the coating of fibres using bridge patterns via the capillary number 
$Ca$, which has long been known to control the coating of many thin structures and flat plates according to the Landau–Levich–Derjaguin law (Levich & Landau Reference Levich and Landau1942; Deriagin & Levi Reference Deriagin and Levi1964). A better understanding of the film deposited by flowing bridges may also allow them to be functionalized to deposit suspended particulate via capillary deposition, a concept already explored in the laboratory (Jeong et al. Reference Jeong, Kvasnickova, Boutin, Cébron and Sauret2020) and used to deliver drugs in a circular geometry (Kim et al. Reference Kim, Guenthart, O'Neill, Dorrello, Bacchetta and Vunjak-Novakovic2017), but unexplored for fibrous systems. Furthermore, the coating of dry fibres with flowing bridges remains unexplored but is expected to yield valuable physical insight akin to that found for beads coating vertical fibres and fibre bundles (Leonard et al. Reference Leonard, Van Hulle, Weyer, Terwagne and Vandewalle2023)
Image sequence of a liquid column between two dry fibres that are being separated at a constant rate shows that the bridge contracts towards its centre of mass and then descends along the fibres due to gravity. For short time 
$\tau < \tau _{delay}$, the bridge's motion predominantly transitions its shape between column-like and drop-like, after which gravity causes net drop motion. This delay in the onset of descending drop motion is analogous to the dynamic behaviour observed in the ‘falling slinky’ phenomenon, where a stretched slinky appears to hover momentarily before collapsing.
Finally, it is worth emphasizing the exceptional degree of control and the myriad of patterns that extend well beyond the scope of what has been highlighted in this paper. For instance, liquid bridges can flow between curved fibres or along a winding trajectory (Gabbard & Bostwick Reference Gabbard and Bostwick2023b). In these scenarios, liquid bridges can be purposely engineered to glide, twist and traverse intricate paths, necessitating the incorporation of additional physics, such as the interplay between adhesion and liquid weight (Aziz & Tafreshi Reference Aziz and Tafreshi2019), which will determine if bridges flow or detach. Furthermore, bridges can flow between arrays of more than two fibres (Gabbard & Bostwick Reference Gabbard and Bostwick2023b), a scenario that approaches plug flow in a tube as the number of fibres increases. The wealth of potential flow structures and their adaptability presents an ideal canvas for innovative system design, and equips creative practitioners with effective means to craft novel and sophisticated solutions to fluid transport problems.
Supplementary material
Supplementary movies are available at https://doi.org/10.1017/jfm.2024.794.
Funding
J.B.B. acknowledges support from NSF grant CMMI-1935590.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Equilibrium length of a column between parallel fibres
Here, we restate the results for the scaling of static liquid bridges between fibres from Princen (Reference Princen1970) and Protiere et al. (Reference Protiere, Duprat and Stone2013) for completeness, which we use to compare with our experimental data shown in figures 5(b) and 7(d).
We begin by considering the force balance on a small volume of fluid in equilibrium between two long, parallel fibres, 
${\rm d}V=A\,{\rm d}L$:
\begin{equation} 4 \sigma r \alpha \,\text{d}L - 4 \sigma R \left(\frac{\rm \pi}{2}\right) \,\text{d}L - \sigma\,\frac{A \,\text{d} L}{R} = 0, \end{equation}
which includes forces at the free surface and fibre–liquid interface, and the force associated with the Laplace pressure in the column 
$\sigma / R$. We define a geometric factor 
$f(\alpha )=2 \alpha - \sin 2 \alpha$, where 
$\alpha$ is the angle between the line connecting the centre of the fibres and the line from the centre of a fibre to the liquid–solid–vapour contact line. Letting 
$\tilde {A} = A/r^{2}$ and 
$\tilde {R}=R/r$, we can relate 
$\tilde {A}$ to 
$\tilde {R}$ and 
$\alpha$ through
\begin{equation} \tilde{A} = \tilde{R}^{2}\left(4 \alpha - {\rm \pi}- f(\alpha)\right) + 2 \tilde{R} \sin 2 \alpha - f(\alpha). \end{equation}The radius of curvature is then given by
\begin{equation} \tilde{R} = \left(\sqrt{\frac{\rm \pi}{f(\alpha)}} - 1\right)^{{-}1}, \end{equation}
and we can define the fibre spacing as 
$\tilde {w}=w/r$ and express it in terms of 
$\alpha$ and 
$\tilde {R}$:
\begin{equation} \tilde{w}=2(\cos \alpha(1+\tilde{R})-1). \end{equation}
The theoretical length of a column is given by (A2)–(A4) since each 
$w$ gives a unique cross-sectional shape such that 
$\ell r_f^2/V=1/\tilde {A}$ as plotted in figures 5(b) and 7(d).
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