It has been known at least since the work of Reynolds and Marangoni in the 1880s that floating particulates strongly affect water surface behaviour, and research involving particle–fluid interactions continues in modern applications ranging from microfluidics and cellular morphogenesis to colloidal dynamics and self-assembly. Here, we report and analyse an unexpected result from a simple experiment: clean water is discharged along an inclined channel into a lower container contaminated with floating particles. Surprisingly, the floating particles are transported both up a waterfall as long as 1 cm, and upstream in channels to lengths of at least several metres. We confirm through experiments and simulations that this upstream contamination is paradoxically driven by the downstream flow of clean water, which establishes a surface tension gradient that sustains the particulate motion. We also show that contamination may occur in practical applications, such as the discharge of a standard pipette or simulated release of waste into larger scale channels.
When water is poured into a teacup, it seems self-evident that material in the cup will not make its way upstream into the teapot. Similarly, when chemicals are pipetted onto a culture plate, it is taken for granted that cells from the plate will not contaminate the chemical source. In this work, we demonstrate that contrary to expectation, floating particles can contaminate upstream reservoirs by travelling in rapid jets at accelerations substantially larger than the gravitational downstream acceleration. This counterintuitive phenomenon was first observed during the preparation of mate tea, when hot water was poured from a pot into a cup containing tea leaves: it was found that when the spout was within 1 cm above the leaves, floating leaves would find their way from the cup into the pot (see the electronic supplementary material, figure S1).
To study the phenomenon under controlled conditions, we first perform the experiment illustrated in figure 1a, where water flows from an upstream reservoir down an inclined channel, off a waterfall and into a receiving vessel. Floating particles of mate tea (Ilex paraguariensis), chalk and other powders in the receiving vessel are observed to travel up the waterfall and through the channel to end up in the upstream reservoir, as shown in figure 1b (see also the electronic supplementary material, video S1). The downstream water flux has been varied up to 16 cm3 s−1 along an 8 cm long channel inclined at a slight angle. Experiments using inclined (figure 1b) and horizontal (figure 1d) channels both generate upstream contamination; effects of channel angle are discussed shortly. The snapshots shown in figure 1 use deionized water, but experiments using tap water yield indistinguishable results; similarly, the upstream flow of floating particles persists in experiments using either boiling or cold water, and tests using fluorescein powder also exhibit upstream contamination.
The flows of particles in the waterfall and channel regions are indicated by arrows in figure 1b and consist of vortices that transport particles upward from the back of the waterfall to the outside of the channel, and then back downwards through the centre of the channel and the front of the waterfall. We dissect the flows in greater detail later.
Upstream flow begins near the waterfall, shown in the snapshot of figure 1e and enlarged in figure 1f. From these panels, we see that a pair of particle-rich circulatory vortices form, one on either side of the waterfall (see also the electronic supplementary material, video S1). Floating particles travel at speeds ranging between 1 and 7 cm s−1, as determined by measuring the lengths of streaklines at a known shutter speed (here, 1/60 s).
Only a fraction of the particles in the recirculating waterfall flow become entrained into the channel region (figure 1b,c), and consequently in this region we see much lower particle densities. For this reason, figure 1e contains a single snapshot, whereas figure 1b requires 30 successive video frames to show a similar number of particles. As indicated by arrows in figure 1c, particles in the channel travel upstream near the flow boundaries, and most particles ultimately change direction and travel down the centre of the stream. Particles here travel at speeds between 4 and 7 cm s−1, measured as before.
We have also quantified velocities using particle tracking velocimetry (PTV). Typical results are shown in figure 1d from an experiment using a horizontal channel. To perform PTV, we use a horizontal channel because this generates slower downstream flow than inclined channels, and this slower speed results in longer and steadier upstream flows, which in turn produce low noise velocimetry data. Flow in a horizontal channel is produced by maintaining a slight head between the upstream and downstream reservoirs. Particle tracking of video images is performed using ImageJ software with the MTrack2 plug-in. Particle tracking confirms the visual appearance that particulate flow is predominantly parallel to the channel walls, upstream in high-speed jets near the channel edges and downstream through the channel centre.
To analyse possible mechanisms underlying the unexpected upstream flows that we report, we begin by noting that it is well established that surface tension depends strongly on the concentration of floating contaminants [1–5]. We therefore hypothesize that the higher surface tension in the clean, upper, reservoir may draw particles from the contaminated, lower, reservoir  against the downstream flow. Indeed, it has previously been reported that amphiphilic surfactants in a receiving reservoir can be drawn up waterfalls to a height of 2 cm through surface tension gradients based on a similar mechanism . Consequently, we measured the surface tension of water as a function of density of floating chalk and mate tea particles (figure 2a). The chalk particles were prepared by rubbing a stick of chalk on rough sandpaper, whereas the tea was obtained directly from commercial yerba mate leaves passed through a no. 25 (0.7 mm) sieve. The chalk or mate was dropped from a small distance onto clean tap water and allowed to equilibrate, after which the surface tension was measured using the du Noüy ring method . As shown in figure 2a, the surface tension of water decreases by nearly a factor of 2 with added chalk, and by a factor of 3 with tea. We note that the upstream contamination effect appears for both soluble and insoluble surfactants. For example, the principal ingredient in chalk, calcium carbonate, is nearly insoluble in water (solubility 0.01 g l−1 at room temperature), while we have also verified that pure fluorescein and sodium dodecyl sulfate (SDS) surfactant powder exhibit upstream contamination, with solubilities ranging from weakly to highly soluble (0.8 g l−1 for fluorescein to 250 g l−1 for SDS).
We can easily estimate the order of magnitude effect of such a change in surface tension on a floating particle, as follows. A particle of radius r on the surface of the receiving reservoir that is ideally exposed to clean water from the waterfall on one side and a fixed concentration of solids on its opposite side will feel a surface tension difference ΔT, which from figure 2b is on the order of 0.01 N m−1. The resulting acceleration of the particle will be ΔT⋅L/m, where L is the characteristic length of a particle, and m is its mass. Taking L=2r and m=4πρr3/3, where ρ is the particle density, gives an acceleration, a=10×2×3/4πρr2∼5/(ρr2). A conservative estimate for a can be obtained by considering a chalk particle with bulk density ρ=2.5 g cm−3 and radius r∼100 μm: this gives an acceleration a∼20 000 cm s−2, or about 20 times gravity. This estimate should be viewed as an upper bound: most particles will likely feel a milder concentration gradient than this approximation suggests. Furthermore, it remains to be established how small the particle concentration needs to be to generate upstream contamination; however, this estimate illustrates that surface tension gradients produced by flowing clean into particulate-contaminated water can provide ample accelerations to drive particles upstream much faster than gravity. This is in keeping with more detailed calculations , as well as with experimental data demonstrating that fine particles disperse explosively when dropped onto a clean water surface .
We challenge the hypothesis that upstream contamination is driven by surface tension gradients by introducing a small amount of liquid surfactant, benzalkonium chloride, to the upper or lower reservoir. As expected, when surfactant is added to the upper reservoir, upstream contamination is abruptly eliminated, while adding surfactant to the lower reservoir causes a transient expansion of the surface layer that initiates upstream contamination if it has not yet begun, or accelerates contamination if it is already present.
To more systematically evaluate whether surface tension gradients account for the observations, we have constructed a simulation of multiple point particles on a two-dimensional domain, where each particle is subject to a competition between two influences: surface elasticity drawing particles upstream, and ambient fluid flow entraining particles downstream. Details are presented in the supplementary material; in overview, elasticity and ambient fluid flow that drive particle motion are defined as follows.
The elasticity acting on a floating particle is taken to depend on the concentration gradient of contaminants that is produced by neighbouring floating particles. Heuristically, we model each particle as being surrounded by a halo of molecules that diminish the local surface tension between that particle and any neighbour. For example, in the inset to figure 2d, we confirm that a halo of fine particulates appears shortly (under 1 s) after a single particle of chalk is dropped into clean water; some powders may well also contain chemical surfactants (e.g. stearic acid  in mate tea).
A cartoon illustrating how this heuristic is used to compute tensile forces in our simulation is shown in figure 2b in the simplest case of three particles in a line, with particle p1 closer to the central particle than particle p3. We make use of the fact that the molecular concentration surrounding a spherical source particle is analytically known  to be a Gaussian function of radius with prescribed variance, σ2g, and amplitude, Ag, so if we assume that all particles are identical spheres and that all Gaussians are defined at the same instant in time, we can solve for the concentration everywhere. In figure 2b, the contaminant concentration at the central particle, p2, due to the Gaussian surrounding p1 would be larger than the concentration due to the more distant p3, and consequently the elastic tension due to p1 would be smaller than that due to p3—resulting in a net elastic force on p2 towards p3. For simplicity, we take the surface tension to decrease linearly with concentration, and since the diffusion equation itself is linear, all concentrations—and so tensile forces—at any point are simple linear superpositions of values due to all other particles. Thus, we determine the tensile force on any given particle by evaluating the concentrations at that location due to the analytically defined diffuse halo of concentrations surrounding all neighbouring particles.
The ambient fluid velocity is calculated independently, using separate calculations in the channel and in the waterfall regions. In the channel region, surface flow is bounded at the sides, and since the shallow water Reynolds number  is under about 8 (see the electronic supplementary material), we take the velocity at the surface in that region to be defined by laminar two-dimensional Poiseuille flow with maximum speed, , at the centre of the channel. In the waterfall region, flow is azimuthally periodic, and so the velocity is readily expressed as a Fourier series. We approximate the flow in the waterfall to be the leading order term in this series, a single cosine, fastest in the front of the waterfall where the surface is free and slowest in its back, where the surface is initially retarded by contact with the channel. We close the problem by noting that the waterfall is fed from the channel, so that if we conserve surface area, the surface flow rate integrated across the channel will equal the flow rate integrated around the circumference of the waterfall. This sets the amplitude of the cosinusoidal waterfall flow to be expressible in terms of , thus reducing the number of parameters to three kinetic terms, , Ag and σ2g, several purely geometric terms (e.g. the widths and lengths of the channel and waterfall) and the number of particles in the simulation. Finally, we define boundary conditions by assuming that particles that impinge on the channel sides reflect specularly, whereas particles travel freely around the azimuthally periodic waterfall. Effects of parameter variations are discussed below; for further discussion, see the electronic supplementary material.
Given a choice of parameters, we calculate trajectories of particles on the surface by applying the elastic force to every particle, taken to have unit mass, and then requiring each particle to follow the ambient fluid velocity. To mimic experimental conditions, particles start in a 1×1 unit reservoir at the bottom of the computational domain, i.e. at y<0, and particles move on a domain x∈[−1,1] and , thereafter. To add verisimilitude, the waterfall region is defined to be narrower than the channel region (again defined in the electronic supplementary material), as shown in figure 2c, although separate computational trials show that geometric details such as this have little effect on the upstream contamination. Likewise, the simulations shown here use 500 particles, but we obtain similar results using between 100 and 1500 particles, with more particles crowding to move upstream as the number of total particles is increased.
More detailed embellishments to this model are certainly possible, for example, accounting for time variations in contaminant concentration surrounding each particle, or including inertial corrections. Despite its simplicity, the model captures both quantitative and qualitative features seen in experiments. Qualitatively, the model reproduces both the rapid vortical motion circulating particles between the front and back of the waterfall seen in experiments, and the upstream flow that entrains a fraction of these circulating particles to flow upstream along the edges of the channel and downstream near its centre (see the electronic supplementary material, video S2). These features are shown in figure 2c at high , and figure 2e at smaller .
Quantitatively, we can compare simulation and experiment by observing that grows in the experiment with the angle of inclination of the channel—that is, water flows faster on steeper surfaces. Thus in figure 2d, we show a comparison between the maximum distance that particles travel up the channel as a function of angle of inclination of the channel (the relation between and angle is derived in the electronic supplementary material). Evidently, experiment and simulation show comparable increases in upstream contamination as the channel angle is decreased. Mechanistically, we note that upstream flow in both simulations and experiments appears where the downstream flow is slowest—at the back of the waterfall and along the sides of the channel.
Finally, we also tested the upstream contamination effect at both smaller and larger scales. At smaller scales, surface tension effects typically grow as scales diminish , so it can be anticipated that upstream contamination may increasingly be encountered in smaller scale applications —e.g. cell culture  or free-surface microfluidics . Indeed, it has long been known that bacteria actively manipulate surface tension gradients to control migration and swarming . To assess upstream contamination in an archetypal bioengineering device, we pipetted deionized water into a Petri dish containing water on which fluorescein powder was sprinkled. In six out of 10 trials at pipette angles of 20°±5° to the horizontal, heights between 3 and 5 mm, and flow rates of 0.7±0.1 ml s−1, significant contamination inside the pipette was seen (shown in the electronic supplementary material, figure S2 and video S3), and in seven out of 10 additional trials, contamination was seen either inside or outside of the pipette. Importantly, in both experiments and simulations, upstream contamination appears to rely on a differential in ambient fluid speed, and correspondingly, upstream flow was not encountered in our experiments if the pipette was held vertically. Nevertheless, we caution that we cannot rule out the possibility that upstream contamination could occur in uniform flows—especially at small scales.
At larger scales, we have performed two sets of tests. First, to study the effects of scale on flow up waterfalls, we repeated the experiments shown in figure 1 using square channels of widths 3, 5 and 10 cm. We confirmed that chalk powder continues to climb small waterfalls (heights less than 1 cm) in all of these channels. Second, to study upstream flow in channels, we used a 4.5 m long, 30 cm wide, flume (described in ) filled with tap water, and found that contamination against a steady downstream flow occurs over this scale as well. In figure 3a, we show a schematic of the flume. We found that inclining the chute bottom very slightly uphill produces a reproducibly steady and uniform flow, so that for all experiments described here, the flume was set at 1° uphill to the horizontal. For the experiments shown in figure 3, this produced a water depth at the downstream end of the flume of 3 cm and a depth at the upstream end of 5 cm. In these experiments, we measured the downstream surface velocity to be 3.6±0.1 cm s−1 by timing how long individual small polystyrene beads dropped in the centre of the flow take to travel the length of the flume.
In this larger system, we metered chalk powder continuously from above at a fixed location near the downstream end of the flume (indicated in figure 3a,c), and we recorded the powder flow from above with a video camera (see the electronic supplementary material, video S4). We did not investigate climbing of a waterfall in the flume because this would have required a large quantity of powder that could foul the recirculation pump. Instead, we used the flume to focus on contamination in a larger scale channel as might occur upstream of a continuous waste discharge into a stream.
As in the smaller channel experiments shown in figure 1, we find that upstream contamination is fastest along the edges of the channel and competes against downstream ambient flow in the centre of the channel to produce recirculation near the channel edges. Unlike the earlier experiments, the powder forms a contiguous raft that travels upstream across the width of the flume. The competing flows and the powder raft are indicated in figure 3b in a snapshot taken 30 s after the start of an experiment. Recirculation produces tendrils of entrained powder-free fluid as seen near the bottom of this figure.
The same can be seen in time-lapse images: in figure 3c, we show the leading edge of a raft as it travels upstream, traced from video images at 5 s intervals. Tendrils in the raft appear here as well: these tendrils break off at times 0.30 and 0.35 (arrowheads in figure 3c) to form recirculating powder-free islands that are highlighted in dark shading at later times. These islands continue to recirculate—and the raft continues to move upstream—as long as powder is metered downstream. When metering is halted, recirculation stops and the raft convects downstream into the holding tank.
Edges in figure 3 are traced by hand. Gradient and Sobel edge-finding algorithms were also investigated, but were found to be ineffective due to the intrinsic low contrast of the powder–water interface, as well as being dominated by spurious edges caused by reflections and flume support members. A typical tracing is shown as a dotted line in figure 3b, and comparison with the video record can be made from the electronic supplementary material, video S4.
We also varied the fluid flow speed by increasing the head in the upstream tank, with the goal of establishing the maximum speed, , that could be defeated by upstream contamination. However, we found that depends on the rate of metering of powder, Qpowder: by increasing Qpowder, we could drive the raft of powder upstream at increasing downstream speeds—at least up to 30 cm s−1, beyond which a prohibitive amount of powder was needed to sustain upstream contamination. Additionally, we investigated fixing Qpowder, but in that case, we found that as was increased, the raft would approach a limiting upstream distance, , and stop, and as the flow speed was increased, would decrease, and vice versa. Thus, for continuous deposition of powder, depends both on Qpowder and on .
Therefore, to produce a unique measure of a threshold fluid speed that would out-compete upstream contamination, we metered a fixed quantity of powder (5.5 g released over 3.5 s) and evaluated the maximum speed, at which the highest upstream points of the raft (asterisks in figure 3b) passed upstream of the deposit location. We found this criterion to be reproducible, yielding a maximum speed cm s−1 measured at the centre of the channel.
In conclusion, we have demonstrated that floating particles can travel upstream as much as 1 cm up a waterfall and several metres up a channel against a downstream fluid flow. We have seen that this effect occurs for both pure ingredients (e.g. fluorescein in deionized water) and more common materials (e.g. tea in tap water). We have shown that upstream flow of contaminant particles can be generated by surface tension gradients that are established by the downstream flow of clean water into a contaminated reservoir, and we have seen upstream contamination in both small (millimetre scale) and large (metre scale) experiments.
The authors acknowledge useful discussions with J. E. Wesfreid, E. Clèment, D. Quéré, J. Gollub, O. Sotolongo-Costa and L. del Río. E.A. and A.L. thank R. Mulet for bringing their attention to the serendipitous findings of S. Bianchini. T.S. and T.S. thank J. Cotton for dedicated laboratory support, and all of the authors thank Qizhong Guo for generously providing the flume for our experiments.
- Received February 1, 2013.
- Accepted June 4, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.