2.1. Paper siphon

For the single-sheet paper siphon shown in figures 1(b) and 1(c), we measure the volumetric flow rate $Q$ under conditions of fixed heights $h$ and $H$, the latter then varied. A long strip of filter paper (Whatman 1004-648 Grade 4 Qualitative Filter Paper, areal density 96 g $\text {m}^{-2}$, pore size 20 to 25 $\mathrm {\mu }$m, width 3.8 cm) is cut to the desired length and hung over the edges of plastic supporting disks (diameter 2.54 cm). One end is immersed in the upper source container, where the water height is maintained by an overflow mechanism (not shown in figure 1c). The other end dips into the lower receiver container, which hangs from a high-precision digital scale (OHAUS model PX323, 0.001 g). The lower container's cross-section is sufficiently large to keep nearly constant water height during each measurement. An enclosure (not shown) over the whole set-up ensures high humidity and prevents evaporation.

Dynamic equilibrium is reached in several hours, as assessed by repeated measurements of the volumetric flow rate $Q={\rm \Delta} m/\rho {\rm \Delta} t$, where ${\rm \Delta} m$ is the measured mass of water (density $\rho$) collected during the time interval ${\rm \Delta} t$ as measured with a stopwatch. For the results reported in figure 1(d), the head $h=3.7$ cm is fixed, and the measurement procedure is repeated several times for a given $H$, and this parameter is then systematically varied. Evidently, the flux $Q$ steeply increases with $H$ before saturating to a plateau. To facilitate comparison to the $Q\sim \sqrt H$ law for conventional siphons, we show a log–log plot of these data in the inset, where the paper siphon can be seen to deviate significantly from a line of slope 1/2. Linear behaviour of slope 1 for low $H$ transitions to the saturated response of slope 0 for high $H$, and these trends are shown in what follows to be general features of capillary siphoning.

2.2. Groove siphon

We next consider a related but simplified capillary siphon that takes the form of an arched groove in a wetting solid, this system being better controlled and whose geometry can be completely characterized. As shown schematically in figure 2(a), our experimental realization consists of two hook-shaped stainless-steel rods that are thoroughly cleaned and then firmly clamped together by procedures that are detailed below. The surface is hydrophilic, allowing water to wet the two narrow grooves between the rods, as shown in the cross-sectional view in figure 2(b). Each groove presents an open capillary channel whose geometry is uniform along its length.

Figure 2. Groove siphon. (a) Experimental realization of open channels or grooves formed by firmly pressing together two hook-shaped stainless-steel rods. (b)  Cross-sectional view of grooves and wetted region. (c)  Measurement apparatus. The flow rate $Q$ through the groove siphon (dark hook shape) is measured by transferring, via a regular siphon, the collected water to a container hung below a digital scale. Mounting the components on vertical slides (not shown) allows the heights $h$ and $H$ to be varied independently.

We employ rods of radius $R=0.24$ cm, and the inner radius of the arch is 1.03 cm. Flux measurements are carried out using the apparatus of figure 2(c), which is similar to the paper siphon system but with two modifications. First, mounting the components appropriately on vertical slides (not shown) allows $h$ and $H$ to be varied independently, thus enabling a more complete characterization. Second, because the collection chamber is deep to allow for a wide range of $H$, and to avoid exceeding the maximum allowable weight for the scale, we opt to transfer the collected water to a shallow container for the flux measurement. This is done via a regular tube siphon, and the flow rate $Q = \dot {m} (1+A_c/A_t)/\rho$ is calculated from the measured mass rate $\dot {m}$ and the cross-sectional areas $A_c$ and $A_t$ of the collector and transfer containers, respectively.

The groove siphon requires careful protocols for cleaning, assembly and priming. Before assembly, the rods and all other wetted components are gently scrubbed down with a soft sponge and soapy water and then rinsed thoroughly and repeatedly with filtered and deionized water. Since the wetting properties of stainless steel are known to be generally hydrophilic but sensitive to surface preparation (Prajitno, Maulana & Syarif Reference Prajitno, Maulana and Syarif2016; Terpiłowski et al. Reference Terpiłowski, Hołysz, Rymuszka and Banach2016), we directly measure the contact angle $\theta = 27^{\circ } \pm 6^{\circ }$ in separate experiments using a horizontal microscope focused on the apex of the arch. For the siphon experiments, assembly is done immediately after cleaning. The rods are firmly clamped by tightening the screws of many two-piece stainless-steel shaft collars (not shown in figure 2) that are distributed along the entire length of the rods. Alignment and uniformity of the contact is checked by inspection of the groove with strong backlighting; no passed light indicates firm contact between the rods. After assembly, we prime the siphon by adding several water drops to the apex to thoroughly wet the grooves. The collars are in contact with the outer portions of the rods but not the inner portions that form the groove, thus avoiding any obstruction of the flow. The siphon then starts, and the system is again enclosed and found to reach steady state in about half an hour.

Figure 3(a) shows how $Q$ depends on $H$ for two values of $h$, and the experimental data (red and blue dots) follow trends similar to those of the paper siphon. In particular, the linear increase in $Q(H)$ for low $H$ transitions to a plateau at higher $H$. Comparison of the two datasets shows that higher values of $h$ lead to weaker flows overall, that is, lower slopes of $Q(H)$ in the linear regime and lower values of the saturated flux. These trends are confirmed by systematic measurements of $Q(h)$ for fixed $H$, as shown by the green dots in figure 3(b) for $H=10.1$ cm. The flux $Q$ decreases monotonically and nonlinearly with the head $h$. Thus, unlike the conventional siphon, whose flux is independent of the head, the flow rate of the groove siphon is strongly modulated by changing  $h$.

Figure 3. Experimental (dots) and model (dashed curves) results for the groove siphon. (a)  Flow rate $Q$ versus siphon height $H$ for head values $h=2.2$ cm (blue) and $h=5.6$ cm (red). Shaded bands represent the variations in model predictions when contact angles are varied over the range $\theta \in [21^{\circ }, 33^{\circ }]$ measured in experiments. (b)  Flow rate $Q$ versus siphon head $h$ for fixed $H=10.1$ cm.

3.1. Governing equations

Our model seeks to determine the flow rate through the groove siphon by considering how the shape of the wetted region varies along its length. The flow is assumed to be steady, laminar and well developed, as justified by the low Reynolds numbers $Re = \rho l_c V/\mu \lesssim 1$ observed in experiments. Here $\rho = 1.0~\text {g}~\text {cm}^{-3}$ and $\mu = 0.89$ cP are the density and dynamic viscosity of water, respectively. The typical dimension of the cross-section is taken to be the capillary length scale $l_c = \sqrt {\gamma / \rho g} = 0.27$ cm, where $\gamma$ is the water–air surface tension coefficient. Assuming a characteristic flow speed $V=Q/l_c^2$, the experimentally measured range of $Q \sim 10^{-5} - 10^{-3}~\text {cm}^3~\text {s}^{-1}$ implies a range $Re \sim 10^{-2} - 10^0$. Further, since the length scale of the liquid in the groove cross-section is typically smaller than the total siphon length by an order of magnitude or more, the slender geometry motivates a simplifying approximation in which the pressure and velocity fields are homogenized by averaging within each cross-section. The section-averaged pressure $P$ and flow speed $V$, together with the shape of the wetted region, are descriptors of the system that vary with the position $s$ along the siphon axis. These quantities are related through three governing equations for mass conservation, the Young–Laplace law and the Stokes equation for low-Reynolds-number flow.

The model incorporates the detailed geometry and parameters relevant to our experimental system, though we expect the phenomenology and trends to hold more generally. Prescribed constants include the fluid density $\rho$, dynamic viscosity $\mu$, contact angle $\theta$, rod radius $R$ as well as the hook-shaped path of the siphon axis, which determines the height function $z(s)$ (see figure 4a). For any arbitrary cross-section at location $s$, the liquid wets the solid through an angle of $\phi (s)$, as shown in figure 4(b). The free surface is assumed to be a circular arc with radius of curvature $r(s)$. Assuming uniform pressure over each cross-section, the Young–Laplace law for capillary pressure relates the internal pressure $P(s)$ to the shape of the wetted region:

(3.1)\begin{equation} P_0 - P(s) = \frac{\gamma}{r(s)}, \end{equation}

where $P_0$ is atmospheric pressure. The slender geometry justifies the inclusion of only the interfacial curvature in the plane perpendicular to the siphon axis. The radius of curvature $r(s)$ relates to $\phi (s)$, $\theta$ and $R$,

(3.2)\begin{equation} r = R \, \frac{1 - \cos \phi}{\cos (\phi+\theta)}, \end{equation}

which can be derived based on the assumed geometry of figure 4(b).

Figure 4. Model of the groove siphon. (a) Side view of the siphon. The arclength $s$ marks position along the siphon axis, and $z$ is the vertical coordinate. (b)  Cross-sectional view. Water wets the solid rods (radius $R$) with contact angle $\theta$. The wetted angle $\phi$ and radius of curvature $r$ of the air–water interface vary with $s$ and are to be determined by the model. (c)  Zoomed-in view of the wetted region. The colour map shows an example numerical solution for the axial velocity field for steady, viscous flow with no-slip conditions on the solid and a stress-free interface. (d)  Cross-sectional area $A$ (blue) of the wetted region and its mean pressure $P$ (red) for a representative solution of the model ($h=5$ cm and $H=7$ cm). The minima in these quantities (dotted line) occur somewhat downstream of the apex of the arch (dashed line).

Mass conservation demands that the product of wetted area $A(s)$ and the mean speed $V(s)$ be uniformly constant along the siphon axis,

(3.3)\begin{equation} A(s)V(s) = Q, \end{equation}

where $Q$ is the volumetric flow rate whose value we seek. The wetted area $A(s)$ at each cross-section also relates to $\phi (s)$, $\theta$ and $R$ via the specified geometry shown in figure 4(b):

(3.4)\begin{equation} A = R^2 (2\sin\phi - \sin\phi\cos\phi - \phi) - r^2\left[\frac{\rm \pi}{2}-\phi-\theta-\frac{1}{2}\sin(2\phi+2\theta)\right]. \end{equation}

This form reflects a particular approach to the geometric calculation, and many equivalent expressions exist, including those that use (3.2) to eliminate $r$ in favour of $\phi$.

Considering the groove as a conduit whose cross-section dimensions vary slowly along the length (Tanner Reference Tanner1966; White Reference White2015), the one-dimensional Stokes equation for fully developed laminar flow takes the form

(3.5)\begin{equation} \frac{\mathrm{d}P(s)}{\mathrm{d}s} + \frac{\mu K V(s)}{2D_h^2(s)} + \rho g \frac{\mathrm{d}z(s)}{\mathrm{d}s} = 0, \end{equation}

which expresses the balance of axial pressure gradients with viscous and gravitational stresses. The second term can be understood as a hydraulic approximation that averages within each cross-section the viscous pressure gradient $\mu \nabla ^2 v$ associated with the axial flow velocity field $v$. The non-circular cross-sectional geometry and its variations with arclength $s$ along the conduit are reflected in the effective hydraulic diameter $D_h(s) = 2A(s)/R\phi (s)$ and the dimensionless friction factor coefficient $K$ (Ayyaswamy, Catton & Edwards Reference Ayyaswamy, Catton and Edwards1974; Suh, Greif & Grigoropoulos Reference Suh, Greif and Grigoropoulos2001; White Reference White2015). The latter only depends on the shape of the wetted region but is found to be nearly identical across the family of shapes encountered here, as described in detail below.

Combining equations (3.1), (3.3) and (3.5) yields a boundary value problem for $\phi (s)$ with $Q$ as an unknown coefficient to be determined:

(3.6)\begin{equation} f_1(\phi) \, \frac{\mathrm{d}\phi}{\mathrm{d}s} + f_2(\phi) \, Q + f_3(s) = 0, \end{equation}

with $f_1(\phi ) = ({\gamma }/{r^2(\phi )})\,{\mathrm {d}r}/{\mathrm {d}\phi }$, $f_2(\phi ) = \frac {1}{8} \mu K R^2 \phi ^2 /A^3(\phi )$ and $f_3(s) = \rho g \,{\mathrm {d}z}/{\mathrm {d}s}$. Here $\phi (s)$ is regarded as the quantity of interest since $r$ and $A$ relate to $\phi$ via (3.2) and (3.4). The third term is prescribed by the siphon centreline and hence implicitly contains $h$ and $H$. The boundary conditions are given by the pressures being atmospheric at the two ends of the siphon, which implies via (3.1) that the cross-sectional interfaces are flat and thus $\phi (0) = \phi (L) = {\rm \pi}/2 - \theta$. Numerical solutions obtained via the MATLAB function bvp4c yield $Q$ and $\phi (s)$, and thus $r(s)$ and $A(s)$; these results will be discussed in detail in the sections that follow.

The friction factor $K$ that appears in the viscous term of (3.5) and (3.6) is determined numerically by solving the open-channel flow problem for geometries of the type shown in figure 4(b). The two-dimensional (2-D) Poisson equation $\mu \nabla ^2 v = -G$ for axial velocity field $v$ and constant applied pressure gradient $G$ is solved using MATLAB's Partial Differential Equation Toolbox with no-slip boundary conditions on the solid boundaries and stress-free conditions on the interface. We systematically vary $\theta \in [21^{\circ }, 33^{\circ }]$ in order to account for the experimental variability in $\theta$ and also explore the full range of $\phi \in [0, {\rm \pi}/2-\theta ]$. Numerical solutions of the 2-D flow fields, an example of which is shown in figure 4(c), allow for the computation of the cross-sectional average speed $V$ across $\theta$ and $\phi$. The friction factor coefficient is then determined as $K = 2D_h^2 G/V$ and is found to vary at most between 26.3 and 28.9. Given this small range, we assume a constant $K = 27.6$, this value being an input to the boundary value problem described above.

3.2. Comparison of model and experiment

The model prediction of the flux $Q$ for prescribed $H$ and $h$ is compared to experiments in figure 3. We investigate a range of values $\theta = 27^\circ \pm 6^\circ$ that spans the uncertainty in the experimental measurement of $\theta$. The results are shown in figures 3(a) and 3(b), where the dashed curves correspond to the mean value $\theta = 27^\circ$ and the shaded bands represent the results of varying $\theta$. The plots and insets show that the major trends are successfully reproduced by the model: $Q$ increases linearly with $H$ for small $H$, plateaus for large $H$, and strongly diminishes with increasing $h$. The model and experiment show fair quantitative agreement for high $H$, but the former tends to overestimate the linear rise of $Q$ for low $H$. Possible sources of this discrepancy are discussed below. Nonetheless, the flux magnitudes, scaling relations and qualitative trends seem to be robust. The strong agreement for large siphon heights and heads suggests that the model predictions may be accurate for values of $H$ and $h$ beyond those explored here.

The disagreement between model and experiment for low $h$ and $H$ is likely to be due to three-dimensional (3-D) effects present in experiments near the ends of the siphon but which are not accounted for in the model. In particular, the meniscus geometry is complex where the rods meet the bath, as the liquid surface not only spans the grooves but also rises up from the bath to form contact lines that wrap around the rods. Consequently, the flow velocity field is expected to have significant non-axial components. Our model, on the other hand, assumes a slender geometry in which the cross-sectional shape and area change only slowly along the length of the siphon, and the flow is purely axial. These assumptions are quite reasonable for locations far from the ends, and indeed the model compares well when $h$ and $H$ are large, conditions for which end effects can reasonably be taken as negligible. An improved model would incorporate the 3-D nature of the meniscus, as has been considered for related geometries (Pozrikidis Reference Pozrikidis2010; Cooray, Cicuta & Vella Reference Cooray, Cicuta and Vella2012).

A benefit of the model is that it offers a detailed view of how the cross-sectional area of the wetted region and its pressure vary along the siphon axis, as shown in figure 4(d) for a representative case of $H = 7$ cm and $h=5$ cm. The area $A(s)$ (blue) and pressure $P(s)$ (red) are maximal at the two ends and minimal at an intermediate arclength location $s$. The pressure curve is qualitatively similar to that of a conventional ideal siphon: the ends being in contact with the atmosphere requires that $P(0)=P(L)=P_0$, and $P(s)$ is otherwise lower than atmospheric pressure primarily because of the gravitational contribution. The area curve can be qualitatively interpreted through the coupling between the shape of the free surface and the pressure. The strongly negative gravitationally induced pressure along the upper portions of the siphon must be resisted by high curvature of the free interface, which is achieved by the surface retreating deeply into the groove and thus shrinking the wetted area. The changes in area are dramatic, steeply dropping upon exiting the source container, thinning by two orders of magnitude, and then rapidly rising in the approach to the collector. For typical parameters explored here, the wetted region shrinks to $A_{{min}} \sim 10^4~\mathrm {\mu }\text {m}^2$, equivalent to a length scale of about 100 $\mathrm {\mu }\text {m}$ or roughly the width of a human hair.

In contrast to the conventional siphon, for which pressure is lowest at the apex, the minimal pressure and minimal area of the groove siphon occur somewhat displaced along the longer arm, as shown in figure 4(d). This fact can be interpreted by analysing the balance of pressures represented by (3.5). At the apex, the gravitational contribution is zero, and thus a negative pressure gradient is needed to drive the flow against viscous resistance. The negative gradient at the apex implies that the minimal value of pressure must occur downstream along the longer arm. The Young–Laplace law of (3.1) ensures that the cross-sectional area reaches a minimum at this same location.

3.3. Characterization via dimensionless quantities

The general characteristics of capillary siphoning can be revealed by recasting the model in dimensionless form and exploring a wide range of the relevant parameters. Non-dimensionalization is carried out by choosing length, velocity and pressure scales based on surface tension and viscous effects (De Gennes, Brochard-Wyart & Quéré Reference De Gennes, Brochard-Wyart and Quéré2013): capillary length $l_c = \sqrt {\gamma /\rho g}$, velocity $V_c=\gamma /\mu$, flux $Q_c = V_c l_c^2$ and pressure $P_c=\gamma /l_c$. The solutions of the model can then be recast in terms of dimensionless volumetric flow rate $\tilde {Q} = Q / Q_c$ generated for given values of $\tilde {H} = H / l_c$ and $\tilde {h} = h / l_c$. For simplicity, we set the radius of the arch to be zero, so that $\tilde {H}$ and $\tilde {h}$ completely specify the siphon path. The colour map $\tilde {Q}(\tilde {H},\tilde {h})$ of figure 5(a) summarizes the numerical solutions and verifies the generality of the observed trends. For fixed $\tilde {h}$, the flux $\tilde {Q}$ increases with increasing $\tilde {H}$ and then saturates to a plateau. For fixed $\tilde {H}$, $\tilde {Q}$ rapidly diminishes with increasing $\tilde {h}$.

Figure 5. Characterization of the groove siphon using dimensionless quantities. (a) Flow rate $\tilde {Q}$ across siphon heights $\tilde {H}$ and heads $\tilde {h}$. Solid black curves are contours of $\tilde {Q}$ whose values are marked by dots on the colour bar. (b)  Representative curves of $\tilde {Q}(\tilde {H})$ for different $\tilde h$ show the transition from linear growth to plateau. Each transition point (circle) is the intersection of two asymptotic lines (dashed). (c)  Transition height $\tilde {H}^{*}$ and saturated flow rate $\tilde {Q}^{*}$ versus $\tilde {h}$. For large $\tilde {h}$, scaling laws of $\tilde {H}^{*} \sim \tilde {h}$ and $\tilde {Q}^{*} \sim \tilde {h}^{-3}$ are observed. (d)  Minimal cross-section area $\tilde A_{{min}}$ versus $\tilde {h}$ for three $\tilde H$ values. The curves nearly overlay on one another, indicating a weak dependence on $\tilde H$. An approximate scaling $\tilde A_{{min}} \sim \tilde {h}^{-3/2}$ holds for large $\tilde {h}$.

The dimensionless form of the model also proves the generality of the transition of $\tilde {Q}(\tilde {H})$ from linear growth to plateau for fixed $\tilde {h}$, as shown in figure 5(b). To characterize this change of behaviour, we define the transition point (circles) as the intersection of best-fitting asymptotic lines (dashed) to the curves $\tilde {Q}(\tilde {H})$ in the linear and plateau regimes. This analysis yields critical values of the height $\tilde {H}^{*}$ and flux $\tilde {Q}^{*}$, both of which vary with the head $\tilde {h}$ as plotted in figure 5(c). The transitional height $\tilde {H}^{*}$ (red curve) tends to increase quasi-linearly with $\tilde {h}$, indicating that the crossover between regimes occurs for $\tilde {H} \sim \tilde {h}$. The log–log plot in the inset confirms a quasi-linear scaling for sufficiently large $\tilde {h}$. The transitional flux $\tilde {Q}^{*}$ is the same as the saturated value and is found to rapidly decrease with $\tilde {h}$, as shown by the blue curves in the linear and log plots of figure 5(c). While $\tilde {Q}^{*}(\tilde {h})$ does not follow a power-law relation across all $\tilde {h}$, an approximate form $\tilde {Q}^{*} \sim \tilde {h}^{-3}$ seems to hold for sufficiently high $\tilde {h}$ and is discussed in more detail in what follows.

3.4. Scaling analysis and hydraulic interpretation

The model permits a scaling analysis that explains the dependence of flow rate $Q$ on the heights $H$ and $h$ and, more generally, offers physical interpretations of our observations. Viewing the siphon as a hydraulic resistor, the driving pressure difference ${\rm \Delta} P$ and current $Q$ are related by the Hagen–Poiseuille law, ${\rm \Delta} P \sim LQ/A^2$, which applies to a conduit of length $L$ and uniform cross-sectional area $A$. For the siphon, the driving is provided by the gravitational contribution to the hydrostatic pressure, ${\rm \Delta} P \sim H$, while the total length is approximately $L \sim H+2h$. The scaling of area $A$ with $H$ and $h$ is more subtle. Recalling figure 4(d), $A$ is not uniform but rather varies with $s$, but we can expect the hydraulic resistance to be dominated by the long segment over which $A$ is exceedingly small. This motivates the approximation $A \sim A_{{min}}$, where the dependence of the minimal area on $H$ and $h$ must be determined. The numerical solutions suggest an approximate scaling $A_{{min}} \sim h^{-3/2}$, as shown by the curves of figure 5(d), where the dimensionless form of the minimal area $\tilde {A}_{{min}}$ is seen to decrease strongly with $\tilde {h}$ but depends only weakly on $\tilde {H}$. Putting all these results together yields the scaling relation $Q \sim H/(H+2h)h^3$ that relates the flux to the height parameters of the siphon.

The above scaling relation $Q(H,h)$ is consistent with and helps to interpret our experimental and numerical results. Namely, for fixed $h$, the linear form $Q \sim H$ arises for $H \ll h$ and then transitions to the saturated form $Q \sim \text {const.}$ for $H \gg h$. This agrees with the $Q(H)$ experimental data and model curves in figures 1(d), 3(a) and 5(b). In essence, the gains in $Q$ for increasing $H$ diminish since both the gravitational driving and viscous resistance increase together. Likewise, for fixed and high $H \gg h$, the saturation flux $Q^* \sim h^{-3}$, which agrees well with the blue curves of figure 5(c). The hydraulic interpretation is that increasing $h$ throttles the flow of the siphon by shrinking its cross-sectional area $A$ and thereby increasing resistance.

Left unexplained by the above argument is the approximate scaling $A_{{min}} \sim h^{-3/2}$, which seems to lack any simple scaling argument and is a consequence of the geometric details. The relation can be derived by noting that $\phi$ is exceedingly small over the long segment for which $A \sim A_{{min}}$, and retaining leading-order terms in Taylor expansions of (3.2) and (3.4) yields $r \sim \phi ^2$ and $A \sim \phi ^3$. The shape of the free surface at the upper portions of the siphon arises from a balance of capillary (${\sim }r^{-1}$) and hydrostatic (${\sim }h$) pressures, indicating $r \sim h^{-1}$ and thus $\phi \sim h^{-1/2}$ and $A \sim h^{-3/2}$. This is consistent with the curves of figure 5(d).