2.1. Experimental protocols

The apparatus was run for an initial period of 30 min to establish a uniform film of viscous fluid on the belt before any levitation experiment was carried out. The thickness of the film was set using the knife edge. Achieving levitation with cylinders was relatively straightforward and involved placing a cylinder on the moving layer and adjusting the speed of the belt to achieve balance. If the belt speed was too high, the cylinder moved up the belt, too low and it travelled down the belt. A given cylinder would always rotate around its own axis at a rate that was found to be independent of the belt speed. A typical set of results illustrating these points are shown in figure 3. It can be seen in figure 3(a) that there is a linear relationship between the belt speed and the cylinder translation velocity, and that the translation speed is zero for a particular belt speed, i.e. the cylinder is balanced at a unique belt speed. The gradient of this linear relationship is approximately $1$. This is because the problem of determining the belt speed required for levitation is equivalent to determining the cylinder fall speed with a stationary belt for the following reason. The cylinder translates at an approximately uniform speed, since inertial terms are small, and so the frame moving with the cylinder is inertial. At this specific belt speed, the cylinder centre is stationary in the laboratory frame while the cylinder continues to rotate about its own axis. This balance was observed to be stable over a time scale of at least 12 h. Note that the balance speed and cylinder rotation rate were dependent on the size and weight of cylinder used. Occasionally, a mild rippling in the free surface is observed downstream of the cylinder, analogous to the classic printer's instability (Pearson Reference Pearson1960), but this does not appear to affect the cylinder motion.

Figure 3.

(a) The translation velocity of a cylinder in the laboratory frame. The dashed black line represents a cylinder that is fixed in the laboratory frame, so below/above this line the cylinder moves down/up the belt. The solid blue line shows a linear best fit of the data. (b) The rotation velocity (rotation rate multiplied by cylinder radius) of the same cylinder, which we see is essentially independent of belt speed. For these data, we use an 8 mm diameter, 12 mm long PTFE cylinder on a 0.3 mm oil layer. The data are collected by measuring the time taken for the cylinder to translate 100 mm, and averaged over 10 repetitions. The error bars correspond to one standard deviation of the data.

The majority of the experiments were performed with cylinders with radii which were larger than the layer thickness by a factor of approximately $10$. In these cases, a small stable tongue was observed to form on the upstream side of the cylinder as in figure 2(a). For smaller cylinders, the tongue could grow to much longer lengths, as shown in figure 2(b). Part of our modelling work will involve understanding the onset of tongue formation. We will consider a two-dimensional model of the fluid problem, neglecting any edge effects near the ends of the cylinder (e.g. deformations in the fluid surface near the ends of the cylinder in figure 2). Even with these assumptions, the model predicts results that agree well with experimental data.

3.1. Asymptotic structure

Since the ratio $\tilde {h}/\tilde {a} =: \epsilon$ is small, we expect to be able to model much of the flow by lubrication theory. However, before we non-dimensionalise the system, it is helpful to understand the asymptotic regions present in the flow problem in the limit $\epsilon \to 0$, in order to focus on the most important regions. As such, we now outline the asymptotic structure of the dimensionless problem with lengths scaled with cylinder radius.

In the limit $\epsilon \to 0$, there are six asymptotic regions in the system, as illustrated in figure 5. The first region (Region I) is the thin layer of oil between the scraper at the bottom of the belt and the tip of any tongue created by the cylinder. Scaling length with cylinder radius, it has thickness $\epsilon$ and length of $\mathit {O}(1)$. In this region, the fluid from the scraper is undisturbed – the fluid moves up the belt with a free surface parallel to the belt.

Figure 5.

A schematic of the asymptotic regions involved in the dimensionless system, assuming no tongue formation. The regions with boundaries denoted as dashed green lines are lubrication regions and those denoted with dotted red lines are transition regions. All lengths are measured relative to the cylinder radius.

Region II is the small upstream coalescence region representing the inlet of the nip region between the belt and the cylinder. The flow is two-dimensional here and so, in the absence of a tongue, its thickness and length are of $\mathit {O}(\epsilon )$. In Region II, the cylinder is physically present in the problem, but its surface appears flat here (not shown in figure 5). This is in contrast to the free surface, whose curvature is apparent at leading order in this region. This inlet region marks the three-way transition between the fluid coming in from over the cylinder (Region VI, discussed below), the fluid entering from upstream (Region I), and the fluid entering the nip region between the belt and the cylinder (Region III, discussed below). The extent of Region II must be determined as part of the solution.

Region III is the thin nip region between the belt and the cylinder. Its thickness is of $\mathit {O}(\epsilon )$ and, since the cylinder cross-section is circular, its length is of $\mathit {O}(\epsilon ^{1/2})$. We will find that the leading-order force and torque acting on the cylinder occur in this region. In Region III, the curvature of the cylinder is apparent and is well-approximated by a parabola to leading order. In addition, there is no free surface between the oil and the air in Region III, but the position of the coalescence and separation regions at either end of this region must be determined as part of the solution. Hence, Region III can be characterised as a free-boundary lubrication problem with two unknown end positions.

The small downstream separation region is Region IV. As the flow is fully two-dimensional here, its thickness and length are of $\mathit {O}(\epsilon )$. This marks the outlet of the nip region between the belt and the cylinder. In a similar way to Region II, the cylinder is physically present in this region, though it again appears to be flat since its length is of the same order as the downstream film thickness. The curvature of the free surface is apparent at leading order in Region IV, and this region marks the three-way transition between the fluid leaving the nip from Region III to head downstream of the cylinder (Region V) and to coat the cylinder (Region VI).

Once the fluid leaves the cylinder behind, it enters Region V. This region is the downstream equivalent of Region I. Its thickness is $\epsilon$ and its length is of $\mathit {O}(1)$. In this region the fluid moves downstream with a free surface parallel to the belt, with the same thickness as the free surface in Region I since the system is in a steady state.

The final Region VI corresponds to the fluid travelling over the surface of the cylinder. Its thickness is of $\mathit {O}(\epsilon )$ and its length is of $\mathit {O}(1)$. Here, the fluid moves around the cylinder, transporting fluid from the downstream separation region (IV) to the upstream coalescence region (II).

While the above regions are all coupled to one another, the most important regions to understand levitation are Regions II, III and IV. Although the importance of Region III is fairly intuitive, the importance of Regions II and IV is best demonstrated after investigating Region III. Thus, in the next section we non-dimensionalise the system using variables appropriate for Region III.

3.2. Dimensionless system

We non-dimensionalise the system using the scalings outlined for Region III in figure 5, the nip region between belt and cylinder. That is, we use the geometric mean of the upstream film thickness $\tilde {h}$ and the cylinder radius $\tilde {a}$ as the $\tilde {x}$-direction length scale and the upstream film thickness as the $\tilde {y}$-direction length scale. In addition, we use a typical belt speed to scale the $\tilde {x}$-direction velocity, and use a $\tilde {y}$-direction velocity scaling that achieves a balance in the lubrication equations. This results in the scalings

(3.7a–e)\begin{equation} \tilde{x} = (\tilde{h} \tilde{a})^{1/2} x, \quad \tilde{y} = \tilde{h} y, \quad \tilde{u} = \tilde{U} u, \quad \tilde{v} = \epsilon^{1/2} \tilde{U} v, \quad (\tilde{p}, \boldsymbol{\tilde{\sigma}}) = (\tilde{\mu} \tilde{U} /\tilde{a} \epsilon^{3/2}) (p, \boldsymbol{\sigma}), \end{equation}

where $\tilde {U}$ is a typical value of the belt velocity.

Using the scalings (3.7a–e) in the dimensional equations (3.1a,b), we obtain the dimensionless governing equations

(3.8a–c)\begin{equation} p_{x} = u_{y y} + \epsilon u_{x x} - \epsilon^{3/2} \varGamma, \quad p_{y} = \epsilon v_{y y} + \epsilon^2 v_{x x}, \quad u_{x} + v_{y} = 0, \end{equation}

where $\varGamma := \tilde {\rho } M/{\rm \pi} \tilde {\rho }_s$ and $M := {\rm \pi}\tilde {\rho }_s \tilde {g} (\tilde {h} \tilde {a}^3)^{1/2}/ \tilde {\mu } \tilde {U}$ is the dimensionless weight per unit length of the cylinder. In the next section, it will become apparent that the parameter $M \leqslant \mathit {O}(1)$ in order for levitation to occur, and it will henceforth be regarded as a control parameter.

The position of the cylinder is defined by

(3.9)\begin{equation} x^2 - 2 (y - h_0) + \epsilon (y - h_0)^2 = 0, \end{equation}

with $h_0 := \tilde {h}_n/\tilde {h}$, where $\tilde {h}_n$ is the nip height as shown in figure 4, and hence $h_0$ is to be determined. On each solid surface, the scaled boundary conditions (3.2) are as follows

(3.10a,b)\begin{equation} \boldsymbol{u} = U \boldsymbol{e}_{x} \quad \text{on the belt}, \quad \boldsymbol{u} = \varOmega \boldsymbol{t} \quad \text{on the cylinder}, \end{equation}

where $U = \tilde {U}_b/\tilde {U}$ is the unknown dimensionless belt speed, $\varOmega = \tilde {a} \tilde {\varOmega }/\tilde {U}$ is the unknown dimensionless rotation rate of the cylinder, $\boldsymbol {e}_{x}$ is the unit vector in the $x$-direction, and $\boldsymbol {t}$ is the unit tangent vector in the anticlockwise direction.

Scaling $\mathrm {d}\tilde {S}$ with $(\tilde {h} \tilde {a})^{1/2}$ in (3.5), the force balance is

(3.11)\begin{equation} \epsilon^{1/2} M \boldsymbol{e}_{x} = \oint_{\partial S} \boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{n} \, \mathrm{d}S. \end{equation}

The torque balance (3.6) is simply

(3.12)\begin{equation} \oint_{\partial S} \boldsymbol{r} \times \left(\boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{n}\right) \mathrm{d}S = \boldsymbol{e}_z \oint_{\partial S} \boldsymbol{t} \boldsymbol{\cdot} \left(\boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{n}\right) \mathrm{d}S = \boldsymbol{0}. \end{equation}

In the simplest configuration, we expect the nip region to extend between two free boundaries with Regions II, IV, so that $x_{-} < x < x_{+}$ where $x_{\pm }$ must be determined as part of the solution. We will address this free-boundary problem in the next section.

4.1. Region III: nip under cylinder

In this region, the fluid is enclosed by the cylinder and the belt, and the region terminates in small menisci at each end, corresponding to free boundaries.

Taking the limit $\epsilon \to 0$, the system (3.8a–c) becomes the usual lubrication equations

(4.1)\begin{equation} p_{x} = u_{y y}, \quad p_{y} = 0, \quad u_{x} + v_{y} = 0 \quad \text{for } x_{-} < x < x_{+}, \quad 0 < y < H(x), \end{equation}

where the dimensionless leading-order cylinder surface is located at $y = H(x)$, where

(4.2)\begin{equation} H(x) := h_0 + x^2/2. \end{equation}

The boundary conditions (3.10a,b) become

(4.3a,b)\begin{equation} u = U, \quad v = 0 \quad \text{on } y = 0, \quad u = \varOmega, \quad v = \varOmega x \quad \text{on } y = H(x). \end{equation}

We proceed in the usual manner to deduce

(4.4)\begin{equation} u = \dfrac{p_{x}}{2} y (y - H) + \dfrac{y}{H} \left(\varOmega - U \right) + U. \end{equation}

We define the unknown flux as

(4.5)\begin{equation} Q_c := \int_0^H u \, \mathrm{d} y ={-}\dfrac{p_{x}}{12}H^3 + \left(\varOmega + U \right) \dfrac{H}{2}. \end{equation}

Rearranging (4.5) yields the following ordinary differential equation (ODE) for the pressure:

(4.6)\begin{equation} \frac{\mathrm{d} p}{\mathrm{d} x} = \dfrac{6\left(U + \varOmega \right)}{H^2} - \dfrac{12 Q_c}{H^3} \quad \text{for } x_{-} < x < x_{+}, \end{equation}

which can be solved analytically as follows

(4.7)\begin{equation} p = 3 \left(U + \varOmega - \dfrac{3 Q_c}{2 h_0}\right) \left(\dfrac{x}{h_0 H} + \sqrt{2} \dfrac{\tan^{{-}1} x/\sqrt{2 h_0}}{h_0^{3/2}}\right) - \dfrac{3 Q_c x}{h_0 H^2} + P_0, \end{equation}

where $P_0$ is an unknown constant of integration. Hence, apart from a constant, $p$ is an odd function of $x$.

Finally, we note that the leading-order stress on the cylinder is

(4.8)\begin{equation} \left.\boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{n}\right|_{y = H} ={-}\epsilon^{1/2} \left.\left(p \frac{\mathrm{d} H}{\mathrm{d} x} + \frac{\partial u}{\partial y}\right)\right|_{y = H} \boldsymbol{e}_{x} + \left.p\right|_{y = H} \boldsymbol{e}_{y}, \end{equation}

where

(4.9)\begin{equation} \left.\frac{\partial u}{\partial y}\right|_{y = H} = \dfrac{2 \left(U + 2\varOmega \right)}{H} - \dfrac{6 Q_c}{H^2}. \end{equation}

Hence, in combination with (3.11), we deduce that $M \leqslant \mathit {O}(1)$ and $\tilde {U}_b = \mathit {O}(\tilde {\rho }_s \tilde {g} (\tilde {h} \tilde {a}^3)^{1/2}/\tilde {\mu })$.

We have now proceeded as far as possible in Region III without invoking any global or matching conditions. At this point, it is worth emphasising the unknowns in the system. They are $\varOmega$, $Q_c$, $h_0$, $P_0$, $x_{-}$, $x_{+}$ and $U$. To determine these seven unknowns, we need seven equations. Three of these will come from force and torque balances on the cylinder, and the remaining four will come from asymptotic matching with the coalescence and separation regions at either end of Region III. It will soon become apparent that it will suffice to solely consider Region IV, even though the local models for the Regions II and IV are similar. For the moment, we expect the flow pattern in Region IV to be as in figure 6 and a solution for this configuration will be presented in § 4.3.

Figure 6.

A sketch of the inner problem for Region IV. The green arrows denote the direction of the flow with dashed line denoting the dividing streamline. The solid black lines are solid boundaries, and the blue curved line denotes the free surface. We note that there may be more than one dividing streamline with accompanying recirculation zones for low capillary numbers (Coyle et al. Reference Coyle, Macosko and Scriven1986).

4.2. Regions IV: the separation zone

To scale into Region IV, we change variables as follows:

(4.10a–d)\begin{equation} x = x_{+} + \epsilon^{1/2} H(x_{+}) X, \quad y = H(x_{+}) Y, \quad (p,\boldsymbol{\sigma}) = \dfrac{\epsilon^{1/2}}{H(x_{+})} (P, \boldsymbol{\varSigma}), \quad v = V / \epsilon^{1/2}. \end{equation}

At leading order, the flow is governed by the incompressible Stokes equations

(4.11a–c)\begin{equation} \frac{\partial P}{\partial X} = \frac{\partial^2 u}{\partial {X}^2} + \frac{\partial^2 u}{\partial {Y}^2}, \quad \frac{\partial P}{\partial Y} = \frac{\partial^2 V}{\partial {X}^2} + \frac{\partial^2 V}{\partial {Y}^2}, \quad \frac{\partial u}{\partial X} + \frac{\partial V}{\partial Y} = 0. \end{equation}

The boundary conditions on the solid surfaces are

(4.12a,b)\begin{equation} u = U, \quad V = 0 \quad \text{on } Y = 0, \quad u = \varOmega, \quad V = 0 \quad \text{on } Y = 1. \end{equation}

On the free surface, surface tension may be important and the dimensional stress condition (3.3) becomes

(4.13)\begin{equation} \boldsymbol{\varSigma} \boldsymbol{\cdot} \boldsymbol{n} = \dfrac{\kappa}{{Ca}} \boldsymbol{n} \quad \text{on the free surface}, \end{equation}

where $\kappa$ is the $\mathit {O}(1)$ curvature of the free surface in the $X$–$Y$ plane and ${Ca} = \tilde {\mu } \tilde {U}/ \tilde {\gamma }$ is the capillary number, which is expected to be at least of $\mathit {O}(1)$.

As briefly mentioned in § 4.1, we need four matching conditions from Regions II and IV to formally close the problem. Two of these will come from matching the pressure, and two from matching the flux. However, the far-field conditions in these inner regions arise from matching into the appropriate outer regions, which involves the unknown flux. Thus, determining the flux through these regions is important in formally closing the system. Solving each of the inner problems numerically yields a result for the scaled flux at each end of Region III.

As shown in figure 6, for Region IV we denote the film thicknesses in the separated regions as $h_{+}^{b}$ and $h_{+}^{c}$ for the fluid near the belt and cylinder, respectively. Then, the total flux into and out of Region IV is

(4.14)\begin{equation} q_{+} = U h_{+}^{b} + \varOmega h_{+}^{c}, \end{equation}

and we note that the flux in Region III is related to this flux through

(4.15)\begin{equation} Q_c = H(x_{+}) q_{+}. \end{equation}

A similar result holds in Region II, where mass conservation again requires that the total flux is $Q_c$. We will return to this region later, but first we will consider what extra information emerges from a study of Region IV.

4.3. Direct numerical simulation of Region IV

Here we describe the full numerical solution to the inner problem (4.11a–c)–(4.13) in Region IV. Hence, we use $H$ to refer to $H(x_{+})$ for notational convenience in this section. The far-field conditions in Region IV are obtained through matching with Region III, Region V and Region VI. They are:

(4.16a)\begin{align} u \to 3 \left(2q_{+} - U - \varOmega \right) Y (1 - Y) + (\varOmega - U) Y + U, \quad V \to 0 \quad \text{as } X \to -\infty, \end{align}

(4.16b)\begin{align} u \to U, \quad V \to 0 \quad \text{as } X \to +\infty, \text{ with } Y \in (0, h_{+}^{b}) \end{align}

(4.16c)\begin{align} u \to \varOmega, \quad V \to 0 \quad \text{as } X \to +\infty, \text{ with } Y \in (1 - h_{+}^{c}, 1). \end{align}

We emphasise that $q_{+}$, $h_{+}^{b}$ and $h_{+}^{c}$ are a priori unknown, but linked through the conservation of flux relationship (4.14). The goal of solving this system is to determine these parameters which will, in turn, allow us to derive a closed system for the motion of the cylinder and the flow structure in Region III.

Since we may reduce the dependence of this inner problem from three dimensionless parameter groupings to two by scaling the pressure and velocity with the lower wall velocity, we henceforth take $U = 1$ without loss of generality. For given values of ${Ca}$ and $\varOmega$, we implement and solve a generalised version of the system formed by (4.11a–c)–(4.13) and (4.16) using the open-source Basilisk platform (see http://basilisk.fr/ or Popinet (Reference Popinet2020) for details). The package provides functionality for the solution of partial differential equations using a second-order time- and space-adaptive finite volume methodology. The generalisation stems from the fact that both liquid and gas environments are accounted for in a time-dependent setting, with fictitious time introduced to allow easy and accurate iterations towards a steady-state solution. The physical properties of the gas are well aligned with the asymptotic framework (hydrodynamically passive, with a density ratio of $10^{-3}$ and viscosity ratio of $10^{-2}$ relative to the reference liquid), while a small dimensionless gravitational acceleration contribution (of magnitude $10^{-2}$) is imposed in the negative $X$-direction at the level of the entire multi-fluid system, even though it will soon be apparent that gravitational effects are negligible in the nip region. Interfacial advection is performed using a conservative scheme, as described by Weymouth & Yue (Reference Weymouth and Yue2010), while the well-known continuum surface force (CSF) model by Brackbill, Kothe & Zemach (Reference Brackbill, Kothe and Zemach1992) is employed to include surface tension effects.

At the grid level, the method lives on a quadtree structure, which ensures efficient multi-scale capabilities, as well as convenient parallelisation features, with aspects surrounding the algorithm and in particular interfacial reconstruction described by Popinet (Reference Popinet2009). For the present investigation, the computational domain is set to occupy a region of dimensionless size $20 \times 1$ (illustrated in figure 7), which extensive testing has shown to be sufficient in allowing an accurate imposition of both inlet and far-field conditions on the basis of quantitative confirmation of converged locally parallel flows in the respective regions. The mesh adaptivity strategy is built around the location of the fluid–fluid interface, as well as changes in the magnitude of the velocity components. It retains a sufficiently large number of degrees of freedom (typically of ${O}(10^{5})$ and with a reduction of a factor of 3–5 over an equivalent uniform mesh) in sensitive regions, such as for example cases with a relatively low value of $\varOmega$, in which the interface may approach the upper boundary to within a distance of ${O}(10^{-2})$. A typical convergence procedure to a steady-state solution involves an adjustment of an initially prescribed interfacial profile (formed by the union of strip regions of prescribed width joined by a semicircular tip) and the inlet condition (restricted by the functional form in (4.16a), but with $q_{+}$ as an additional allowed degree of freedom), such that the interfacial tip location no longer changes in time. This can be summarised as a feedback relaxation method where we slowly vary $q_{+}$ in response to the interface location until the free boundary stops translating in $X$. The procedure requires ${O}(10^2)$ central processing unit (CPU) hours for each individual run (executed in parallel, most often over $16$ CPUs, on local high performance computing facilities). Video supplementary material for a particular case is provided for reference, using the parameters indicated in figure 7. The method ultimately yields values of $q_{+}$, $h_{+}^{b}$ and $h_{+}^{c}$ for a steady solution, for given ${Ca}$ and $\varOmega$.

Figure 7.

Full computational domain constructed in Basilisk, with the background colour showing the norm of the calculated velocity field $\mathcal {U} = (u^2 + V^2)^{1/2}$ in both liquid and gas regions, and the interface separating them highlighted in a thick black line. Thin black lines are used for presenting selected streamlines inside the liquid region. This particular case study is described by parameter values $Ca = 2.0$, $U = 1.0$ and $\varOmega = 0.6$. Enhanced video supplementary material (Movie 3, available at https://doi.org/10.1017/jfm.2021.284) expands on the set-up and individual contributions of specific flow components.

We find that increasing ${Ca}$ increases both $h_{+}^{b}$ and $h_{+}^{c}$, but this increase is bounded above as ${Ca}$ becomes large (figure 8). Conversely, as ${Ca}$ is decreased, $h_{+}^{b}$ and $h_{+}^{c}$ both become small. This requires a significant decrease in mesh size to resolve, which makes the simulations computationally prohibitive for smaller values of ${Ca}$. Increasing $\varOmega$ decreases $h_{+}^{b}$ but increases $h_{+}^{c}$, as more of the fluid shifts towards the cylinder surface. When $\varOmega = 1$, we find that $h_{+}^{b} = h_{+}^{c}$, as one would expect from the symmetry of the problem. When $\varOmega$ becomes small, the singular nature of the problem ($h_{+}^{c} \to 0$ in this limit) means that the simulations become much more computationally expensive, for similar reasons as for small ${Ca}$.

Figure 8.

The key results from the DNS of the inner problem we derive in § 4.2: (a) $h_{+}^{b}$, the height of the film on the belt and (b) $h_{+}^{c}$, the height of the film on the cylinder. The different colours represent different values of ${Ca} \in \{ 0.2, 0.4, 0.7, 1, 2, 5, 10, 20\}$, in the direction indicated by the arrow.

Strikingly, we find that the ratio of film heights is related to the ratio of boundary velocity through a two-thirds power-law. That is, we find that in all DNS the results collapse onto the following curve:

(4.17)\begin{equation} h_{+}^{c}/h_{+}^{b} = \left(\dfrac{\varOmega}{U}\right)^{2/3}, \end{equation}

as shown in figure 9. This is similar to the observation in Coyle et al. (Reference Coyle, Macosko and Scriven1986), where the authors solve a similar Region IV (separation) inner problem numerically and state that the relationship $h_{+}^{c}/h_{+}^{b} = (\varOmega /U)^{0.65}$ appears to hold for ${Ca} > 0.1$. While the small-${Ca}$ limit is difficult to resolve numerically and is not relevant to our experiments, the power-law relationship does continue to hold in this limit, as we show in appendix A through an asymptotic analysis.

Figure 9.

All DNS data from figure 8 approximately collapses onto the curve $h_{+}^{c}/h_{+}^{b} = (\varOmega /U)^{2/3}$. For each value of $\varOmega /U$, there are eight different individual runs that appear to have collapsed onto a single point, so there are 80 simulations represented in this figure in total. The eight values of ${Ca} \in \{ 0.2, 0.4, 0.7, 1, 2, 5, 10, 20\}$ correspond to different colours, as described in figure 8.

The computational results we have presented in figure 8 provide the two remaining closure conditions required to understand the cylinder motion. In the next section, we bring together all seven equations needed to solve the problem.

4.4. Closing the problem

There are seven unknowns in the problem: $\varOmega$, $Q_c$, $h_0$, $P_0$, $x_{-}$, $x_{+}$ and $U$. Of the seven equations required to determine these, three arise from the global force and torque balances (3.11)–(3.12). The leading-order contributions from these global conditions all emanate from Region III, so using the stress results from this region (4.8), and noting that $(1, \epsilon ^{1/2} x)$ is tangent to the cylinder yields

(4.18a)\begin{gather} \int_{x_{-}}^{x_{+}} p \, \mathrm{d}x = 0, \end{gather}

(4.18b)\begin{gather}\int_{x_{-}}^{x_{+}} x p + \left.\frac{\partial u}{\partial y}\right|_{y = H} \, \mathrm{d}x + M = 0, \end{gather}

(4.18c)\begin{gather}\int_{x_{-}}^{x_{+}} \left.\frac{\partial u}{\partial y}\right|_{y = H} \, \mathrm{d}x = 0. \end{gather}

Here, (4.18a) is the force balance in the $y$-direction, (4.18b) is the force balance in the $x$-direction and (4.18c) is the moment balance. We note that imposing (4.18c) will simplify (4.18b). A consequence of (4.18b) is the physical interpretation that the force levitating the cylinder depends on the pressure variation within the nip. This contrasts with the result for the block in Mullin et al. (Reference Mullin, Ockendon and Ockendon2020) where levitation resulted from the shear forces.

Two additional equations arise from asymptotically matching the pressure in Region III with the pressure in the coalescence and separation Regions II and IV. Since the pressure in Region III is asymptotically larger than the pressure in the coalescence and separation Regions II and IV, as can be seen from the pressure scaling in (4.10a–d), matching yields the conditions

(4.18d)\begin{equation} p(x_{-}) = 0, \quad p(x_{+}) = 0. \end{equation}

As (4.18) only consists of five equations in total, it remains to determine two additional equations to close the system. These two degrees of freedom correspond to our current lack of information regarding the positions of the free boundaries $x_{-}$ and $x_{+}$. This is a well-studied problem in lubrication theory, see, for example, Swift (Reference Swift1932), Stieber (Reference Stieber1933), Coyne & Elrod (Reference Coyne and Elrod1970), Coyne & Elrod (Reference Coyne and Elrod1971), Savage (Reference Savage1977), Ruschak (Reference Ruschak1982), Taroni et al. (Reference Taroni, Breward, Howell and Oliver2012). We note that the existence and uniqueness of solutions to the relevant inner problem remain open questions.

However, as noted above, it is possible to use two simple conditions in conjunction with our analysis in §§ 4.2 and 4.3 to close the problem. Firstly, we note that both inner regions are linked through a globally conserved quantity. Due to the steady assumption, the fluid flux into Region I is the same as the fluid flux out of Region V in their respective far-fields. Hence, the film thickness is the same in these regions. In particular, this means that the fluid leaving Region IV on the belt has a dimensionless thickness of $1$ in Region V. This means

(4.19a)\begin{equation} h_{+}^{b} = 1/H(x_{+}), \end{equation}

using the scaling in (4.10a–d). We are able to use the DNS results summarised in figure 8 to determine the left-hand side of (4.19a), and $H(x_{+}) = h_0 + x_{+}^2 / 2$.

Secondly, using the two-thirds power-law relationship (4.17) in conjunction with the scaled flux condition (4.14), the flux scaling (4.15), and the first closure condition (4.19a), results in the following relationship:

(4.19b)\begin{equation} Q_c = U + \varOmega \left(\dfrac{\varOmega}{U}\right)^{2/3}. \end{equation}

The conditions (4.19) therefore represent two independent equations to close the system in Region III. To summarise the flow of information here: solving the flow problem in Region IV yields $h_{+}^{b}$ and $h_{+}^{c}$. The former can be used in (4.19a) to obtain one closure condition. Then the power-law relationship (4.17) yields the second closure condition (4.19b). The solution will then automatically solve the equivalent Region II inner problem.

The equations (4.18) and (4.19) close the entire system, and their investigation will allow us to determine the belt speed required for balance, the rotation of the cylinder and the flow structure in Region III. This analysis is the subject of the next section.

4.5. Investigating the closed system

In this section, we investigate the closed system we have derived, (4.18) and (4.19), to understand the critical belt speed, the motion of the cylinder and the flow structure in Region III. Although (4.19b) is derived from DNS through (4.17), we do not require the actual DNS results to implement (4.19b). Hence, the only explicit appearance of the DNS results in (4.18) and (4.19) is through the left-hand side of (4.19a). Thus, to avoid errors in iterated interpolation, we begin by solving the remaining six equations (4.18), (4.19b) by sweeping through different possible values of $x_{+}$ to generate a one-family parameter of solutions for the six remaining unknowns: $x_{-}$, $h_0$, $\varOmega$, $P_0$, $Q_c$ and $U$, then use (4.19a) to relate $x_{+}$ to the experimental parameters. We also note that it is possible to scale out the mathematical dependence of the system on the parameter $M$ by solving for $\varOmega /U$, $Q_c/U$ and $U/M$ instead of $\varOmega$, $Q_c$ and $U$.

The resulting nonlinear algebraic system from (4.18), (4.19b) is

(4.20a)\begin{gather} \alpha \left[I_2 \right]_{-}^{+} - \beta \left[\dfrac{x}{H^2} \right]_{-}^{+} = 0, \end{gather}

(4.20b)\begin{gather}P_0 + \alpha I_2(x_{+}) - \dfrac{\beta x_{+}}{H^2(x_{+})} = 0, \end{gather}

(4.20c)\begin{gather}\left(\dfrac{\alpha}{3} + 2 \varOmega \right) \left[I_1 \right]_{-}^{+} - \beta \left[\dfrac{x}{H} \right]_{-}^{+} = 0, \end{gather}

(4.20d)\begin{gather}\dfrac{\alpha}{2 h_0} \left[x I_1 \right]_{-}^{+} + \beta \left[\dfrac{1}{H} \right]_{-}^{+} + P_0 \left[x \right]_{-}^{+} = 0, \end{gather}

(4.20e)\begin{gather}\alpha \left[H I_2 - I_1 \right]_{-}^{+} - 2 h_0 \beta \left[I_2 - \dfrac{x}{h_0 H} \right]_{-}^{+} + \dfrac{P_0}{2} \left[x^2 \right]_{-}^{+} + M = 0, \end{gather}

(4.20f)\begin{gather}U + \varOmega \left(\dfrac{\varOmega}{U}\right)^{2/3} - Q_c = 0. \end{gather}

Here, we use $\left [f \right ]_{-}^{+}:= f(x_{+}) - f(x_{-})$ for any function $f$, and introduce the following notation

(4.21)\begin{equation} \left.\begin{gathered} I_1(x) := \int_0^{x} \dfrac{\mathrm{d}s}{H(s)}, \quad I_2(x) := \int_0^{x} \dfrac{\mathrm{d}s}{H^2(s)}, \quad I_3(x) := \int_0^{x} \dfrac{\mathrm{d}s}{H^3(s)}, \\ \alpha = 6\left(U + \varOmega \right) - 3 \beta, \quad \beta = \dfrac{3 Q_c}{h_0}, \end{gathered}\right\} \end{equation}

where the integrals are evaluated explicitly in appendix B.

We note that (4.20) comprise six equations for seven unknowns but, as noted above, we solve for six of these unknowns in terms of $x_{+}$ before closing the problem fully by interpolating the DNS results. We first note that $x_{+} = - x_{-}$ and $P_0 = 0$, so $p$ must be odd in $x$. This follows from the symmetry of $p_{x}$ in $x$ and the fact that $p$ has only two turning points, both of which follow from (4.6), in conjunction with (4.18a) and (4.18d). A typical solution for the pressure is shown in figure 10. It is helpful to note the large-$x_{+}$ asymptotic solutions for the remaining variables $\varOmega /U$, $h_0$ and $M/U$. We find

(4.22)\begin{equation} \varOmega/U \sim \dfrac{2 \sqrt{3}}{{\rm \pi} x_{+}}, \quad h_0 \sim \dfrac{3}{2} - \dfrac{3 \sqrt{3}}{{\rm \pi} x_{+}}, \quad M/U \sim \dfrac{4 {\rm \pi}}{\sqrt{3}} - \dfrac{12}{x_{+}} \quad \text{as } x_{+} \to \infty. \end{equation}

These provide accurate initial guesses to solve the system (4.20) for large $x_{+}$, and then we are able to search for a solution to (4.20) with a slightly smaller $x_{+}$ by iterating using the previously found solution as an initial guess, all the way down to small values of $x_{+}$.

Figure 10.

Typical pressure profile from (4.7), using $x_{+} = 5$ to solve (4.20) yielding the solutions: $x_{-} = -5$, $\varOmega /U = 0.282$, $h_0 = 1.33$, $M/U = 4.43$, $P_0 = 0$ and $Q_c/U = 1.12$, each correct to three significant figures.

We find that $\varOmega /U$ and $U/M$ both monotonically decrease as $x_{+}$ increases from zero to infinity (grey lines in figure 11). However, $h_0 \in (h_{{min}}, 2)$ is a non-monotonic function of $x_{+}$, where the minimum value $h_{{min}} \approx 1.327$ occurs at $x_{+} \approx 4.675$.

Figure 11.

Solutions to the nonlinear system (4.20) as functions of $x_{+}$ in terms of (a) the scaled cylinder rotation rate $\varOmega /U$, (b) the nip height $h_0$ and (c) the scaled belt velocity $U/M$. The solid grey line represents the numerical solution. The dashed black lines represent the large-$x_{+}$ asymptotic solutions (4.22).

To validate our model, we compare our theoretical results with the experimental data from Eggers et al. (Reference Eggers, Kerswell and Mullin2013) in figure 12, where both the film thickness and the cylinder radius are varied. We are able to do this by using $x_{+}$ as the parameter when we plot $U/M$ and $\varOmega /U$. We see good agreement between theory and data across a wide range of film thicknesses and cylinder radii. That is, the experimental data collapses onto the one-parameter family of solutions that we derive. The only data set that does not appear to collapse onto the theory curve is for a film thickness of 0.2 mm, one of the thinnest films tested, for which our model over-predicts the rotation rate. As noted in Eggers et al. (Reference Eggers, Kerswell and Mullin2013), we expect surface roughness to affect the results more for thinner films which could explain this small discrepancy. We also note from figure 12(a) that the largest experimentally observed value of $\varOmega /U$ is just below $0.67$, as predicted by our model, which we will discuss below.

Figure 12.

Comparison between the experimental data of Eggers et al. (Reference Eggers, Kerswell and Mullin2013) and our theoretical predictions in terms of the relationship between $\tilde {\varOmega }$ and $\tilde {U}_b$. In panel (a), we present the data in the same manner as in Eggers et al. (Reference Eggers, Kerswell and Mullin2013) (with different notation), and in panel (b), we separate $\tilde {\varOmega }$ and $\tilde {U}_b$ onto each axis. The markers correspond to the experimental data, with the legend entries corresponding to film thickness. Within each group of marker type, the cylinder radius varies.

We now use (4.19a) to determine the full solution of the system in terms of the control parameter $M$, which is determined by the experimental set-up. We do this by interpolating the DNS results from figure 8(a) onto the one-parameter family of solutions to (4.20) that we derive above. We show the route this one-parameter family of solutions takes through the DNS-determined $H(x_{+})$ in figure 13(a). We note that figure 13(a) predicts that $\varOmega /U$ can never be larger than $0.67$ (to two significant figures), which occurs at the point at which the solution branch drops below the DNS data for large ${Ca}$. Combining these DNS results with the condition (4.19a) provides a relationship between ${Ca} \,M$ and $x_{+}$ (figure 13b). In general, $x_{+}$ decreases as ${Ca} \,M$ increases, but this is bounded below by $x_{+} \approx 2.3$, which appears to be the minimal possible value of $x_{+}$ for this problem. As ${Ca} \,M$ becomes small, the value of $x_{+}$ becomes large. This corresponds to small values of $h_{+}^{b}$ and $h_{+}^{c}$ in the DNS, which means that this limit is more difficult to resolve numerically. We will discuss this limit in more detail in § 5.

Figure 13.

The closure results from DNS of the inner problem we derive in § 4.2: (a) $H$, the splitting height in Region III, where the eight different colours correspond to different values of ${Ca} \in \{ 0.2, 0.4, 0.7, 1, 2, 5, 10, 20\}$ as described in figure 8, with the one-parameter family of solutions derived in § 4.5 overlaid. Interpolating our DNS results along this solution curve and combining it with the results from § 4.5 yields (b) $x_{+}$ in terms of ${Ca} \,M$, which is specified by the experimental set-up.

The analysis above has implicitly relied on a vital constraint that must be satisfied for its validity, namely that $x_{+}$ is finite. When we plot the response diagram for the scaled belt speed as a function of $M$, we find that this function is nearly linear, as in figure 14(a) (and the same applies to the scaled rotation rate in figure 14b). In particular, we note that for the experiments in figure 3, for which $M = 2.2$, figure 14(a) predicts that $U = 1.4$ so that $\tilde {U}_b = 14\ \textrm {mm}\ \textrm {s}^{-1}$, which is reasonable agreement with the measured value of $10\ \textrm {mm}\ \textrm {s}^{-1}$. The results of figure 14 provide a solution for the dimensional belt speed $\tilde {U}_b$ and rotation rate $\tilde {\varOmega }$ for any given cylinder.

Figure 14.

The dimensionless (a) belt velocity and (b) cylinder rotation rate, as functions of the dimensionless weight per unit length of the cylinder, $M$, specified by the experimental set-up. We use the typical belt velocity $\tilde {U} = 10\ \textrm {mm}\ \textrm {s}^{-1}$; different choices will result in differently scaled axes consistent with the definitions of the dimensionless parameters.

More importantly, from figure 13(b), we see that the limit $M \to 0$ results in $x_{\pm } \to \pm \infty$. Additionally, from figure 14, the same limit results in both $U \to 0$ and $\varOmega \to 0$. In this limit of small cylinder mass our previous analysis is invalid; we now describe a scenario for the levitation of cylinders of small mass.

5.1. The limit $M \to 0$ in the lubrication regime

We now consider what happens as $x_{-} \to -\infty$ in (3.8a–c), recalling that $U = \mathit {O}(M)$ and $\varGamma = \mathit {O}(1)$ as $M \to 0$. The gravitational term becomes relevant when $y = \mathit {O}(M^{1/2} \epsilon ^{-3/4})$ and, since the film thickness is of $\mathit {O}(x^2)$, $x = \mathit {O}(M^{1/4} \epsilon ^{-3/8})$. As we require $x$ to be large, we will only consider cylinders whose mass is such that $M \gg \epsilon ^{3/2}$. Hence, we rescale

(5.1a–e)\begin{align} x = M^{1/4} \epsilon^{{-}3/8} \bar{x}, \quad y = M^{1/2} \epsilon^{{-}3/4} \bar{y}, \quad u = M \bar{u}, \quad v = M^{1/4} \epsilon^{{-}3/8} \bar{v}, \quad p = M^{1/4} \epsilon^{9/8} \bar{p}, \end{align}

which leads to a model that differs from that in the nip only in the $x$-momentum equation in (3.8a–c), which at leading order becomes

(5.2)\begin{equation} \frac{\partial \bar{p}}{\partial \bar{x}} = \frac{\partial^2 u}{\partial {\bar{y}}^2} - \varGamma \quad \text{for } 0 < \bar{y} < \dfrac{\bar{x}^2}{2}. \end{equation}

The solution must match with that in the nip as $\bar {x} \to 0$ and, since the nip solution gives that $U/M \sim \sqrt {3}/4 {\rm \pi}$ and, from (4.22) $\varOmega = \mathit {O}(M^{3/4} \epsilon ^{3/8})$ as $M \to 0$, the boundary conditions are

(5.3a)\begin{gather} \bar{u} = \dfrac{\sqrt{3}}{4 {\rm \pi}}, \quad \bar{v} = 0 \quad \text{on } \bar{y} = 0, \end{gather}

(5.3b)\begin{gather}\bar{u} = 0, \quad \bar{v} = 0 \qquad \text{on } \bar{y} = \dfrac{\bar{x}^2}{2}. \end{gather}

Hence, we find that

(5.4)\begin{equation} \bar{p} ={-} \varGamma \bar{x} - \dfrac{2 \sqrt{3}}{{\rm \pi} \bar{x}^3} + C, \end{equation}

for some constant $C$ that would be determined through matching with higher-order terms in the nip region.

We note that the effect of this longer lubrication region on the force and moment balance equations (4.18a)–(4.18c) is of $\mathit {O}(M^{-1/4} \epsilon ^{3/2})$ compared with the forces exerted in the nip and hence it can be neglected to leading order. From (5.4), we see that it is now possible for the pressure to vanish at two points $\bar {x} = \bar {x}^{*}_{-}$, $\bar {x}^{\circ }_{-}$ in $\bar {x} < 0$, as shown in figure 15. Here, we have set the pressure to be zero when $\bar {x}$ exceeds the value $\bar {x}_{+}$ because there is no experimental evidence that the upper meniscus ever moves much beyond the nip region where $x = \mathit {O}(1)$. The response diagram for $|x_{-}|$ as a function of $M$ in the extended region is as in figure 16, the physically relevant branch tending to infinity as $M \to 0$, as discussed at the end of § 4.

Figure 15.

Pressure profiles in the flooded region where $p < 0$. The solid lines show pressure profiles, for different values of $M$, and the red dashed line marks zero. The $+$ symbols mark $\bar {x}^{*}_{-}$ and the $\times$ symbols mark $\bar {x}^{\circ }_{-}$. For the grey line, these symbols mark the same point; this occurs when $\bar {p}_{\bar {x}}(\bar {x}^{*}_{-}) = 0$. In the inset we show the same pressure profiles over the entire domain, which reveal very little difference between profiles in the nip region – the major difference is their horizontal extent.

Figure 16.

Schematic response diagram for the spatial positions in $\bar {x}<0$ where the pressure vanishes. As $M$ decreases, there is a critical value at which the pressure can vanish at two upstream values. These mark the approximate start and end of the flooded region.

Although we do not determine the constant $C$ in (5.4), we note that the pressure minimum in the flooded regions in figure 15 occurs when

(5.5)\begin{equation} \bar{x} = \bar{x}_m :={-}\left(\dfrac{6 \sqrt{3}}{{\rm \pi} \varGamma} \right)^{1/4}, \end{equation}

and that $M = M^*$ in figure 16 when $\bar {p} = 0$ at this point. As $M$ decreases below $M^*$, the region under the cylinder floods more and more fluid down the belt and this suggests a possible explanation for the tongues observed in figure 2. In the next section, we briefly recapitulate a simple two-dimensional theory for such tongues.

5.2. Tongue modelling

A simple model for a tongue when levitating a block has been described in Mullin et al. (Reference Mullin, Ockendon and Ockendon2020). By including gravity effects into the lubrication equations, and allowing for recirculating flow, the uniform tongue thickness, $\tilde{H}_t$, including the fluid initially on the belt, satisfies the dimensional cubic equation

(5.6)\begin{equation} \tilde{U}_b \tilde{H}_t - \dfrac{\tilde{\rho} \tilde{g}}{3 \tilde{\mu}} \tilde{H}_t^3 = \tilde{Q}, \end{equation}

where $\tilde {Q}$ is the dimensional net mass flux; for a steady tongue, $\tilde {Q}$ is the mass flux past the scraper.

Following § 5.1, we scale the tongue thickness by writing $\tilde {H}_t = \epsilon ^{1/4} M^{1/2} \tilde {a} h_t$, so that (5.6) becomes

(5.7)\begin{equation} \dfrac{\sqrt{3}}{4 {\rm \pi}} h_t - \dfrac{\varGamma}{3} h_t^3 = \mathit{O}(\epsilon^{3/4}), \end{equation}

and

(5.8)\begin{equation} h_t = \left(\dfrac{3 \sqrt{3}}{4 {\rm \pi}\varGamma}\right)^{1/2}, \end{equation}

to lowest order. This suggests that, if a tongue exists, it will terminate at the cylinder near the point

(5.9)\begin{equation} \bar{x} = \bar{x}_t ={-}\left(\dfrac{12 \sqrt{3}}{4 {\rm \pi}\varGamma}\right)^{1/4}, \end{equation}

which is where the upper surface of the tongue intersects the cylinder.

5.3. Matching the nip and tongue regions

The discussion of § 5.1 suggests that, as $M$ decreases, the flooded region below the nip will extend down to the increasingly remote point $\bar {x}^{\circ }_{-}$. In the vicinity of this point, we expect a local region to occur in which a fully two-dimensional Stokes flow model is included, as in Region IV. This region will join the lubrication region in $\bar {x}^{\circ }_{-} < \bar {x}$, with the coating flow coming around the cylinder and the incoming film on the belt. However, to confirm this would require an analysis similar to that in Region IV, which we will not pursue here. A further two-dimensional model would be needed when $\bar {x}^{\circ }_{-}$ reaches $\bar {x}_t$, which is larger in modulus than the minimum pressure point given by (5.5). Then, a tongue will form and we expect a configuration as in figure 17 where the transverse length scale is $\mathit {O}(\tilde {a} \epsilon ^{1/4} M^{1/2})$.

Figure 17.

Schematic of the putative inlet coalescence region when a tongue is present.

It is reassuring that, for all values of $M$, the dominant levitation forces still occur in the nip, so that our prediction that there is always a unique belt velocity for any given $M$ still holds. This is in contrast to the levitation of blocks (Mullin et al. Reference Mullin, Ockendon and Ockendon2020) where, although the force and moment balances could be satisfied without a tongue for a unique belt velocity, a continuous range of belt velocities were possible in the presence of a slowly evolving tongue.

We remark that when the cylinder is made lighter by reducing its radius rather than its density, an even more complicated asymptotic analysis becomes necessary. As mentioned in the Introduction, experimental evidence suggests that a downward propagating tongue is more likely to occur for smaller radii cylinders.

We summarise this section by listing our main predictions in the flooding regime when $M \ll 1$:

1. (i) The zero-gravity theory in § 4 holds for $M = \mathit {O}(1)$ and, to lowest order, it predicts an odd pressure distribution that vanishes at $x = x_{-}, x_{+}$, as in figure 10 and figure 15 (grey line in inset). However, as $M \to 0$, $|x_{\pm }| \to \infty$ and, as shown in § 5.1, gravitational effects become comparable with viscous effects when the dimensional nip extent in the $x$-direction is of $\mathit {O}(M^{1/4} \epsilon ^{1/8} \tilde {a})$. As long as $\epsilon ^{3/2} \ll M \ll 1$, this exceeds the nip length of $\mathit {O}(\epsilon ^{1/2} \tilde {a})$ in Region III: the regime in which $M = \mathit {O}(\epsilon ^{3/2})$ requires further study.

2. (ii) The incorporation of gravity means that the nip pressure distribution is no longer an odd function of $x$ as it was in figure 10 and figure 15 (grey line in inset). This allows flooding, in which the nip region extends downwards to the green crosses marked in figure 15. There is then an extended nip region in which gravity is included but the dominant lubrication forces still occur in the original nip region of length $\mathit {O}(\epsilon ^{1/2} \tilde {a})$. Hence, the prediction that the dimensionless belt velocity is $U = \sqrt {3} M / (4 {\rm \pi}) \ll 1$ remains valid to lowest order, and the corresponding dimensional value is $\tilde {U}_b \sim (3 \tilde {h} \tilde {a}^3)^{1/2} \tilde {\rho }_s \tilde {g}/4 \tilde {\mu }$.

3. (iii) A more precise estimate of the dependence of the length of the flooded region on $M$ can be obtained from the dependence of $x_{\pm }$ on $M$. Since scaling the velocity and pressure in Region IV with $U \ll 1$ in § 4.2 leads to the same problem except with an effective capillary number of $U \,{Ca}$, the results of appendix A allow us to deduce that $M = \mathit {O}(U) = \mathit {O}((h_{+}^{b})^{3/2}) = \mathit {O}(|x_{-}|^{-3})$ in the limit of small $M$, which is consistent with figure 13(b). Thus, $M$ is correlated to $\epsilon$ by $M \sim M^{-3/4} \epsilon ^{9/8}$, i.e. $M \sim \epsilon ^{9/14}$. This gives the dimensional extent of the flooded region to be of $\mathit {O}(\epsilon ^{2/7} \tilde {a})$, and the dimensional angular velocity to be of $\tilde {\varOmega } = \mathit {O}(\epsilon ^{3/14} \tilde {U}_b/ \tilde {a})$.

4. (iv) All the above comments apply when ${Ca}$ is $\mathit {O}(1)$, but if the surface tension is sufficiently small so that ${Ca} \,M$ is large, then $x_{-}$ will remain finite and no flooding will occur.