2.1 Experimental procedure

In all experiments the apparatus was run for an initial period of half an hour to establish a uniform film of viscous fluid on the belt. The thickness of the film was set using the knife edge. In earlier experiments with cylinders and spheres, achieving levitation was very straightforward in practice. This involved placing each levitating object on the moving layer and adjusting the speed of the belt to achieve balance. In these cases it was found that when the belt speed matched the fall speed of the sphere or cylinder, a balance was achieved. If the belt speed was too high, the levitating object moved up the belt, too low and it travelled downwards.

Levitating a rectangular block on the layer was considerably more challenging. The blocks were accurately machined pieces made primarily from plastic although a few results were obtained using metal cubes. If the block was simply placed on the belt it typically rotated about a horizontal axis perpendicular to the belt. However, if the block was held in tweezers until a small tongue of fluid developed, a steady levitated state could be achieved. If the belt speed was sufficiently high, the fluid tongue did not grow significantly. Determining the correct speed was an iterative procedure, so that obtaining a steady balanced state could take several hours. However, with due diligence we were able to levitate blocks, which ranged in shape from cubes to rectangular parallelepipeds, for many hours. It was initially difficult to obtain systematic results. A sample set of images of four different blocks levitated on a 0.2 mm layer is given in figure 3. In each case the tongue was very small and did not grow. However, the refined technique described in the following paragraph enabled the collection of reliable results for the growth of the tongue at lower belt speeds.

A systematic approach was developed which involved holding the block in a pair of tweezers with the lower face of the block immersed in the fluid layer as before. This produced a short tongue of fluid as in the left image in figure 4(b). This tongue then grew in length as in the next two images in figure 4. Once the tongue was established such that it was as long as the block, the block was released from the tweezers and this resulted in a levitating block which remained in situ for at least an hour. A practical limitation was that when the tongue reached the knife edge the incoming flow conditions changed. A further set of experiments was performed in which the speed of the belt was adjusted to investigate the growth rate of the tongue as a function of belt speed.

The tongue propagated slowly down the belt and typically took thirty minutes to an hour to reach the fluid bath. The speed of propagation of the tongue was estimated from still images which were captured every 30 s using a Nikon D400 SLR camera. Data used to estimate the tongue propagation speed were only taken in the period before the tongue reached the bath. A typical set of results for the tongue propagation are shown in figure 4(a) and three illustrative examples of the tongue development are shown in figure 4(b). In this case a $20\times 10\times 10~\text{mm}$ cuboid was levitated on a 0.5 mm film and the belt speed was set at $5.5~\text{mm}~\text{s}^{-1}$. It can be seen that a straight line fits the data in figure 4(a) reasonably well. There is a clear departure from this for a short period and this arose when the block moved a small distance up the belt.

An important observation was that, in situations where small bubbles appeared on the free surface of the tongue, the bubbles always moved down from the block to the tip of the tongue.

3.1 Flow between the block and the belt

Lubrication theory can be applied to the fluid flowing between the belt $y=0$ and the block $y=H(x)=H_{0}(1+(mx/L))$ where $H_{0}$ and $m$ are to be determined. We assume that $H$ is comparable to the aperture at the knife-edge $H_{k}$, and $H_{k}/L=\unicode[STIX]{x1D6FF}$ is small, where $L$ is the length of the block in the direction of motion of the belt. The reduced Reynolds number is of $O(10^{-4})$ so that, to lowest order in $\unicode[STIX]{x1D6FF}$, the lubrication equations (Ockendon & Ockendon Reference Ockendon and Ockendon1995) for $p$, the pressure relative to atmospheric, and the velocity $(u,v)$ are

(3.1)$$\begin{eqnarray}\displaystyle & \displaystyle 0=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}-\unicode[STIX]{x1D70C}g+\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}y^{2}}, & \displaystyle\end{eqnarray}$$

(3.2)$$\begin{eqnarray}\displaystyle & \displaystyle 0=\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}, & \displaystyle\end{eqnarray}$$

and

(3.3)$$\begin{eqnarray}0=\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y},\end{eqnarray}$$

with

(3.4)$$\begin{eqnarray}u=U,v=0\quad \text{on }y=0\end{eqnarray}$$

and

(3.5)$$\begin{eqnarray}u=v=0\quad \text{on }y=H(x).\end{eqnarray}$$

We assume that $p$ at the top corner of the block is zero to lowest order. This assumption will be discussed later when we will see that the pressures at $x=0$ and $x=L$ are both small, but the former is an order of magnitude greater than the latter. Hence, without loss of generality,

(3.6)$$\begin{eqnarray}p=0\quad \text{at }x=L.\end{eqnarray}$$

Equations (3.1)–(3.5) give that

(3.7)$$\begin{eqnarray}u=\frac{1}{2\unicode[STIX]{x1D707}}\left(\frac{\text{d}p}{\text{d}x}+\unicode[STIX]{x1D70C}g\right)y(y-H)+U\left(1-\frac{y}{H}\right),\end{eqnarray}$$

and $p$ satisfies

(3.8)$$\begin{eqnarray}-\frac{1}{12\unicode[STIX]{x1D707}}\left(\frac{\text{d}p}{\text{d}x}+\unicode[STIX]{x1D70C}g\right)H^{3}+\frac{1}{2}UH=Q_{b},\end{eqnarray}$$

where $Q_{b}$ is the constant flux per unit width between the block and the belt. Hence, from (3.6), (3.8),

(3.9)$$\begin{eqnarray}p=(L-x)\left[\unicode[STIX]{x1D70C}g+\frac{6Q_{b}\unicode[STIX]{x1D707}\left(2+m\left(1+{\displaystyle \frac{x}{L}}\right)\right)}{H_{0}^{3}\left(1+{\displaystyle \frac{mx}{L}}\right)^{2}(1+m)^{2}}-\frac{6\unicode[STIX]{x1D707}U}{H_{0}^{2}\left(1+{\displaystyle \frac{mx}{L}}\right)(1+m)}\right].\end{eqnarray}$$

The equations for the equilibrium of the block are, to first order in $\unicode[STIX]{x1D6FF}$

(3.10)$$\begin{eqnarray}\displaystyle & \displaystyle \text{horizontal force balance: }\int _{0}^{L}p\,\text{d}x=0, & \displaystyle\end{eqnarray}$$

(3.11)$$\begin{eqnarray}\displaystyle & \displaystyle \text{vertical force balance: }W\int _{0}^{L}\left(p\frac{H_{0}m}{L}+\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right)_{y=H}\,\text{d}x=-Mg, & \displaystyle\end{eqnarray}$$

(3.12)$$\begin{eqnarray}\displaystyle & \displaystyle \text{torque balance: }W\int _{0}^{L}xp\,\text{d}x=-{\textstyle \frac{1}{2}}MgD, & \displaystyle\end{eqnarray}$$

where $D$ is the width of the block as shown in figure 5 and $W$ is its width in the third dimension. Thus we have three transcendental equations for $Q_{b}$, $H_{0}$ and $m$. When we neglect the right-hand side of (3.12) and the gravity term in (3.1), which is reasonable for the block sizes considered here, algebraic manipulations reveal that the only real solution is

(3.13a-d)$$\begin{eqnarray}p=0,\quad m=0,\quad Q_{b}=\frac{\unicode[STIX]{x1D707}U^{2}LW}{2Mg},\quad H_{0}=\frac{\unicode[STIX]{x1D707}ULW}{Mg}.\end{eqnarray}$$

Thus, to lowest order in $\unicode[STIX]{x1D6FF}$, the pressure is zero and the block is parallel to the wall in figure 5. Moreover, the flow under the block is a simple shear.

In standard lubrication theory the pressure is $O(\unicode[STIX]{x1D707}U/L\unicode[STIX]{x1D6FF}^{2})$ and the inclination of the block is $O(\unicode[STIX]{x1D6FF})$ but here the right-hand side of (3.12) means that both of these estimates are multiplied by $\unicode[STIX]{x1D6FF}$. This can be seen by reference to the dimensionless form of the problem in appendix A, which also contains the corresponding lowest-order correction to (3.13). The calculation reveals that the pressure at $x=0$ takes the positive value $(6MgD)/(WL^{2})$, the implications of which will be discussed in § 3.3.

Figure 6.

Relationship between $Q$ and $H$ for the selected value of $U=4.0~\text{mm}~\text{s}^{-1}$ using (3.16). The heavy curve is for $UH>3Q$ and indicates the range of $H$ over which recirculating flow exists.

Figure 7.

Plot of $Q_{b}$ and $Q_{k}$ against $U$; $U^{\ast }$, the value of $U$ at which $Q_{b}=Q_{k}$, is indicated.

3.2 Flow between the block and the knife

The lubrication model (3.1)–(3.4) also applies to the region with a free surface below the block but (3.5) is replaced by

(3.14a,b)$$\begin{eqnarray}p=0,\quad \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}=0\quad \text{on }y=H.\end{eqnarray}$$

Hence, $p\equiv 0$, $H=\text{const.}$ and

(3.15)$$\begin{eqnarray}u=U+\frac{\unicode[STIX]{x1D70C}g}{2\unicode[STIX]{x1D707}}y(y-2H),\end{eqnarray}$$

so that the constant flux $Q$ satisfies

(3.16)$$\begin{eqnarray}Q=UH-\frac{\unicode[STIX]{x1D70C}g}{3\unicode[STIX]{x1D707}}H^{3}.\end{eqnarray}$$

Assuming that (3.15)–(3.16) apply all the way down to the knife edge, $Q_{k}$, the flux from the bath, is given by setting $H=H_{k}$ in (3.16). From figure 7 we see that there are two values of $U$ for which $Q_{b}=Q_{k}$, one of which is very small. We define

(3.17)$$\begin{eqnarray}U^{\ast }=\frac{Mg}{\unicode[STIX]{x1D707}LW}\left(H_{k}+\sqrt{H_{k}^{2}-\frac{2\unicode[STIX]{x1D70C}LWH_{k}^{3}}{3M}}\right)\end{eqnarray}$$

as the larger of these two values. We note that this value is within ∼5 % agreement for the blocks shown in figures 2–3. However, we have seen in § 2 that the block can remain nearly stationary for long periods of time over a range of values of $U$ and the explanation for this depends crucially on a study of the tongue observed in some of these images.

3.3 Flow with a tongue beneath the block

Now we suppose that a tongue, as in figure 8, is generated immediately beneath the block and has constant thickness $H_{t}$. We also assume that the tongue terminates at a well-defined tip that is above the knife. The equations (3.15), (3.16) still hold and thus, in a steady flow, it is now the flux in the tongue that must be equated to the flux at the block so that

(3.18)$$\begin{eqnarray}Q_{b}=UH_{t}-\frac{\unicode[STIX]{x1D70C}gH_{t}^{3}}{3\unicode[STIX]{x1D707}};\end{eqnarray}$$

this gives two positive values for $H_{t}$ when $Q_{b}$ is prescribed, as shown in figure 6. When we consider the corresponding velocity profile (3.15), we find that it is only when $H_{t}$ takes the larger of these values that the flow can recirculate, with fluid moving upwards near the belt but downwards near the film surface in accordance with the observation made at the end of § 2.

We also note that the response diagram for $H_{t}(U)$, obtained using (3.13c) and (3.18), is

(3.19)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D707}U^{2}LW}{2Mg}=UH_{t}-\frac{\unicode[STIX]{x1D70C}gH_{t}^{3}}{3\unicode[STIX]{x1D707}},\end{eqnarray}$$

which limits the permissible values of $U$ as shown in figure 9. The lower branch gives values of $H_{t}$ that are considerably smaller than those observed in § 2. We therefore conjecture that the upper branch indicated in bold in figure 9(a) is the one that is physically relevant in the tongue.

Figure 8.

Schematic diagram illustrating the tongue which forms below the block.

Figure 9.

(a) Plot of $H_{t}$ versus $U$ evaluated using (3.19) with respective physical parameter values. (b) An expanded view over the parameter range which is relevant to the experiments. The location of $U^{\ast }$ for the film thickness $H_{k}=0.5~\text{mm}$ is indicated.

A comprehensive model for a recirculating tongue will require a more detailed analysis of the flow near $x=0$ where the lubrication pressure is of $O(\unicode[STIX]{x1D707}U/\unicode[STIX]{x1D6FF}L)$. The velocity in such a region would be much smaller than $U$ away from the belt and thus could reduce the local capillary number, $U\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70E}$ to be of $O(1)$, allowing a bulge to form in which surface tension and viscous terms are in balance. In such a region the pressure would have to be of $O(\unicode[STIX]{x1D70E}K)$, where $K$ is the curvature of the bulge, and $K$ would thus be of $O[(L\unicode[STIX]{x1D6FF})^{-1}Ca]$. However the local flow would lead to a free boundary problem for a fully two-dimensional Stokes flow which we will not attempt here. A similar local analysis would also be needed to reveal the flow details near the tongue tip.

This scenario changes dramatically when we allow the tongue tip to have a non-zero upward velocity $V$ while still assuming the block to be stationary. Then $Q_{b}$ is no longer equal to $Q_{k}$ and conservation of mass at the tip requires that

(3.20)$$\begin{eqnarray}V=\frac{Q_{k}-Q_{b}}{H_{k}-H_{t}},\end{eqnarray}$$

which is negative if $H_{t}>H_{k}$, $Q_{b}<Q_{k}$ and, as seen from figure 7, this occurs when $U<U^{\ast }$. Upward-moving tongues are theoretically possible if $U>U^{\ast }$ but experiments reveal that they always lead to block instability over short time scales.

The situation changes yet again when the tongue tip reaches the knife so that $Q_{k}$ is no longer a control variable. The tongue thickness then remains constant throughout at the larger root of (3.18) and, as mentioned in § 2, this configuration is observed to be stable over time scales of hours.

We show a quantitative comparison between theoretical and experimental results for the propagation speed of the tongue as a function of the belt speed in figure 10. The relationship was calculated using (3.20) and the experimental estimates of the tongue speed were made by analysing sets of image sequences as described in the experimental section with reference to figure 4. The slopes of the time history plots were found to be linear although the block tended to tilt to the left or right at higher belt speeds and on occasions the motion was weakly time dependent. When $(U^{\ast }-U)$ is small, the accord between the experimental results and the predictions of the two-dimensional was very satisfactory considering the three-dimensional effects at higher speeds in the experiment. However, it becomes more and more difficult to estimate $V$ as $(U^{\ast }-U)$ increases.

In summary, our simple two-dimensional model suggests that there is a range of values of $U$ for which the block can remain stationary with $Q_{k}>Q_{b}$ as long as there is a tongue whose downward velocity is of the order of a few mm/min. On the other hand, for small values of $U^{\ast }-U$, the time scale for the tip motion is long compared with the transit time of the film through the device. We conjecture that the deviation between theory and observations as $U^{\ast }-U$ increases happens because a travelling wave is not attained before the tip reaches the knife edge.

Figure 10.

Plot of tongue speed versus belt speed for $20\times 10\times 10~\text{mm}$ cuboid on a 0.5 mm film. The solid line has been calculated using (3.19), (3.20).