A squeezed tube is held open by its own fluid

The pump that is better the more it squeezes wrote down the energy account of a peristaltic pump — a channel whose walls carry a travelling wave — and found a closed form for its best efficiency at every amplitude. The efficiency rose as the wave closed the channel, towards one at complete closure, where the pump stops leaking and becomes a travelling piston. The wave in that essay was prescribed: the wall went where it was told, whatever the fluid did.

Its last section named the pumps that do not work that way. A roller pump presses a flexible tube against a track; a ureter or a gut contracts a ring of muscle round a soft tube; neither dictates the tube’s shape. They apply a squeeze, and the fluid inside pushes back. The shape near the narrowest point is decided by the balance between the squeeze and the fluid’s pressure, and the essay asked whether a pump driven by a stated squeeze ever closes at all — or whether the fluid holds the gap open by an amount that sets a ceiling on the efficiency.

A squeeze and a spring #

The channel is the earlier one: two-dimensional, its half-width a small fraction of the wavelength, the fluid slow enough that inertia plays no part, so the velocity across it is parabolic at every station and the pressure gradient along it is fixed by the local width and the flux. In the frame moving with the wave the flow is steady and the walls are streamlines, which is continuity’s whole contribution: the lab-frame flow is the wave-frame flux plus the volume the moving wall shape sweeps past.

The wall is now a spring. Where it is pressed with nothing inside, it moves inward by the squeeze’s own displacement, $S\,g(\theta)$ — a bump peaked at the middle of the wavelength, whose height $S$ is measured in relaxed half-widths, so that a squeeze of one would just close an empty tube and a squeeze of 1.3 would push the wall three-tenths of a half-width past the centreline. The fluid’s pressure pushes it back out, by the pressure times a compliance $\kappa$:

$$h(\theta) = 1 - S\,g(\theta) + \kappa\,\tilde p(\theta).$$

The compliance is measured in the lubrication pressure unit, $\mu c\lambda/a^2$, so a faster wave, a more viscous fluid or a softer wall all make it larger. The pressure $\tilde p$ the wall answers is the local swing — the fluid’s pressure less the steady rise the pump sustains along its length, which the tube’s mean state takes up slowly. The lubrication equation then ties the pressure to the width, the width depends on the pressure, and the flux and the pressure level are found together by shooting: integrate the equation once round the wavelength and adjust both until the pressure swing returns to where it started and averages to zero.

Three checks come first. With the compliance set to zero and a sinusoidal wall, the solver must return the earlier essay’s closed-form flow and wall power, and it does to two parts in $10^7$. The lab-frame flux computed at a fixed station matches the wave-frame flux plus the mean width to rounding. And halving the integration step moves the narrowest gap and the flow by less than $10^{-7}$.

Past closing, and still open #

The first figure is the answer to the question. The dashed curve is the squeeze alone: it would push the wall 0.3 of a half-width past the centreline over a sixth of the wavelength. With fluid in the tube the wall does not get there. Ahead of the narrowest point the pressure rises sharply — fluid is being driven into a narrowing gap faster than it can escape — and that pressure pushes the wall back. At the stiffest wall drawn, a compliance of 0.003, the gap stays open at 0.013 of the half-width; at 0.03, at 0.045; at 0.1, at 0.089. The narrow region also lengthens and flattens, like a tyre pressed on a road, because along it the pressure and the squeeze very nearly balance.

The pressure that does it is largest at the narrowest point itself, as it is in a squeeze film under a load, and its size there is fixed by the wall’s own arithmetic: the spring must push the wall back from where the squeeze would put it, $1 - S$, to the gap, so the compliance times the pressure swing at the throat is exactly $S - 1 + h_{\min}$. At a compliance of 0.01 that is a swing of 33 units above the mean; at 0.003, 105. Just behind the throat, on the side the squeeze has left, the pressure falls instead — 11 and 25 units below the mean — and the wall there is sucked inward by a tenth of a half-width. In a real tube at a low mean pressure that is where the wall would collapse on its own or the liquid would tear and still pull, and neither is in a model whose pressure can go as far below the mean as the flow asks.

This is soft lubrication — the same physics as the shape of a gap in a bearing, and as a ball that bounces in water and not in oil, whose rebound is decided by how far the film’s pressure deforms the surfaces before they touch, where a converging film carries a load by its pressure, with the gap now chosen by the load rather than by the machinist. The fluid in the throat is a bearing between two surfaces the squeeze is pressing together, and a bearing’s film does not close.

The gap never closes #

The second figure follows the gap as the squeeze grows. While the squeeze is gentle the fluid hardly resists, and the narrowest gap is the squeeze’s own: one minus the squeeze. Past a squeeze of about 0.8 the curves bend away from that line and level off. At a squeeze of 1.5 — half as far again as closing an empty tube — the gap is 0.012 of the half-width at the stiffest wall and 0.074 at the softest, and it is still falling only slowly. No finite squeeze closes the channel. As the squeeze grows the pressure at the throat grows with it, and the pressure a narrowing film can generate grows without limit as the film thins, so there is always a gap at which the two balance.

That removes the premise of the earlier essay’s best case. There the efficiency approached one because the channel approached closure and the leak back through the throat approached zero. Here the channel cannot approach closure except by making the wall infinitely stiff, and the question becomes how the held gap depends on the wall.

The square root of the wall’s give #

The third figure answers it. Across three decades of compliance the held gap grows as a power of it, the 0.54 power at a squeeze of 1.2 and the 0.53 power at 1.5 — close to a square root. A wall four times as stiff, or a wave four times as slow, holds about half the gap.

The square root has a simple reading. The throat is a lubricating film of thickness $h$ over a contact whose length is set by where the wall’s spring and the squeeze balance; the pressure a film carries rises as the inverse square of its thickness at a fixed flux, and the width over which that pressure acts rises as the compliance lets the wall flatten. The two together make the held gap depend on the compliance and the squeeze in a combination whose exponent the calculation measures rather than assumes — and the measured value near one half is the same kind of number that soft elastohydrodynamic lubrication, of tyres on wet roads and cartilage in joints, is known for.

Squeezing harder pumps less #

The fourth figure is the first practical consequence, and it runs against intuition. The flow pumped against no pressure — the flux that mass having nowhere to go fixes as the wave-frame flux plus the volume the moving wall sweeps past — rises with the squeeze while the tube is open, as the earlier essays found for a prescribed wave. It peaks at a squeeze of about 1.2 — beyond nominal closure — and then falls: at the stiffest wall, 0.655 of the wave speed times the half-width at a squeeze of 1.2, and 0.577 at 1.5.

A harder squeeze does not close the throat further; it lengthens the flattened region, pressing a longer stretch of the tube against the held gap. The fluid that a peristaltic wave carries is the bolus between squeezes, travelling with the wave, and a longer flattened region leaves a shorter bolus. Past the peak, the extra squeeze costs flow and buys nothing.

For a roller pump this is the familiar operating advice — set the occlusion “just past closed”, because over-occluding shortens the tube’s life and does not raise the flow — and here it has a mechanism and a number.

A displacement pump until the throat is blown open #

The fifth figure pumps against a pressure. The flow hardly falls as the pressure rises — at a compliance of 0.01 it is 0.616 against no pressure and 0.608 against a pressure rise of 40 units per wavelength — because the held gap is too narrow for much to leak back through it. That is how a positive-displacement pump behaves, and it is why roller pumps are used to meter fluids: their delivery barely depends on what they pump against. On the map that sorts pumps by specific speed, displacement pumps sit at the far low end, where no rotodynamic machine can deliver a small flow against a large pressure, and a squeezed tube is the simplest of them.

Then the curve falls off a cliff. At a pressure roughly the inverse of the compliance — between 250 and 630 units for a compliance of 0.003, between 100 and 250 for 0.01, between 40 and 100 for 0.03 — the pressure behind the throat is enough to push the wall there outward by as much as the squeeze pushes it in. The throat is blown open, the gap widens several-fold, and within a factor of two or three in pressure the flow has collapsed and reversed. The pump’s pressure limit is set not by its fluid mechanics but by its wall: a stiffer wall, or a slower wave, holds a higher pressure in proportion.

The held gap keeps the efficiency short of one #

The sixth figure is the answer to the essay’s last question. The efficiency rises with the pressure pumped against, as a displacement pump’s does, since the useful power grows with the pressure while the flow hardly changes. It peaks just before the throat blows open, at 0.959 for a compliance of 0.003, 0.904 for 0.01 and 0.807 for 0.03, and then falls away with the flow.

The prescribed-shape pump of the earlier essay approached one as its wave closed the channel. A squeezed pump cannot close its channel, and the held gap costs it twice. Some flow leaks back through it: at the best operating point the flow is 0.016 short of free pumping for the stiffest wall and 0.063 short for a wall ten times as compliant. And the film in the throat, sheared between a wall moving at the wave’s speed and fluid nearly at rest, dissipates far more than its share of the length — the price of a gradient is paid as its square, and the gradients are steepest where the gap is narrowest: the stretch where the gap is under three times its narrowest occupies a fifth of the wavelength and dissipates between a third and two-fifths of everything the squeeze puts in. So the ceiling the earlier essay suspected exists, and it is set by the wall. Its shortfall from one grows faster than the compliance’s square root — from 0.04 to 0.10 to 0.19 as the compliance goes from 0.003 to 0.01 to 0.03 — and a wall stiff enough to make the pump nearly perfect is also one that the fluid can barely hold open, so its throat is the most easily blown.

One more check belongs here, because the energy account is where a subtle error nearly got through. The squeeze’s work on the fluid has to be computed from the travelling part of the pressure alone: in the laboratory the pump’s steady rise is fixed along the tube and does no net work on a wall that returns to where it started each period, and counting it gives a total that depends on where the wavelength is started. Computed from the travelling part, the work less the useful power equals the viscous dissipation integrated from the parabolic profiles to five parts in a million at four operating points, rigid and squeezed.

What was checked #

The ledger holds four checks and two results. A rigid sinusoidal wall reproduces the earlier essay’s closed-form flow and wall power to $2\times10^{-7}$. The energy account closes to $5\times10^{-6}$. The flux measured at a fixed station equals the wave-frame flux plus the mean width, which is continuity. And halving the step moves the held gap and the flow by less than $10^{-7}$.

What the long-wave spring leaves out #

The wall is a bed of independent springs. A real tube wall bends and stretches, so its response at one station depends on its neighbours’, which smooths the narrow region and changes the exponent near one half. A membrane or a thick elastic tube would give a different number, and the square root here is a property of a Winkler wall.

Two dimensions. A tube is round, and squeezed between rollers it flattens into a slot rather than narrowing uniformly; the lubrication in a flattened tube’s edges is not in a two-dimensional channel.

Contact. The model’s gap never closes, and a real tube’s walls may touch through a roughness or a boundary layer of mucus; the held gap here is the limit set by the fluid alone.

The mean pressure. The wall answers the local pressure swing and not the slow rise along the pump. A long tube pumping against a large pressure does inflate progressively along its length, which this model assigns to the tube’s mean state.

The convention: the lubrication pressure unit #

Lengths across the channel are in the relaxed half-width $a$, positions along it in wavelengths, velocities in the wave speed $c$, pressures in $\mu c\lambda/a^2$. The squeeze $S$ is the inward displacement an empty tube’s wall would suffer at the squeeze’s peak, in half-widths; the compliance $\kappa$ is the wall’s outward displacement per unit of that pressure. Flows are time-mean lab-frame fluxes per half-channel in units of $ca$; efficiency is useful power, flow times pressure rise, over the work the wall does on the fluid.

Who worked it out #

Peristaltic pumping with a prescribed wall wave was set out by Shapiro, Jaffrin and Weinberg in 1969. Soft lubrication — films between surfaces that deform under their own pressure — was developed for elastomeric seals and biological joints from the 1960s onward, by Dowson and Higginson among others, and the travelling-contact problem of a roller squeezing a compliant tube has been studied for blood pumps and for swallowing since the 1980s; the square root of the compliance appears in soft-contact scalings for a spring-backed layer.

Still open: a gut that squeezes with a wave of force, not a wave of place #

The squeeze here travels at a fixed speed with a fixed shape, as a roller’s does. A gut’s muscle contracts in a wave of force, and where the fluid pushes back the contraction can stall or be delayed, so the wave’s shape and even its speed are part of the solution. The next calculation lets the squeeze’s strength rise and fall over a stated time at each station as the wave passes, rather than holding a fixed profile, and asks whether the pressure the fluid builds ahead of the contraction slows the wave, whether a gut propelling a viscous bolus has a speed at which its efficiency is largest, and whether that speed lies near the speeds at which contraction waves are measured to travel along the small intestine.