The liquid that climbs the rod

Put a rod in a beaker of water and spin it. The water is flung outward, and the free surface dips at the axis and rises at the wall. Put the same rod in a solution of a little polymer in the same water and spin it at the same rate, and the liquid climbs the rod.

The concentration that does this is small — a tenth of a per cent of a long-chain polymer is enough — and the viscosity is barely changed. So the effect cannot be a matter of degree, and it is not.

What shear does to a Newtonian liquid, and what it does not #

Shear a liquid: slide one plane over another with fluid between. Momentum is carried sideways and the result is a stress on the planes, in the direction of the sliding, proportional to the rate.

That is the entire content of Newtonian behaviour, and it is worth saying what it excludes. In such a fluid the three normal stresses — the pushes along the flow direction, along the gradient direction, and along the third direction — are all equal to minus the pressure. There is no tension along the streamlines. There is no difference between the push along the flow and the push across it.

No value of the viscosity changes this. Glycerol is a thousand times as viscous as water and produces exactly the same kind of stress, a thousand times larger. Spin a rod in glycerol and it dips.

The two stresses, and their two slopes #

In a polymer solution the same experiment produces stresses of a second kind. Sheared in the flow direction, the fluid develops a tension along the streamlines that exceeds the push across them.

The measured quantity is the first normal stress difference, the excess of the stress along the flow over the stress along the gradient. Two facts about it decide everything downstream.

It grows as the square of the shear rate. The fitted slope on the figure is $1.994$ against the shear stress’s $0.997$. A difference of one power means the normal stress is negligible at low rates and dominant at high ones, with a crossover rather than a competition at every rate.

It is identically zero for a Newtonian liquid. Not small — zero, at every rate, for every viscosity. That is why the effect cannot be reached by thickening water.

The mechanism, in one sentence: the polymer molecules are long chains that a shear flow stretches and orients along the streamlines, and a stretched chain pulls its ends together. Summed over the chains, that is a tension along the flow direction, and the amount of stretching goes as the rate while the tension goes as the stretching, which is where the square comes from. The restoring force is entropic rather than energetic — a coiled chain has vastly more configurations than a stretched one — which is why the elastic modulus of these materials rises with temperature instead of falling.

There is a second normal stress difference too, the excess of the gradient-direction stress over the neutral-direction one. It is small — about a tenth of the first — and negative, and its being small rather than zero turns out to matter later.

One number for which stress is in charge #

Two stresses with different powers of the rate means there is a rate at which they are equal, and that rate is the reciprocal of a time.

The time belongs to the fluid: it is how long a stretched chain takes to relax back. The Weissenberg number is that time multiplied by the shear rate, and it compares how fast the flow is deforming the polymer with how fast the polymer can recover.

Below one, the fluid is a viscous liquid with an elastic correction. Above one, it is an elastic material with a viscous correction, and every intuition trained on water is a liability.

This is a different comparison from the one the Reynolds number makes, and the difference is worth being clear about. A Reynolds number weighs inertia against viscosity, both of which appear in the equation of motion. A Weissenberg number weighs two stresses the material itself produces, and says nothing about inertia at all. Rod climbing happens at Reynolds numbers of a thousandth, where inertia is nearly absent — which is exactly why so little of it is needed to be beaten.

It is also a different comparison from the Deborah number, which weighs the relaxation time against the duration of the observation rather than against the rate of the flow. The two coincide in steady shear and separate in anything that starts and stops.

Why a tension along a curved streamline lifts #

The streamlines around a rotating rod are circles. A tension along a circle pulls inward, in the way the tension in a stretched rubber band round a parcel does, and the fluid it pulls inward has nowhere to go but up.

That is the picture, and it can be made a calculation. In creeping flow the velocity around a rotating cylinder in a large bath is exactly $\Omega R^2/r$, so the shear rate falls as $1/r^2$ and the normal stress, going as the square of the rate, falls as $1/r^4$. Integrating the radial force balance outward from the rod, with the surface at atmospheric pressure, gives

$$h(r) - h_\infty = \frac{\Omega^2R^4}{\rho g}\left[\frac{\Psi_1 + 4\Psi_2}{r^4} - \frac{\rho}{2r^2}\right].$$

Two terms, falling off at different rates. The first is the hoop tension, which lifts. The second is the centrifugal term, which is what water does on its own and which lowers. And because the two have different powers of $r$, which one wins depends on where the fluid is — that is, on how thick the rod is.

A number puts the size of the effect in perspective. At one revolution a second on a ten-millimetre rod the shear rate at the rod’s surface is about thirteen reciprocal seconds, which for a fluid with a one-second relaxation time is a Weissenberg number of thirteen: deep in the elastic regime, while the Reynolds number based on the same numbers and on water’s density is about one. Both the climbing and the smallness of the inertia it has to beat come from the same pair of numbers, and neither is available to a liquid without a relaxation time in it.

The rod that is too thick #

Setting the bracket to zero gives a radius, and it is a property of the fluid alone:

$$r_c = \sqrt{\frac{2(\Psi_1 + 4\Psi_2)}{\rho}}.$$

For the solution in the figure it is $34.6$ millimetres. Rods thinner than that are climbed and rods thicker are not, at any speed, because the speed multiplies both terms equally and cancels out of the comparison.

The numbers are $2.21$ millimetres of climb on a $10$ millimetre rod, $1.16$ on a $25$ millimetre one, and a $4.83$ millimetre dip on a $60$ millimetre one — same fluid, same rotation rate, three outcomes.

This is where the second normal stress difference earns the attention it was given above. It enters the criterion as $\Psi_1 + 4\Psi_2$, multiplied by four, so a quantity that is a tenth of the first in size contributes nearly half of it to the thing that decides the sign. A model that sets $\Psi_2$ to zero — which the simplest viscoelastic models do — predicts a critical radius forty per cent too large.

Two competing terms with different dependences and a crossing where one gives way to the other is the shape of the rod-climbing criterion. The elastic stress that drives the climb falls with distance from the rod at one rate and the centrifugal stress that flings the liquid outward falls at another — so a thin rod climbs and a thick one throws, with a crossover diameter that depends on the fluid. That is why the demonstration needs a stirrer rather than a paddle.

The same stress, doing other things #

Once a tension along the streamlines is admitted, several other observations stop being separate curiosities.

Die swell. Polymer melt extruded through a die emerges wider than the die. In the die the melt is sheared and the chains are stretched along the flow; on emerging, the tension is released and the chains recoil, which shortens the stream and therefore thickens it. Ratios of three are ordinary, and every extrusion die is cut smaller than the product it is meant to make.

The syphon that needs no tube. Start a polymer solution flowing out of a beaker and lift the tube out of the liquid: the stream continues, drawing the liquid up out of the beaker with nothing round it. The column is held together by the same tension along the streamlines.

The jet that will not break up. A Newtonian thread pinches off because surface tension drives a growing instability. In a polymer solution the thinning region is a strong extensional flow, the chains there stretch enormously, and the resulting tension resists further thinning — so the drop stays connected by a long thread that persists for seconds. It is why an unwanted thread follows a drop of paint and a drop of water leaves none, and it is the reason the instability that breaks a jet is arrested rather than removed.

And the flow that goes the wrong way in a mixer. In a Newtonian liquid, stirring a suspension moves particles outward. In an elastic one they migrate inward and collect at the axis, which is a nuisance in some processes and the working principle of others.

What it took to see it as a stress rather than a curiosity #

Rod climbing was not discovered; it was noticed repeatedly and explained wrongly for a long time, and the reason is instructive about what a constitutive law is for.

Every classification in rheology has the same shape: a ratio of two times, or two stresses, deciding which of two behaviours is seen. A damped oscillator’s three regimes are distinguished by exactly one such ratio, and the analogy is worth making because it explains why rheological “types” are not types of material at all — they are regions of a dimensionless parameter, and one fluid moves between them as the conditions change.

The trouble is that “viscosity” is not one thing that can be made larger. It is the coefficient in a particular relation between stress and rate of deformation, and that relation is an assumption about the material, not a definition of stress. Newton’s assumption is that the stress depends on the instantaneous rate of deformation, linearly, and on nothing else — not on the history, not on the direction. Everything a Newtonian liquid does follows, and so does everything it cannot do.

Weissenberg’s contribution in 1947 was to say what an alternative would have to specify: not one number but three material functions of the shear rate, of which the viscosity is one and the two normal stress differences are the others. The climbing rod was the demonstration, and its point was that the extra functions are not corrections — they can be the dominant term.

The measurement side followed. A cone-and-plate rheometer measures the shear stress from the torque and the first normal stress difference from the thrust pushing the cone away from the plate, which is a force at right angles to the one being applied, and which for a Newtonian liquid is zero at every rate. That thrust is now a routine instrument reading, and it is the same tension the rod climbs on.

What has not become routine is the second normal stress difference. It requires measuring a pressure distribution across the gap rather than a single force, the signal is a tenth of the first, and published values for the same material disagree by more than that. The rod-climbing criterion depends on it with a weight of four, which makes the critical radius one of the more sensitive things anybody predicts from these constants.

The additive that makes a pipe wider #

The economically important consequence of polymer elasticity is not that anything climbs a rod. It is that a few parts per million of the same additive can cut the friction in a turbulent pipe by three quarters.

Toms reported it in 1948 and it was disbelieved for some years, because it sounds like a violation of something. Dissolve a very long-chain polymer in water at a concentration of tens of parts per million — too dilute to change the viscosity measurably — and the pressure needed to drive a given flow through a pipe falls, by as much as eighty per cent at high flow rates.

Two features make it clear what is happening. It works only in turbulent flow: in laminar flow the same solution costs slightly more pressure, not less, because its viscosity is fractionally higher. And there is a ceiling — a universal curve, the same for every polymer and every solvent, which no concentration and no molecular weight beats.

The mechanism is the tension of this essay, acting where the flow stretches hardest. Turbulence near a wall is organised into streamwise vortices that pump slow fluid away from the wall and fast fluid toward it, and the flow between and around them is strongly extensional. A polymer chain caught in it is stretched, and the elastic stress it develops opposes the stretching — taking energy out of exactly the structures that carry momentum to the wall. The near-wall layer thickens, the momentum transport falls, and so does the drag.

The applications are unglamorous and large. The Trans-Alaska pipeline has injected a drag-reducing polymer since 1979, and the resulting increase in throughput was the equivalent of building additional pump stations that were never built. Fire services use polymer additives to reach further from the same pump. Municipalities inject them into sewers during storm surcharge to raise the capacity of pipes already in the ground.

There is a characteristic operational problem and it follows from the physics. The chains are broken by the very shear that makes them useful, so the effect degrades as the fluid travels and cannot be restored by anything but injecting more. A pipeline’s dosing stations are spaced by how far the polymer survives, not by how far its benefit is needed.

The drop that will not let go #

The thread that follows a drop of shampoo was mentioned in passing above and deserves its own section, because it is simultaneously a nuisance, a design constraint and a measuring instrument.

A Newtonian drop pinching off does so quickly and cleanly: surface tension drives the neck to zero in a way that depends on the fluid’s viscosity and density and on nothing else. Add a trace of polymer and the neck stops collapsing and starts thinning exponentially instead, at a rate fixed by the fluid’s relaxation time. The thinning region is an intensely extensional flow, the chains in it are stretched to nearly their full length, and the resulting tension holds the filament open.

What is left is a long thin thread joining the drop to whatever it came from, often decorated with a row of smaller drops — the beads-on-a-string structure — which persists for a second or more where a Newtonian thread would have gone in milliseconds.

That exponential rate is a measurement, and an instrument is built on it. Pull a small sample into a filament, let it thin under surface tension, and record the neck diameter against time; the slope of the logarithm is one third of the reciprocal relaxation time. It is the standard way of measuring the relaxation time of a dilute solution, and it works on fluids far too thin for a rotational rheometer to give a normal-stress signal at all.

The engineering matters in both directions. Inkjet printing requires a drop to detach cleanly and fly straight; a filament produces a tail, satellite drops and misplaced ink, so the molecular weight of anything dissolved in an ink is capped tightly — a formulation constraint that comes directly from this thinning behaviour.

Agricultural spraying wants the opposite. A nozzle producing a spectrum of droplet sizes sends the smallest ones drifting away from the field, and drift is the industry’s central regulatory problem. Adding a small amount of high-molecular-weight polymer suppresses the fine end of the spectrum, because the ligaments that would have broken into fine droplets are instead held together by the elastic tension and end up as fewer, larger drops. Anti-drift adjuvants are sold by the drum and are polymer solutions doing exactly what the rod-climbing fluid does, in a flow that lasts a millisecond.

Where the model runs out #

The material functions here are constants at low rate and are not in general. $\Psi_1$ is treated as a constant with a thinning correction, which is a fair description of a dilute solution over a couple of decades of rate and no more. Real polymer melts have normal stress coefficients that fall by orders of magnitude across their working range, and a criterion computed from low-rate values does not apply at high ones.

A polymer chain with nothing stretching it takes the shape of a random walk, and the elastic stress in these fluids is the entropic cost of pulling such a coil out straight. That is where the stress comes from: not from bonds being stretched but from arrangements being removed, which is why the modulus is proportional to temperature and why it disappears when the chains are short.

Creeping flow is assumed and it fails first at the rod. The velocity profile $\Omega R^2/r$ is the exact Stokes solution for an unbounded fluid, and near a fast rod inertia and secondary flows both matter. The climbing height computed here is the first term of an expansion in $\Omega^2$, so it is quantitative for a slow rod and qualitative for a fast one.

The surface deflection is assumed small. The derivation puts the free surface at atmospheric pressure and treats the column as hydrostatic, which requires the climb to be small compared with the rod’s radius. The spectacular demonstrations — several centimetres of climb — are well outside that.

And surface tension is left out entirely. Near a thin rod the meniscus alone lifts the liquid by a millimetre or two, and separating that from the elastic climb is the main experimental difficulty. The usual method is to measure the change in height with rotation rate rather than the height, since the meniscus does not know how fast the rod is turning.

At the rod diameters where climbing is easiest to see, a meniscus lifts the liquid by about as much as the effect being measured — so the two have to be separated before anything is claimed. That is a practical caution rather than a limit of the theory, and it is why the published demonstrations use rods well above the capillary length and photograph the difference between rotating and still.

The ladder from here #

Later rungs on this anchor: extensional viscosity, which is the material function the thread and the die-swell experiments actually probe and which can be thousands of times the shear viscosity; elastic instabilities, where a curved streamline with tension along it becomes unstable and produces turbulence at a Reynolds number of one; the tube model, which derives the relaxation time from the chain’s motion among its neighbours rather than fitting it; and thixotropy, where the material functions themselves depend on how long the fluid has been sheared.

The neighbouring ladders are the fluid that answers back, where the viscosity stops being a constant, the liquid that remembers, where the same relaxation time is weighed against an observation instead of a rate, and momentum going sideways, which is what shear stress is in a fluid with no memory at all.