The sand whose friction depends on how fast it flows

The angle that does not know the size of the heap found that a pile of sand stands at a slope set by friction alone, independent of its size, and that it has two such slopes: a steeper one at which it starts to avalanche and a shallower one at which a running avalanche stops. Between the two, it said, a heap is stable if still and unstable if moving. What it did not say is how a moving heap moves — how fast an avalanche runs on a given slope, how the speed varies through the flowing layer, why some slopes support a steady flow and others an accelerating one.

For most of the twentieth century there was no answer that worked across conditions. A granular material is not a solid, because it flows; not a liquid, because it holds a slope; not a gas, because its grains are in contact. The answer that emerged in the early 2000s, from experiments on grains flowing down rough inclines and simulations of grains being sheared, is short. A dense granular flow is a frictional material whose friction coefficient depends on one dimensionless number.

A race between two times #

Shear a layer of grains at a rate $\dot\gamma$ under a confining pressure $P$. Each grain is pushed over the one beneath it, and as it rides over, a gap opens below it. Two times compete. The shear opens a gap in about $1/\dot\gamma$. The pressure pushes the grain down into the gap in about $d\sqrt{\rho/P}$, the time a grain of size $d$ and density $\rho$ takes to fall a grain diameter under an acceleration set by the pressure on its area. Their ratio is the inertial number,

$$I = \frac{\dot\gamma\, d}{\sqrt{P/\rho}}.$$

When $I$ is small, the pressure wins: grains drop into every gap as soon as it opens, the packing stays tight, and the material deforms through a sequence of nearly static arrangements, like the grains in a slowly sheared box. When $I$ is large, the shear wins: gaps open faster than grains can fall into them, the packing dilates, and the grains begin to fly and collide — the regime of the gas that cools itself into clumps. Dense flows, which include every avalanche on a sand dune and most flows in silos and hoppers, sit between: $I$ from about a thousandth to a few tenths.

Rough numbers show where real flows sit. Sand avalanching down the slip face of a dune, in a layer a centimetre deep of grains a third of a millimetre across, is under a pressure of about 150 pascals at the bottom of the layer, so $\sqrt{P/\rho}$ is about a quarter of a metre per second; shearing at ten per second, it has $I$ near a hundredth. A rock avalanche of half-metre blocks thirty metres deep, under nearly half a megapascal, runs at tens of metres a second and still has $I$ of a few hundredths. Grains streaming past the wall of a silo sit in the same range. Very different flows — a centimetre of sand and thirty metres of rock — land on the same part of the curve, which is the point of a dimensionless number: it measures the state of the flow, not its size.

The pressure in the denominator is the same pressure that the sound sand carries faster the deeper it goes found stiffening the contacts between grains, so that sound travels faster in sand under load. There the pressure pressed the contacts flatter; here it sets the time a grain takes to drop into a gap. Both say that a granular material’s properties are not fixed numbers but functions of how hard it is squeezed, which is what distinguishes it from a solid and a liquid alike.

The inertial number is not a property of the grains. It is a property of a place in a flow — of the local shear rate, pressure and grain size together — and the claim of the rheology built on it is that the local state of a dense flow depends on nothing else.

A friction that rises with the race #

The friction of a granular flow is the ratio of the shear stress it supports to the pressure on it, which the force that takes what it needs identified as the defining property of a frictional contact. For a dense flow, measured against $I$, it rises from a static value to a ceiling.

The form used for glass beads, from measurements by Pierre Jop, Yoël Forterre and Olivier Pouliquen in 2006, is

$$\mu(I) = \mu_s + \frac{\mu_2 - \mu_s}{I_0/I + 1},$$

with $\mu_s = \tan 20.9°$, $\mu_2 = \tan 32.76°$ and $I_0 = 0.279$. At low $I$ the friction is the static value — the angle a heap stops at. As $I$ grows the grains are pushed over one another before they can settle, more of them are lifted out of the hollows between their neighbours, and resisting the shear takes more force for the same pressure. At high $I$ the friction levels off: a grain that is barely touching its neighbours as it flies past cannot be made to resist more by moving faster.

Why one number should be enough follows from what is missing. Rigid, dry, cohesionless grains have no viscosity, no stiffness that matters and no intrinsic time scale of their own. The only quantities available to describe a sheared layer are the grain size, the grain density, the pressure and the shear rate, and from those four only one dimensionless combination can be made: $I$. If the friction depends on anything, it can depend on nothing else. The argument is the same kind of dimensional reasoning that fixes the drag on a sphere as a function of the Reynolds number alone, and like that argument it says what the friction depends on without saying how.

The law is empirical; nobody has derived it from the contacts between grains. What makes it more than a fit is that, with these three numbers and nothing else, it describes flows in quite different geometries: down slopes, in rotating drums, between rotating cylinders, out of silos, collapsing columns.

Every depth at the same friction #

On a slope, the law makes a striking prediction. A steady, uniform layer of grains flowing down an incline at angle $\theta$ carries, at every depth, a shear stress and a pressure that are both set by the weight of the grains above. Their ratio — the friction the layer must supply — is $\tan\theta$ at every depth, top to bottom. So the inertial number must be the same at every depth too, the value at which $\mu(I) = \tan\theta$:

$$I = I_0\,\frac{\tan\theta - \mu_s}{\mu_2 - \tan\theta}.$$

The pressure, though, grows with depth below the surface. If $I$ is fixed and $P$ grows, the shear rate must grow as $\sqrt P$: deeper layers shear faster, because the pressure pins them more firmly and they need more shearing to reach the same balance. Integrating the shear rate from the base up gives the velocity profile Ralph Bagnold measured in the 1950s,

$$u(z) = \frac{2}{3}\frac{I}{d}\sqrt{\varphi g\cos\theta}\left[h^{3/2} - (h - z)^{3/2}\right],$$

for a layer of depth $h$ with grains packed at volume fraction $\varphi$.

The profile differs from a liquid’s in a way that can be seen in the experiments. A viscous liquid flowing down a slope has a shear rate proportional to the depth below the surface — the stress grows linearly and the viscosity is fixed — and a parabolic profile, as the fourth power in a pipe found for liquid in a tube. A granular layer’s shear rate grows only as the square root of the depth below the surface, so the shear is spread more evenly, the profile is closer to a straight line near the base, and it is a three-halves power rather than a parabola. The steepness of the slope enters through $I$ alone: three degrees of slope multiply the speed several times over.

A window of slopes #

The formula for $I$ has a numerator and a denominator that can each go to zero, and between them they fence off the slopes on which a steady flow exists.

Below $\tan^{-1}\mu_s = 20.9°$ the static friction is enough to hold every layer, and a layer at rest stays at rest. Above $\tan^{-1}\mu_2 = 32.8°$ no shear rate gives a friction large enough to balance gravity, and the layer accelerates without limit until it is no longer dense — until the grains are flying and colliding and the dense-flow law no longer applies. Between, every slope has its steady flow, and the inertial number rises steeply across the window, from barely creeping near the lower angle to nearly collisional near the upper.

The lower bound of the window is the stopping angle of the angle that does not know the size of the heap. The starting angle, at which a heap at rest begins to move, lies inside the window and is not given by $\mu(I)$ at all: it depends on how the grains were packed and how much the heap must dilate before it can flow, a history the local law does not know. That is one of the places where the theory is incomplete.

Speed against depth #

A deeper layer flows faster, because more of it shears, but the dependence on depth is not the liquid’s.

The surface speed grows as $h^{3/2}$, against $h^2$ for a viscous liquid. Ralph Bagnold, a soldier and desert explorer who spent his career on the physics of blown sand, found the underlying scaling in 1954 by shearing grains between rotating cylinders: in rapid granular flow, both the shear stress and the pressure grow as the square of the shear rate, because every collision transfers momentum in proportion to the speed and collisions happen at a rate in proportion to the speed too. A stress that grows as $\dot\gamma^2$ and a pressure that grows with depth together give the three-halves power. The inertial-number law contains Bagnold’s scaling in its ratio — $I$ is $\dot\gamma$ times $d/\sqrt{P/\rho}$, and fixing $I$ fixes $\dot\gamma^2 d^2\rho/P$ — and extends it to the slower, denser flows where grains slide rather than collide.

The practical consequence is that avalanches on natural slopes are slower for their depth than liquid flows would be, and their speed is less sensitive to how much material is moving. A layer four times deeper runs eight times faster, not sixteen.

A flow rule found before it was explained #

The law was not first found as a law of friction. It was first found as a rule about whole flows.

Olivier Pouliquen in 1999 measured the speed and depth of glass beads flowing down a rough incline at many slopes and found that a single relation collapsed all of his data: the Froude number of the flow, its mean speed over $\sqrt{gh}$, was proportional to its depth, with a slope that depended on the angle through the thickness at which a layer on that slope would stop. The inertial-number law, proposed afterwards by the French granular groups to explain this flow rule locally, reproduces the proportionality exactly — Bagnold’s profile averaged over depth gives a Froude number of $\tfrac25 I\sqrt{\varphi\cos\theta}\,h/d$ — which is the reason it was believed. It does not reproduce the stopping thickness, below which a layer on a given slope will not flow at all, and that is a second place where the local theory fails.

The same law in other shapes #

The test that persuaded the field was that the same three constants work in geometries with nothing in common with an inclined plane. In a slowly rotating drum half full of grains, a thin layer flows down the free surface while the rest rotates as a solid, and the law predicts the layer’s thickness and profile. In the collapse of a column of grains released onto a table, a problem in which the grains first fall, then spread, then stop, simulations that put the law into the equations of fluid flow reproduce the shape of the final deposit and how far it runs out, measured in experiments over a wide range of initial column shapes. In silos and hoppers the law gives the flow profile near the walls, where the silo that does not weigh what it holds found the walls carrying much of the weight. No new constant had to be fitted for any of them, which is the property that separates a constitutive law from a curve through one set of data.

Where the local law fails #

Both failures have the same origin. A local law says that the state at a point depends only on the stresses and shear at that point. In grains, it does not quite: a flowing region drags its neighbours, because grains are big, and a layer only a few grains thick, or a region next to a flowing one, behaves differently from the bulk. Thin layers stop when the bulk law says they should flow; regions the bulk law says are static creep slowly when a flowing region is nearby. Nonlocal extensions of the theory, which add a length over which the flow’s influence spreads — a few grain diameters — repair both, and are an active subject.

Other failures are larger. The law is for dry, rigid, roughly spherical grains of one size, and real flows have sizes that segregate as they move — the big one comes to the top found the large grains rising to the surface of a shaken or flowing mixture — and shapes that interlock. Wet grains add cohesion; grains in water or air add the fluid’s own viscosity and pressure, which matter when the grains are small. The law was also found to be ill-posed as a set of equations at very small and very large $I$, where small disturbances grow without limit in the mathematical model even though real flows are stable, a defect corrected by modifications near the ends of its range.

What the pictures leave out #

The figures use one set of constants, for glass beads half a millimetre across packed at sixty per cent, and assume a steady, uniform layer on an infinitely long, rough incline. Real avalanches start, accelerate and stop, have fronts and tails, and run over beds that are themselves partly mobile; the uniform layer is the cleanest case and the one in which the law was tested. The predicted speeds of thick layers on steep slopes are large because nothing in the local law limits them except the slope; in real flows, air drag on the surface, the dilation of the flow and the transition to collisional flow all intervene. The domain of the drawings is dry glass beads, dense flow, slopes between 20.9° and 32.8°, and layers from a few to a hundred grains deep.

Still open: where the dense law hands over #

The law describes dense flows between two limits it cannot reach. At the slow end, below $I$ of about a thousandth, grains in a slowly sheared material move in bursts and avalanches that a smooth friction law averages away, and the heap that becomes a solid described the jamming transition below which the material is rigid; how the creeping flow between jamming and dense flow should be described is unresolved. At the fast end, beyond $I$ of a few tenths, the flow becomes a granular gas, and the kinetic theory that describes gases has to be adapted to collisions that lose energy. How the dense law and the kinetic theory join — how a single description could cover a landslide from its slow start to its fast collisional runout — is the main open question of granular flow, and it matters for predicting landslides, pyroclastic flows and industrial silos alike, as the hourglass that keeps time found in the narrow case of grains leaving an orifice.

The core of it is a ratio of times. Dense grains flow with a friction set by the inertial number $I = \dot\gamma d/\sqrt{P/\rho}$, rising from tan 20.9° to tan 32.8° for glass beads, so a steady layer exists only on slopes between those angles; at a fixed slope every depth shares one I, the shear rate grows as the square root of the pressure, the profile is Bagnold’s three-halves power, and the surface speed grows as h^(3/2) — 2.57 cm/s at three grains deep on a 25° slope, 75.2 at thirty. Sand on a slope is neither a solid nor a liquid; it is a friction that knows how fast it is being asked to move.