The fastest finger is set by the gap and one number

The exact theory drawn by viscosity found that a Hele-Shaw cell — two plates a millimetre apart with a viscous liquid between them — obeys an averaged law identical to Darcy’s: the gap-averaged velocity is $-(b^2/12\mu)\nabla p$, with $b$ the gap. That makes the averaged pressure satisfy Laplace’s equation, which is why a cell draws ideal flow around an obstacle, and the cell draws a larger cylinder measured how far that picture is off near a wall.

The first essay then turned to two fluids in the cell. It described in words why a less viscous fluid pushing a more viscous one is unstable — a bump that gets ahead sits in fluid that resists less, so it is pushed harder and gets further ahead — said that surface tension stops the shortest bumps, and that “balancing the two gives a most-unstable wavelength”. It then went to the question of how wide the eventual finger is, which is a nonlinear selection problem. This essay computes the balance it named: the linear instability, exactly, in a straight channel and around a growing circle.

The growth rate #

A displacing fluid of viscosity $\mu_1$ pushes a displaced fluid of viscosity $\mu_2$ at speed $U$ through the cell. In each fluid the pressure satisfies Laplace’s equation, the interface moves with the fluid on both sides, and across the interface the pressure jumps by the surface tension $T$ times the interface’s curvature in the plane. Put a ripple of wavenumber $k$ on a flat interface and linearise: the ripple grows as $e^{\sigma t}$ with

$$\sigma(k) = \frac{k\,\big[U(\mu_2 - \mu_1) - \tfrac{b^2}{12}T k^2\big]}{\mu_1 + \mu_2}.$$

The first term is the instability: proportional to the viscosity contrast, to the speed, and to $k$. The second is the tension: it grows as $k^3$ and wins at short wavelengths. The denominator is the sum of the viscosities, the total resistance a perturbation has to move against. The rate is zero at $k = 0$ and at the cut-off $k_c = \sqrt{12U(\mu_2-\mu_1)/b^2T}$, and largest at $k_c/\sqrt3$.

Why the drive is proportional to the wavenumber #

The linear growth has a picture behind it. Ahead of the interface the displaced oil needs a steep pressure gradient to move at $U$, because it is viscous; behind it the air needs almost none. Push a crest of the interface forward a small distance $\varepsilon$ and it now sits where the oil’s pressure was lower by its gradient times $\varepsilon$, while the air behind it has carried its own, nearly uniform pressure forward with it. The crest is at a higher pressure than the oil around it expects, by the difference of the two gradients times $\varepsilon$, and that excess has to be relieved by a flow. In a medium obeying Laplace’s equation the pressure adjusts everywhere at once, and a disturbance of wavenumber $k$ decays over a distance $1/k$; the excess pressure is therefore spent over a distance $1/k$, so the velocity it drives, and with it the crest’s further advance, is proportional to $k$. A short ripple pushes its excess over a short distance and moves fast, a long one slowly. The tension, a pressure proportional to curvature, which is $k^2\varepsilon$, spent over the same $1/k$, opposes it in proportion to $k^3$.

Long waves grow, short ones are held, one grows fastest #

For air pushing an oil of a tenth of a pascal-second at a millimetre a second, with a surface tension of 0.02 newtons a metre, the cut-off is 0.245 per millimetre and the fastest wavenumber 0.141: a wavelength of 44.4 millimetres, growing by a factor $e$ in eleven seconds. Doubling the speed raises the whole curve and moves the fastest wavelength shorter by $\sqrt2$; halving it does the reverse.

The shape of the curve is the point. The instability is a long-wave one — every wavelength longer than the cut-off grows — but the growth of a very long wave is slow, because its rate is proportional to $k$. So the interface does not break up at its longest available scale, the channel width, or at its shortest, the gap: it breaks up at the scale where the drive, linear in $k$, and the tension, cubic in $k$, are best balanced. Whatever disturbance seeds the interface — dust on the plates, a vibration, the way the flow started — the components near that wavelength outgrow the rest, and the fingers that appear have that spacing whatever the seed.

One number sets the fastest finger #

Written in gap widths, the fastest wavelength depends on one number. With the capillary number $\mathrm{Ca} = U(\mu_2 - \mu_1)/T$, the ratio of the viscous stress the contrast creates to the tension that resists it,

$$\lambda_{\rm fastest} = \frac{\pi\,b}{\sqrt{\mathrm{Ca}}},$$

exactly, and the cut-off is $\sqrt3$ shorter. At $\mathrm{Ca} = 10^{-3}$ the fastest finger is 99 gaps across; at $10^{-2}$, 31. The gap enters only as the unit of length, through the mobility $b^2/12$, so a cell twice as wide at the same capillary number makes fingers twice as wide. The fastest wavelength reaches the gap itself only at a capillary number near $\pi^2$, far beyond where the averaged law can be trusted, since a finger narrower than the gap has a three-dimensional meniscus the averaged law does not contain.

This is the one-number structure that recurs throughout slow viscous flow, and here it is unusually clean: no fitted constant, no correlation, a square root and a $\pi$. It has the same origin as Darcy’s law itself — the cell is a porous medium of known permeability — and it is why the cell has been used as a model of oil displaced by water in a reservoir, where the same instability makes a flood break through early along fingers.

A channel chooses its number of fingers #

A real cell has side walls. With impermeable walls a distance $W$ apart the allowed ripples are $\cos(n\pi y/W)$, so the longest is half a wavelength across the channel and the wavenumbers come in steps of $\pi/W$. The interface is stable until the first of them, $n = 1$, crosses the cut-off, which happens at

$$\mathrm{Ca} = \frac{\pi^2}{12}\Big(\frac{b}{W}\Big)^2:$$

0.0021 for a channel twenty gaps wide, $8\cdot10^{-5}$ for one a hundred gaps wide. A narrow channel can be displaced flat at speeds at which a wide one breaks up. That threshold is the reason a slow enough displacement in a narrow channel does not finger at all, and it scales steeply: halving the channel width quadruples the speed at which it can be pushed.

Above the threshold, the right-hand panel counts. In a channel a hundred gaps wide the fastest-growing wall mode is the first at onset, the second by a capillary number of $10^{-3}$, and the ninth by $2\cdot10^{-2}$: the number of fingers that first appear is the channel width divided by half the fastest wavelength, rounded to a mode the walls allow. The staircase is the discreteness of a bounded interface, and at each step two neighbouring modes grow at nearly the same rate, which is why the count observed at a given speed scatters by one.

A growing bubble of air #

The commonest demonstration is not a channel at all: air injected through a hole in the middle of a cell full of oil, growing as a circle that breaks into a flower. The circular interface has its own linear theory, in which mode $n$ of the boundary — $n$ lobes around it — grows at

$$\sigma_n = \frac{U}{R}\,(A n - 1) - \frac{b^2 T}{12(\mu_1 + \mu_2)}\,\frac{n(n^2-1)}{R^3}, \qquad A = \frac{\mu_2-\mu_1}{\mu_2+\mu_1},$$

with $U$ the edge’s speed and $R$ its radius. The $-1$ is geometry: a circle that grows stretches any ripple on it, which on its own makes ripples decay relative to the radius. The figure plots the rate as a multiple of the circle’s own growth rate $U/R$, so a value above zero means a lobe that outgrows the bubble carrying it. At one centimetre the fastest mode is the third; at five, the seventh; at twenty, the fourteenth.

The fingers multiply as the square root of the radius #

At a steady injection rate the edge slows as the bubble grows, $U \propto 1/R$, so the local capillary number falls and the fastest wavelength lengthens as $\sqrt R$. The circumference grows as $R$. Their ratio, the number of fingers, grows as $\sqrt R$: from 2 at half a centimetre to 22 at fifty. The circular mode count follows the planar estimate closely once the bubble is a few fastest wavelengths around, which is the circular rate’s planar limit showing.

This is the arithmetic of the familiar flower. Each finger that grows widens at its tip, and once its tip is wider than the fastest wavelength for its own speed it splits, so the number of fingers keeps rising as the pattern spreads. The linear theory cannot follow a split, but it says when the tip becomes wide enough to be unstable to one, and the count it predicts for the young bubble is what experiments record before the fingers begin to compete.

A bench demonstration, in numbers #

The numbers are within reach of two sheets of glass and a bottle of glycerol. With a one-millimetre gap, glycerol at 1.4 pascal-seconds and a surface tension against air of 0.063 newtons a metre, air pushed in at a millimetre a second makes a capillary number of 0.022. The fastest finger is then 21 millimetres across and grows by a factor $e$ in five seconds, so the fingering is plain to the eye within half a minute. In a channel ten centimetres wide the fastest wall mode is the ninth. A photograph taken in the first ten seconds, before the fingers have begun to shade one another, should therefore show eight to ten of them; one taken a minute later shows a few long fingers that have won and many short ones left behind, which is the nonlinear stage and is no longer a test of this calculation. To see the flat interface the onset condition promises, the same channel must be pushed at less than 0.0037 millimetres a second — about thirteen millimetres an hour, which is why the stable regime is rarely demonstrated and the unstable one always is.

Gravity in a standing cell #

Most demonstration cells stand upright, and then gravity enters the drive. A heavier fluid above a lighter one in a vertical cell adds $(\rho_2 - \rho_1)g\,b^2/12$ to the viscous term, with the same factor of $k$ and the same denominator: the averaged law treats a density difference across the interface exactly as it treats a viscosity difference across a moving one. Air injected from below, displacing oil upwards, is therefore more unstable than the same displacement sideways, and air pushed down into oil from above is stabilised by gravity against a modest viscous drive, the light fluid resting on the heavy one. It is the Rayleigh–Taylor instability of a ceiling that drips, filtered through Darcy’s law, and the fastest wavelength again comes from balancing a drive linear in $k$ against a tension cubic in it.

A more viscous invader #

Air is the extreme invader, with almost no viscosity. Water pushing oil, or a thin oil pushing a thick one, is less unstable in two ways at once. The contrast $\mu_2 - \mu_1$, which drives the instability, is smaller; and the sum $\mu_1 + \mu_2$, which every perturbation must move against, is larger. At a viscosity ratio of a half the fastest growth rate is 0.24 of air’s and the fastest wavelength 1.41 times as long; at nine-tenths, 0.016 and 3.2. A displacing fluid of the same viscosity as the displaced one is neutrally stable at every wavelength, and a more viscous one is stable — which is why oil recovery adds polymer to the water it pumps in, thickening it towards the oil it is meant to push.

What was checked #

The planar rate is not read off its formula. The linear problem — perturbation pressures decaying away from the interface on each side, the kinematic condition on each side and the pressure jump at the displaced interface — is set up as three linear equations and solved for the rate by Cramer’s rule, and it agrees with the closed form to $10^{-15}$ of the fastest rate at four wavenumbers. A direct search on a grid of two hundred thousand wavenumbers finds the fastest at $k_c/\sqrt3$ to five parts in ten million, and the fastest wavelength computed from the rate equals $\pi b/\sqrt{\mathrm{Ca}}$ to rounding. The circular rate, evaluated at a radius of fifty metres with the mode number set to $kR$, approaches the planar rate at the same $k$ to two parts in ten thousand, the residual being the curvature terms that vanish as $1/R$.

The convention: in-plane tension, Darcy’s law, and the capillary number from the difference #

The surface tension acts on the interface’s curvature in the plane of the cell; the meniscus across the gap, which also has a curvature, is taken as fixed. The flow in each fluid obeys the averaged law with the same mobility $b^2/12$. The capillary number is formed with the viscosity difference, which is what drives the instability, and the growth rates are those of the linear stage, when the ripple is small compared with its own wavelength.

What the picture cannot show #

The in-plane picture leaves out the meniscus. A finger of air advancing into oil leaves a thin film of oil on the plates, whose thickness grows with the capillary number as Landau and Levich found for a coated plate, and the pressure jump across the meniscus depends on the speed through it. Park and Homsy showed in 1984 that this changes the effective tension by a factor near $\pi/4$ and adds a speed-dependent term, which shifts the fastest wavelength by tens of per cent at the capillary numbers of most experiments. The linear theory also says nothing about what happens once the ripples are as large as their wavelength — the competition in which one finger wins in a channel and the tip-splitting that makes a flower — and the finger’s width, the subject of the first essay, is outside it altogether. Every fluid here is Newtonian. An oil whose viscosity falls with shear rate — most polymer solutions — is thinner where it is sheared hardest, which is at a finger’s tip, so the tip advances more easily than the linear theory says and the fingers come out narrower and more branched; a viscosity that depends on the question is the reason one number cannot describe such a fluid, and why the capillary number here would have to be evaluated at the tip’s shear rate rather than at the mean. Viscoelastic oils, which store some of the deformation, fracture rather than finger at high enough speeds.

Who found it, and when #

Saffman and Taylor published the instability and its single-finger solution in 1958, from experiments in a cell; Chuoke, van Meurs and van der Poel derived the same growth rate with surface tension in 1959, in the context of oil recovery, and the fastest wavelength is often called after both. Paterson worked out the circular case in 1981 for an inviscid invader and showed the fastest mode number growing with radius; Miranda and Widom’s rate of 1998, used here, allows any viscosity ratio. Homsy’s review of 1987 gathered the experiments, which agree with the linear selection in the early stage to within the meniscus corrections.

Still open: the meniscus in the rate #

The weakest number in every figure is the surface tension’s effective value, because the true pressure jump across the interface includes the meniscus across the gap and the film it leaves behind. The next calculation replaces the fixed tension with Park and Homsy’s matched meniscus — an effective tension of $\pi/4$ of the true one plus a term growing as the capillary number to the two-thirds — and asks how far the fastest wavelength and the channel’s onset move at the capillary numbers experiments use, whether the onset in a narrow channel is delayed or advanced by the film, and whether the corrected fastest wavelength brings the linear theory’s finger count into the scatter of the measured ones.