The meniscus slows the finger and hardly moves it

The fastest finger is set by the gap solved the linear stability of a less viscous fluid pushing a more viscous one between two plates: a bump that gets ahead finds less resistance and gets further ahead, surface tension holds the shortest bumps back, and the fastest-growing wavelength is $\pi b/\sqrt{\mathrm{Ca}}$, with $b$ the gap and Ca the capillary number. On a bench cell of glycerol and air pushed at a millimetre a second that is a finger 21 millimetres across. The essay was candid about its weakest number: the surface tension was applied to the interface’s curvature in the plane of the cell, as though the interface were a curve, and it is not — it is a meniscus spanning the gap, with a film of the displaced liquid left on the plates behind it.

Park and Homsy worked out what that meniscus does to the pressure jump in 1984, by matching the meniscus region to the film and to the flow in the plane. Putting their jump into the same linear problem answers the question the earlier essay left, and the answer has a cancellation in it that is worth knowing about before reading any comparison of the theory with an experiment.

The jump across a moving meniscus #

A static meniscus spanning a gap of width $b$ between wetted plates has a radius of half the gap and a pressure jump of $2T/b$. A moving one does not: as the air advances it leaves behind a film of liquid whose thickness grows with speed, and the meniscus’s curvature at the front of that film is set by the film, as Bretherton showed for a bubble in a tube. Park and Homsy’s jump is

$$\Delta p = \frac{2T}{b}\left(1 + 3.80\,\mathrm{Ca}_m^{2/3}\right) + \frac{\pi}{4}\,T\kappa,$$

with $\mathrm{Ca}_m = \mu_2 U/T$ the displaced liquid’s capillary number and $\kappa$ the interface’s curvature in the plane. Both terms differ from the earlier essay’s. The transverse jump is larger than static, 1.30 times at $\mathrm{Ca}_m = 0.022$, and rises with speed. And the tension acting on the in-plane curvature is $\pi/4$ of the true one, because the meniscus is not a sheet perpendicular to the plates but a curved surface whose in-plane bending is shared across the gap.

The film the air leaves #

The speed-dependent part of the jump comes from the film, and the film is an old acquaintance. A plate drawn out of a bath carries a film whose thickness grows as the capillary number to the two-thirds — Landau and Levich’s law, which what a plate takes with it derived — and an air finger advancing between two wetted plates is the same problem seen from the other side: the meniscus moves and the plates stay, and the film left behind on each plate is set by the same balance between viscosity, which drags liquid into the film, and the meniscus’s curvature, which pushes it back. Bretherton’s version for a bubble in a tube gives a film of $1.34\,\mathrm{Ca}^{2/3}$ times the radius, and on the plates’ half-gap that is about fifty microns at the bench’s capillary number of 0.022.

That film does two things to the jump. It shifts the meniscus’s tip away from the plates, so the radius of the curved surface is set by the film’s thickness rather than by the half-gap alone, which is where the $3.80\,\mathrm{Ca}^{2/3}$ comes from. And it thickens with speed, so a bump that runs ahead lays down more film and pays more pressure, which is the resistance. A thin fibre coats by its own radius is the same two-thirds law on a cylinder; the film in a Hele-Shaw cell is its flat version, and it is invisible in every picture of fingering because it is a few per cent of the gap thick.

Two corrections, opposite in sign #

Each term enters the linear problem differently. The in-plane part simply replaces $T$ by $(\pi/4)T$ in the restoring term, so it weakens the stabilisation of short waves: the cut-off wavenumber moves out by $\sqrt{4/\pi}$ and every finger, fastest included, is $\sqrt{\pi/4} = 0.886$ of its earlier width. The transverse part depends on the interface’s local speed, and a bump that moves ahead faster carries a larger pressure jump in front of it — a resistance to getting ahead, proportional to how fast it gets ahead. Writing $D$ for the jump’s speed sensitivity, the growth rate becomes

$$\sigma(k) = \frac{k\left[U(\mu_2 - \mu_1) - \tfrac{\pi}{4} M T k^2\right]}{\mu_1 + \mu_2 + M k D}, \qquad M = \frac{b^2}{12},$$

with the speed-dependent jump appearing as an extra viscosity that grows with the wavenumber. It damps short waves more than long ones, so it pushes the fastest finger the other way, wider.

Which way the finger moves #

The two pull against each other, and which wins depends on the capillary number. The reduced tension shortens every finger by the same 11.4 per cent. The speed-dependent jump’s damping grows with the capillary number, because its sensitivity grows as $\mathrm{Ca}^{-1/3}$ while the wavenumbers it acts on grow as $\mathrm{Ca}^{1/2}$. At a capillary number of $10^{-4}$ the net effect is a finger 5.8 per cent narrower; at 0.0131 the two cancel exactly; at 0.3 the finger is 6.1 per cent wider.

The cancellation sits in the middle of the range bench experiments use. A test of the uncorrected $\pi b/\sqrt{\mathrm{Ca}}$ against measured fingers at capillary numbers near a hundredth will find it right, and right for the wrong reason: two corrections of ten per cent each, in opposite directions, have been left out together. The same theory tested at $10^{-4}$ or at $0.3$ is off by six per cent in opposite directions, and the drift of the error with capillary number is the signature of the two terms, not of a fitted constant.

At the bench’s capillary number #

The bench cell of the earlier essay — a one-millimetre gap, glycerol of 1.4 pascal-seconds, a surface tension of 0.063 newtons a metre against air, air pushed in at a millimetre a second — is at a capillary number of 0.022, just past the cancellation. Its fastest finger moves from 21.2 to 21.4 millimetres, a change no photograph could detect. Its growth rate does not stay put: the fastest mode grows at 0.150 per second rather than 0.196, an e-folding time of 6.7 seconds rather than 5.1. The fingering is still plain within half a minute, but later, and a measurement of growth rates rather than of wavelengths would see the meniscus where a measurement of wavelengths cannot.

Four oils at one speed #

The cancellation is easy to walk through by changing the liquid. Push air at a millimetre a second into the same one-millimetre cell filled in turn with silicone oils of 0.01, 0.1, 1 and 10 pascal-seconds, all with a tension of 0.021 newtons a metre against air: capillary numbers of $4.8\times10^{-4}$, $4.8\times10^{-3}$, 0.048 and 0.48, and uncorrected fastest fingers of 144, 46, 14 and 4.6 millimetres. With the meniscus, the thinnest oil’s fingers are 4.4 per cent narrower and the thickest’s 7.2 per cent wider; the middle two are within 2.3 per cent of the uncorrected widths. The growth rates fall by 10, 18, 27 and 36 per cent. An experiment that varied the oil to test the scaling $\lambda \propto \mathrm{Ca}^{-1/2}$ would find its exponent slightly shallower than a half — about 0.48 across these four — and might read that as a flaw in the theory, when it is the meniscus’s two terms changing places.

The growth rate is where the meniscus shows #

Across the range the speed-dependent jump slows the fastest finger by 12 per cent at a capillary number of $10^{-3}$, 23 per cent at 0.022 and 34 per cent at 0.3. The slowing does not cancel against anything: the reduced tension raises growth rates only by letting shorter waves grow, and at the fastest wavenumber that gain is smaller than the jump’s damping. So the meniscus is, at bench speeds, mainly a brake on the instability rather than a change to its pattern — the opposite of what the reduced tension alone would suggest, and the opposite of the way the correction is usually described.

The mechanism is the same one the exact theory drawn by viscosity relies on for the averaged flow: in a gap, the pressure gradient and the velocity are tied together by the cell’s permeability $b^2/12$, and anything that costs pressure in proportion to velocity behaves as extra viscosity. A meniscus that needs more pressure to move faster is exactly that, concentrated at the interface.

A pattern is a ratio, a clock is not #

The shape of the result is general enough to carry elsewhere. The fastest wavelength of a linear instability is where the drive’s growth with wavenumber is balanced by the restoring term’s, and it depends on the ratio of the two; a correction that weakens the restoring term and another that damps short waves move that balance in opposite directions, and a near-cancellation between them is unremarkable. The growth rate is an absolute quantity, set by the size of the drive against everything that resists, and corrections that add resistance always reduce it. So a correction can leave a pattern’s scale almost untouched while changing its clock substantially, and the clock is the harder thing to measure.

The jet that breaks into drops shows the same structure from the other side. Viscosity lets a jet break, later and into bigger drops found viscosity stabilising no wavelength of a liquid thread but slowing the short ones most, which lengthens the fastest ripple — exactly what the meniscus’s speed-dependent jump does here. The air shortens a jet’s fastest ripple found the air’s pressure doing the opposite, by adding to the drive at short wavelengths. A Hele-Shaw cell with a meniscus is the case where both kinds of correction act at once, on the same interface, and nearly balance.

The same reading applies to the bench cell’s photograph. A picture taken in the first ten seconds shows eight to ten fingers in a ten-centimetre channel with or without the meniscus; a stopwatch on when they first become visible would differ by a third. The earlier essay’s estimate that the fingering is plain within half a minute survives; its estimate of the e-folding time, five seconds, should be read as nearly seven.

A channel goes unstable sooner #

In a channel of width $W$ the interface is flat until the first wall mode, wavenumber $\pi/W$, crosses the cut-off. The cut-off is where the growth rate’s numerator vanishes, and the denominator — where the speed-dependent jump sits — cannot move it. So the onset sees only the reduced tension, and the meniscus lowers the onset capillary number by $\pi/4$ at every width: a channel a hundred gaps wide fingers above $6.5\times10^{-5}$ instead of $8.2\times10^{-5}$. The meniscus makes a flat interface unstable at 79 per cent of the speed the earlier calculation gave. For the bench’s ten-centimetre channel that brings the largest speed at which the interface stays flat down from about thirteen millimetres an hour to about ten.

That is an onset advanced, not delayed, and it is a cleaner test of the meniscus than any wavelength: the onset depends on the in-plane tension alone, with no speed-dependent term to cancel it. It is also the regime in which the cell’s flow is closest to the exact potential flow that makes it a teaching instrument — a cell that draws a larger cylinder uses the same slow flow to draw ideal streamlines — and the meniscus says that the flat interface such a demonstration relies on is a little more fragile than the uncorrected arithmetic suggests. One number decides which physics applies in most flows; here two capillary numbers, nearly equal, decide two different things.

In a growing circle #

The radial geometry of the earlier essay, air injected at a point and spreading as a disc, inherits both corrections in the same way. The mode number at which the disc’s edge first breaks up is set by the restoring tension, so it rises by $\sqrt{4/\pi}$ at a given radius; the rate at which the modes grow is reduced by the speed-dependent jump, which there varies around the disc’s edge as the edge slows with radius. The edge’s speed falls as the inverse of the radius at a constant injection rate, so the capillary number falls as the disc grows and the disc passes through the cancellation from above — its fingers wider than the uncorrected theory early and narrower late. The cell draws a larger cylinder is the same cell’s steady flow around an obstacle, where no meniscus moves and none of this arises.

How the corrections were checked #

With both corrections switched off, the growth rate is the earlier essay’s exactly at three wavenumbers and the golden-section fastest wavelength is $\pi b/\sqrt{\mathrm{Ca}}$ to $5\times10^{-9}$. The speed sensitivity $D$ agrees with a central difference of the jump to $10^{-9}$. And the growth rate at the corrected cut-off is zero with and without the speed-dependent term, which is the statement that the onset sees only the tension.

The convention: two capillary numbers #

The finger’s capillary number is $U(\mu_2 - \mu_1)/T$, on the viscosity difference, as in the earlier essay; the meniscus’s is $\mu_2 U/T$, on the displaced liquid alone, because the film is that liquid’s. For air displacing glycerol the two differ by a part in a hundred thousand. Wavelengths are of the fastest linear mode, not of fingers that have grown and competed.

Where Park and Homsy’s jump stops #

Large capillary number. The jump is the leading terms of an expansion in $\mathrm{Ca}^{1/3}$, valid when the film is thin against the gap; at a capillary number of 0.3 the film is a substantial fraction of the gap and the curve’s right-hand end is an extrapolation. Wetting. The film exists because the displaced liquid wets the plates; a non-wetting pair, or a partially wetting one with contact-line hysteresis, has a different jump and no film. Gap variation. Real plates are not flat to a fraction of the film’s thickness, and a film of tens of microns is affected by the flatness of the glass. Gravity in a cell standing on edge adds a drive or a restoring force proportional to the density difference, which shifts the cut-off in a way the meniscus does not touch; the corrections here are for a horizontal cell. The nonlinear stage, where fingers compete and the winners approach Saffman and Taylor’s half-width finger, is untouched: the meniscus changes that width too, and that is the measured quantity the correction was first proposed to explain.

Park, Homsy, and the film #

Saffman and Taylor’s 1958 paper set the linear problem with the tension on the in-plane curvature. Park and Homsy’s 1984 analysis matched the meniscus to the film and to the outer flow and gave the jump used here, building on Bretherton’s 1961 film; Reinelt, and Tabeling and Libchaber, then used it to explain why measured finger widths depart from the uncorrected theory at small capillary numbers, where its effect on the steady finger is largest. The cancellation in the linear wavelength, and the growth rate and onset where the correction is not small, follow from putting the two terms into the linear problem together.

Still open: the finger that has won #

The linear modes grow, compete, and leave a few fingers that keep growing; in a channel one finger wins and travels steadily at a width close to half the channel’s at large capillary numbers. Its width is set by a solvability condition on the tension, and the meniscus changes the tension in two ways that may not cancel there as they do here. The next calculation takes Saffman and Taylor’s steady finger, applies the selection rule with the corrected in-plane tension and the speed-dependent jump, and asks whether the corrected width tracks measured fingers across the range of capillary numbers where the uncorrected selection rule fails, and whether the cancellation that hides the meniscus in the linear wavelength appears again in the finger that wins.