A post holds liquid back by its radius, a stripe by its length

Twice as slippery along as across computed the slip of the simplest patterned surface there is: parallel stripes, alternately solid and gas, under a shear. A flow far above sees the stripes as a flat wall some distance below the real one, and Philip’s closed forms of 1972 give that distance: the period over π times the logarithm of sec(πφ/2), φ the gas fraction, along the stripes, and exactly half of it across. The logarithm is why a striped surface, however sparse its solid, never becomes very slippery: at a gas fraction of 0.99 the slip length is only 1.3 periods.

The essay ended on the texture that superhydrophobic surfaces are actually made of. Posts are points, not lines, and a liquid flowing past an array of them should be held back far less — so the slip length of a post array should grow as one over the square root of the solid fraction, a much stronger dependence. Computing it needs a pattern that varies in two directions, where the substitution that gave the stripes their exact factor of two has no analogue. This essay computes it, finds the square root and where it comes from, and finds a limit to the claim that posts are always better.

A post array, solved #

The liquid fills the space above a plane and is sheared far above at a rate γ. The plane is impermeable everywhere. Where it is gas — a flat, clean meniscus pinned between the posts, which can hold no shear stress — the liquid slides freely; where it is the top of a post, the liquid does not slip at all. The posts are discs in a square array of period L, and their share of the surface is the solid fraction φ.

The unknown is the traction the liquid exerts on the plane, which is zero on the gas and something to be found on each post. For a traction applied to an impermeable plane under a half-space of slow viscous flow, the velocity it produces at the surface has an exact form in Fourier space:

$$\hat{\mathbf u}(\mathbf k) = \frac{1}{\mu|\mathbf k|}\left(\mathsf I - \tfrac12\hat{\mathbf k}\hat{\mathbf k}\right)\hat{\boldsymbol\tau}(\mathbf k).$$

The traction’s component across each Fourier wave drives a pure shear that decays upwards as $e^{-kz}$ with no pressure; its component along the wave has to move liquid up and down to keep the volume, and answers half as strongly — the same factor of two the stripes found between across and along, appearing here wave by wave. The mean traction is the shear stress far above, μγ, and the mean surface velocity is the slip. No slip on the posts then says: on every post, the slip minus the velocity the traction holds back is zero. That is a symmetric, positive linear system for the traction on the posts, and it is solved by conjugate gradients, with the Fourier operator applied by fast transforms on a periodic grid of one period.

The traction has a square-root singularity at every edge between a no-slip and a shear-free surface, so a grid resolves it only approximately and the answer converges slowly. Each slip length here is computed on three grids and extrapolated to zero spacing. The posts are digitised exactly in area — the right number of cells nearest each centre — because a digitised disc whose area wanders as the grid is refined would leave the extrapolation chasing the area rather than the flow.

First, the stripes again #

The same solver, given stripes instead of discs, must reproduce Philip. It does: at half solid it extrapolates to 0.11032 periods along the stripes, against Philip’s 0.11032, and to 0.05516 across, exactly half. At a fifth solid it is within one per cent, the edge singularity being harder to extrapolate on narrow stripes. With the same calculation producing the stripes’ exact answers, its answers for posts can be read as a statement about posts rather than about the method.

The slip length of a post array #

The figure is the result. The posts’ slip length, in periods, is 6.17 at a quarter of a per cent solid, 2.87 at one per cent, 1.02 at five, 0.597 at ten and 0.066 at fifty. Against the solid fraction on logarithmic axes it runs close to a line of slope minus one half — one over the square root — steepening as the posts crowd. The stripes, along and across, rise only as the logarithm, and at a hundredth solid the posts slip 2.2 times as far as liquid running along the stripes.

The dashed line is the dilute limit, which can be written down without solving anything. If the posts are far apart, each one is a lone disc lying flat in a shear-free plane, dragged edgewise through the liquid at the slip velocity. A disc of radius a moving edgewise through an unbounded liquid has a drag of $\tfrac{32}{3}\mu a U$, and the shear-free plane through it is a plane of symmetry, so the liquid on one side feels half: $\tfrac{16}{3}\mu a U$. The drags of all the posts in a period’s area must balance the shear stress far above, so

$$\frac{16}{3}\mu a\,U_s = \mu\gamma L^2, \qquad b = \frac{U_s}{\gamma} = \frac{3L^2}{16a} = \frac{3}{16}\sqrt{\frac{\pi}{\varphi}}\;L,$$

using $\varphi = \pi a^2/L^2$.

Why a root, and why a logarithm #

The square root is Stokes’ law. A body dragged slowly through a viscous liquid has a drag proportional to its size, not to its area, because the disturbance it makes spreads to a distance set by its size and the velocity gradient it has to sustain falls as one over that distance. Halving a disc’s radius halves its drag, though it quarters its area. A surface of posts whose solid fraction is quartered has posts of half the radius, each holding liquid back half as hard, and the slip length doubles: one over the square root of the fraction.

The figure makes the argument quantitative. Multiplying the slip length by the root of the fraction removes the dilute law’s dependence and leaves its constant, $\tfrac{3}{16}\sqrt\pi = 0.3323$. Fitted through the three sparsest arrays, the computed values extrapolate to 0.3294 at zero fraction — within one per cent of a lone disc — and fall with a slope of −0.43. That slope is the posts’ interaction. A post in an array is not alone: its neighbours slow the liquid it meets, so it holds back less than a lone disc would, and the effect grows as the posts crowd. The fit, $b/L = 0.329/\sqrt\varphi - 0.43$, is the form Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet found from simulations in 2007, with coefficients of 0.325 and 0.44.

The logarithm of the stripes is the other half of the same story, and it is older. A stripe is a line, and a line dragged through a viscous liquid in two dimensions has no finite drag at all: the disturbance a line makes decays so slowly with distance that the drag depends on how far away the liquid is held still, and grows without limit as that distance grows. That is the flow with no solution — Stokes’ paradox, the fact that slow flow past a cylinder cannot be solved. In an array of stripes the neighbours are what hold the liquid still, at a distance of a period, and the drag per stripe depends on the logarithm of the period over the stripe’s width. Invert that and the slip length grows as the logarithm of one over the solid fraction. Stripes slip logarithmically because a line in slow flow is Stokes’ paradox; posts slip as a root because a disc is Stokes’ law.

Where the liquid runs #

The liquid does not slip uniformly over a post array. Along a line through the posts’ centres it stops on each post and, between the posts of a row, climbs back only to about its average. Along the lane midway between rows, which never meets a post, it runs at 1.3 to 1.4 times its average the whole way along. The flow is partly channelled into those lanes, and this is the part of the argument the dilute law leaves out. A lone disc sees liquid approaching from every direction equally; a disc in a square array sees liquid that has already found the unobstructed lanes and is going round it.

Where the posts grip #

The traction on a post shows where the holding back happens. Averaged over the posts it is exactly ten times the shear stress far above, because a tenth of the surface carries all of it. Across a post it is least at the centre, about five times the far stress, and climbs towards the rim, reaching 45 in the cell next to the edge — the square-root singularity every junction between a no-slip and a shear-free surface has, the same one the stripes had at their edges. A post grips the liquid mostly at its rim, which is another way of saying that its drag is set by its perimeter-scale size rather than its area, and it is why a real post’s edge — rounded, or with the meniscus bulging over it — matters more to its slip than its top.

Sparse posts beat stripes, dense ones do not #

The essay’s lead claimed that posts are held back far less than stripes and give a surface far more slippery at the same solid fraction. The figure tests the claim across the whole range, and it holds only half of it. Sparse, the posts slip more than stripes in either direction: at a hundredth solid, 2.2 times the along-stripe length and 4.3 times the across-stripe length. But the ratio to lengthwise stripes falls as the solid fraction rises and crosses one near a tenth. Past that, stripes lined up with the flow are more slippery than posts, and at half solid they slip 0.110 periods against the posts’ 0.066.

The reason is the lanes. A lengthwise stripe pattern has lanes of gas that run unbroken the whole length of the surface, and liquid over a lane never meets a solid at all; the posts’ lanes are interrupted, left and right, by posts in the next row, and the liquid must weave. At low solid fraction the Stokes-law advantage of points over lines dominates; at high fraction, the geometry of uninterrupted lanes does. Crosswise stripes, which force every stream of liquid over every solid, are the worst of the three at every fraction, and posts stay between the stripes’ two limits once they are dense. For a surface that must work whatever the flow’s direction, posts are the better bet; for one whose flow direction is known, stripes aligned with it are better above about a tenth solid.

What the wall has been told #

A slip length is a boundary condition, and it is worth being precise about what kind. How many things a flow must be told counted the conditions a viscous flow needs at a wall: two, the normal velocity and one more, and the no-slip condition is the usual choice of the second. The post array tells the liquid something different at every point — no slip on the posts, no stress on the gas — and the far flow cannot see any of it. What it sees is a single substitute condition, the Navier slip condition, in which the velocity at the wall is the slip length times the shear rate. The post calculation is the computation of that one number from the detailed conditions it replaces, and the reason the substitution is exact at a distance is that every disturbance the posts make decays upwards as $e^{-kz}$ with k at least one period’s wavenumber: a period above the surface, the pattern has faded by a factor of $e^{2\pi}$, more than five hundred.

The substitute also changes what the wall does to the flow’s vorticity. A wall puts in exactly its own speed found that a no-slip wall makes vorticity at a rate set by its acceleration, and that the vorticity it has put into the fluid is its speed. A slipping wall puts in less: the liquid next to it keeps a velocity of its own, and the vorticity at the wall is the shear rate rather than the full velocity difference spread over nothing. A textured wall that slips 2.9 periods at a hundredth solid is, to a flow a few periods away, a wall that has moved 2.9 periods downward and turned partly shear-free — which is why such surfaces lower friction in laminar flow by the ratio of the slip length to the flow’s own thickness, and why they matter most in thin channels, where that ratio is largest.

The Stokes-law argument of the dilute limit has its own boundary, the one how small is small enough drew for a sphere: inertia is negligible only while the Reynolds number on the obstacle’s size is well below one. For posts a few micrometres across in a millimetre channel that is comfortably true. For a textured hull at sea it is not, and there the posts sit inside a turbulent boundary layer’s viscous sublayer only if they are smaller than about five wall units.

What was checked #

Three checks. The same solver on stripes reproduces Philip’s closed forms along and across to a part in ten thousand at half solid and a per cent at a fifth. The sparse posts, extrapolated to zero fraction in the form $b\sqrt\varphi$, meet the lone edgewise disc’s constant to 0.9 per cent. And the slip length of an array a tenth solid on grids of 64, 128 and 256 cells a side — 0.605, 0.602, 0.600 periods — extrapolates to 0.597, a correction of half a per cent from the finest grid, which is the size of the uncertainty the extrapolation carries. The tests refuse posts so large they overlap, a pattern the solver does not know, and a tolerance of zero.

What the picture cannot show #

Flat menisci. Every gas surface here is flat and pinned at the posts’ edges. A real meniscus bulges in or out with the pressure difference across it, and a bulging meniscus changes the slip substantially — a meniscus bowed into the gaps adds drag, and one bulging out over the posts can remove it — which is the subject the earlier essay flagged as the one the slip-length idea hides.

A clean interface. A trace of surfactant immobilises a meniscus, which then behaves as a no-slip solid and destroys the slip altogether. The shear-free condition is the ideal, and measured slip lengths on superhydrophobic surfaces are often well below the ideal’s for that reason.

Square arrays and circular posts. A hexagonal array spaces the posts more evenly and slips slightly more at the same fraction; square posts grip more at their corners. Both change the constants and neither changes the root.

Slow flow. The whole calculation is Stokes flow, valid while the Reynolds number on the post’s size is small; in a turbulent flow over such a surface the relevant quantity is the slip in viscous wall units, and the posts must then be small compared with the viscous sublayer.

Two different lengths a wall can hide #

Slip has now come up three times, and the three are different in kind. Slip is a memory of one mean free path at a molecular scale: a gas remembers the wall’s velocity only over the distance between collisions. A wall that is not quite there replaced a porous wall’s structure with the depth at which a flow would find a solid one. And a textured wall slips because most of it is gas. What they share is that a flow far away sees only one number, the slip length, and cannot tell which of the three made it. What separates them is how the number scales: with the mean free path, with the pore size, and here with the texture’s period divided by the root of its solid fraction — so a superhydrophobic surface’s slip can be made as large as its posts can be made sparse, until its menisci fail.

Who worked it out #

Stokes’ drag on a sphere is from 1851, and the edgewise disc’s $\tfrac{32}{3}\mu aU$ follows from Oberbeck’s solution for an ellipsoid of 1876. The paradox of the cylinder is Stokes’s too. Philip found the stripes’ slip lengths in 1972. The post array’s scaling was established by Ybert and colleagues in 2007, from simulations and a scaling argument, and by Davis and Lauga in 2010 with a model of the menisci; the dilute disc’s constant used here is the leading term of their analyses. The surface Green’s function for a traction on a half-space of Stokes flow is the viscous counterpart of Cerruti’s elastic solution of 1882.

Still open: a meniscus that bows #

Every gas surface here is flat. A real one is curved by the pressure difference across it, and the liquid flowing over a curved meniscus meets a surface that dips between the posts. The next calculation gives each meniscus the small spherical sag its pressure and surface tension set, solves the flow over it to first order in the sag, and asks at what pressure the slip length has fallen by half — which would say how much of the slip a superhydrophobic surface loses before the liquid wets its texture, and whether posts or stripes keep their slip better under pressure.