Twice as slippery along as across

A wall that is not quite there replaced a porous wall’s whole structure with a single length, the depth below the real surface at which a flow far above would find a solid wall. It closed by saying that over a patterned surface the replacement is not a length but a tensor — different along the pattern and across it — and that in a turbulent flow the difference between the two is what matters. It did not compute the tensor.

This essay does, for the simplest patterned surface there is. Lay parallel stripes on a wall, alternately solid and gas: the solid ones impose no slip, and the gas ones — each a narrow clean interface, which can hold no shear stress — impose none. Call the period $L$ and the fraction of it that is gas $\varphi$. A shear flow far above the stripes sees none of that pattern, only a flat wall some distance below the real one, and the question is how far.

Two answers, and the ratio between them #

Philip found both answers in closed form in 1972. For a shear along the stripes, the effective wall is

$$b_\parallel = \frac{L}{\pi}\,\ln\sec\frac{\pi\varphi}{2}$$

below the surface, and for a shear across them it is

$$b_\perp = \frac{L}{2\pi}\,\ln\sec\frac{\pi\varphi}{2} = \tfrac12\,b_\parallel .$$

The dots do not use Philip’s formulas. They come from solving the two flows directly: the velocity above the stripes is written as the uniform shear plus the slip length plus a sum of the pattern’s harmonics, each decaying upwards, and the coefficients are chosen so that the velocity vanishes on the solid and the shear vanishes on the gas. The along-stripe answer agrees with Philip’s to five parts in ten thousand at the fractions drawn, and the across-stripe answer is half of it — not approximately, but in every digit the arithmetic carries.

Why the ratio is exactly two #

That exactness has a reason, and it is short enough to give in full.

Along the stripes the flow is a single velocity component $w(x,y)$, and in Stokes flow it satisfies Laplace’s equation. The harmonics that decay upwards are $\cos(kx)\,e^{-ky}$, so

$$w = y + b_\parallel + \sum_n c_n \cos(k_n x)\,e^{-k_n y}.$$

On the solid the velocity is zero; on the gas its gradient is zero, which is $1 - \sum k_n c_n \cos k_n x = 0$.

Across the stripes the flow has two components and a streamfunction $\psi$, which satisfies the biharmonic equation. The harmonics that decay upwards and leave no flow through the surface are $y\cos(kx)\,e^{-ky}$, so

$$\psi = \tfrac12 y^2 + b_\perp y + \sum_n B_n\, y\cos(k_n x)\,e^{-k_n y}.$$

The velocity along the surface is $b_\perp + \sum B_n \cos k_n x$, and it must vanish on the solid. The shear at the surface is $1 - 2\sum k_n B_n\cos k_n x$ — with a two that the along-stripe problem did not have, because differentiating $y e^{-ky}$ twice at $y = 0$ gives $-2k$ where differentiating $e^{-ky}$ once gives $-k$ — and it must vanish on the gas.

Now set $B_n = c_n/2$ and $b_\perp = b_\parallel/2$. The solid condition becomes half the along-stripe one, which is still zero. The gas condition becomes $1 - \sum k_n c_n \cos k_n x$, which is the along-stripe one exactly. Both sets of conditions are met, so that is the solution, and the across-stripe slip length is half the along-stripe one for any arrangement of stripes at all — any widths, any spacing, any number of different stripes in a period — because nothing in the substitution used the positions. The physical reading is that a flow across the stripes has to rise over each solid ridge and fall into each gas trough to keep its volume, and the returning part of that motion doubles the shear each unit of slip costs.

What the far flow is averaging #

The liquid does not slip uniformly. On the solid it does not slip at all, and over each gas stripe it slips most at the centre, furthest from the ridges holding it back, with a square-root rise from each edge. The average over a period of that surface velocity, divided by the shear rate, is exactly the slip length: the uniform part of the velocity at the surface is $b$, and every harmonic averages to zero.

How uneven the slip is depends on the gas fraction, and not in the direction one might guess. At a gas fraction of a half, the centre of each gas stripe slips at 0.280 of the shear rate times the period while the average is 0.110: the peak is two and a half times the average, because half the surface is holding the liquid still. At nine-tenths gas the centre slips at 0.813 against an average of 0.594, a peak only 1.4 times the average. The more of the surface is gas, the more nearly the liquid slides as a single sheet over it, and the less the average hides.

That makes plain what the effective condition throws away. A flow far above sees the average and not the peaks; a particle carried along the surface, or a chemical reaction happening on it, sees the peaks. The slip length is a correct summary for the first and a meaningless one for the second.

How far above the stripes the pattern is felt #

The summary is correct only far enough above the surface, and the harmonics decaying upwards say exactly how far.

Each harmonic decays as $e^{-2\pi n y/L}$, so the pattern fades on its own wavelength. Half a period above the stripes the largest departure from the averaged flow is six per cent of the slip length, and a full period up it is a quarter of a per cent. An effective slip condition is exact for any flow whose own length scale is more than a period or two, and for a flow in a channel only a period deep the channel’s height and the pattern interact, and the slip length is smaller than the semi-infinite answer.

Where the logarithm comes from #

A logarithm in a Stokes-flow answer is a signature, and it is worth recognising. Stokes flow in two dimensions has no length of its own, so the disturbance a body makes decays only very slowly with distance, and quantities that depend on how the flow far away is matched to the flow near the body pick up the logarithm of the ratio of the two sizes. The best-known case is a cylinder in a flow with no inertia, whose drag has a logarithm in it and which has no solution at all without one.

The solid stripes on a nearly all-gas surface are the same kind of object. As the solid fraction shrinks, each stripe becomes a thin strip of no slip in a surface that otherwise holds nothing back, and the liquid flowing over it is disturbed over a region as wide as the period. The strip’s hold on the liquid depends on the ratio of that region to the strip’s own width, and it enters as a logarithm for the same reason the cylinder’s drag does: two lengths, a two-dimensional Stokes flow between them, and nothing else to set a scale. The slip length is inversely related to that hold, and so it grows as the logarithm of the period over the strip.

The logarithm is therefore not a feature of stripes being badly designed. It is what any line-shaped holding structure does in a slow flow, and only a structure that is not a line — a post — escapes it.

A logarithm, not a fraction #

The formula’s most consequential feature is the logarithm, and it contradicts the obvious design rule for a slippery surface — that more gas means more slip, without limit.

As the gas fraction approaches one, $\sec(\pi\varphi/2)$ grows as $4/\pi(1-\varphi)$, so the slip length grows as the logarithm of one over the solid fraction. A surface that is nine-tenths gas slips 0.59 of a period. Making it 99 per cent gas — a tenth of the solid — adds 0.73 of a period. Making it 99.9 per cent gas, which is a structural feat, adds another 0.73. The slip is bought by the period: the formula is proportional to $L$, so doubling the pattern’s size doubles the slip at any gas fraction.

The period cannot be doubled for free either, and the reason is the gas. Each gas stripe is held in place by a meniscus spanning the gap between two ridges, and a meniscus of width $w$ can hold back a pressure difference of about $2\sigma/w$ before the liquid pushes into the gap. A liquid’s hold on a pore is the same arithmetic in the other direction. For water a gap of ten micrometres holds about fourteen kilopascals and a gap of a hundred about one and a half, so a surface designed for more slip by coarsening its pattern floods at a lower pressure. The design trade is period against pressure, and the gas fraction is a weak third variable.

What the numbers mean in a channel #

Slip lengths in periods are easy to compare and hard to feel, so it is worth putting one surface into one device. Take stripes with a period of twenty micrometres, nine-tenths of each period gas. The along-stripe slip length is 0.59 of a period, or 11.8 micrometres, and the across-stripe one is 5.9.

Put that surface on one wall of a channel a hundred micrometres high, with an ordinary solid wall opposite, and drive a liquid through it with a pressure gradient. The channel is five periods high, comfortably inside the range in which the averaged condition is exact, and for a single slipping wall with slip length $b$ the flow rate rises by the factor

$$\frac{Q}{Q_0} = \frac{H + 4b}{H + b},$$

which follows from the parabolic profile with one end moved to $-b$. Along the stripes that is 1.32 — a third more flow for the same pressure — and across them 1.17. The same wall, the same liquid and the same pressure give a flow rate an eighth larger in one direction than the other, and a gradient applied at forty-five degrees to the stripes drives a flow turned about three and a half degrees towards them.

The effect falls away quickly as the channel grows. In a channel a millimetre high the along-stripe gain is three and a half per cent, and in one a centimetre high it is a third of one per cent. A slip length is a length, and it matters only where the flow’s own length is not many times larger — which is why the whole subject of superhydrophobic drag reduction for laminar flow lives in microfluidics, and why its promise for ships lives entirely in the turbulent argument below, where the comparison is with a viscous length of tens of micrometres rather than with the size of the hull.

A boundary condition that turns the flow #

A scalar slip length can only make a flow faster in the direction it is already going. A tensor can turn it.

Written as one object, the condition is $\mathbf{u}_s = \mathsf{b}\cdot\partial\mathbf{u}/\partial n$, with $\mathsf{b}$ a two-by-two tensor whose axes lie along and across the stripes and whose values on those axes are $b_\parallel$ and $b_\perp$. Rotating the stripes rotates the tensor and changes nothing else; its trace and its determinant, $\tfrac32 b_\parallel$ and $\tfrac12 b_\parallel^2$, are properties of the surface, not of how it is laid.

For a shear at an angle θ to the stripes, the slip velocity has a component $b_\parallel\cos\theta$ along them and $b_\perp\sin\theta = \tfrac12 b_\parallel\sin\theta$ across them, so it points closer to the stripes than the shear does. The liquid at the surface is steered along the pattern.

The turning is largest for a shear at 54.74° to the stripes, where $\tan\theta = \sqrt2$, and there it is 19.47°. That number contains no gas fraction and no period: it comes entirely from the ratio of two, so every striped surface of this kind, at every scale, turns a flow by the same largest angle. In a thin channel with a striped wall set at an angle to the pressure gradient, that turning drives a flow across the channel as well as along it, which is how striped and grooved walls are used to stir a flow too slow to mix itself. A pattern that needs to turn a flow harder has to change the ratio, and flat stripes cannot.

The difference a turbulent flow cares about #

The essay before this one gave the turbulent version of the argument in Luchini’s form: over a textured wall, the shift in a turbulent boundary layer’s velocity profile — which is its drag reduction — depends not on either slip length but on the difference between them, $h_\parallel - h_\perp$, measured in viscous lengths.

For flat stripes that difference is now known exactly. It is the same kind of virtual origin that a rough wall shifts the profile by, measured in the direction that matters. It is $b_\parallel - b_\perp = \tfrac12 b_\parallel$, half the along-stripe slip length, provided the period is small compared with the viscous length that sets the thickness of the layer nearest the wall, so that the flow over one period is still Stokes flow. A striped surface whose along-stripe slip length is four viscous lengths offers a difference of two, and a turbulent layer over it sees its profile shifted by about two viscous velocity units — a laminar calculation feeding a turbulent result, with the factor of two carried straight through.

Checked by a calculation that never uses the answer #

The collocation converges slowly, and the rate is itself information. Each doubling of the number of harmonics halves the error — first-order convergence — where a smooth problem would converge far faster. The slowness is the stripe edges: the velocity rises from a no-slip solid into a shear-free gas with a square-root corner that no finite sum of cosines can reproduce, and the error is dominated by that corner. Because the rate is known, the extrapolation from two counts removes most of it, and the extrapolated values meet Philip’s formula to a few parts in a hundred thousand.

What the flat-stripe model leaves out #

The meniscus is flat. A real gas pocket’s surface bulges into the liquid or sags into the gap depending on the pressure, and a curved meniscus changes the slip — a meniscus bulging outwards slips less than a flat one, because the liquid has to flow round it.

The meniscus is clean. The whole calculation assumes the gas–liquid interface transmits no stress. Surfactants sweep to the downstream end of each gas stripe and immobilise it exactly as they immobilise a bubble’s surface, and measured slip on real superhydrophobic surfaces often falls far below the clean prediction for that reason.

The pattern is stripes. A two-dimensional array of posts rising through a gas layer is a different problem, in which the solid fraction appears as a square root rather than a logarithm, and posts slip much further than stripes of the same solid fraction.

The flow is semi-infinite and slow. A channel a few periods high sees a smaller slip length, and a flow fast enough that inertia matters over one period is no longer Stokes flow there. And the gas is taken to be at rest in its pocket, where a real one circulates slowly and transmits a small stress.

Philip, and the tensor #

Philip’s 1972 paper solved flows with mixed no-slip and no-shear conditions — the most extreme version of the second condition a wall can be given, switched between its two limits along the surface — for stripes in two orientations, and derived the logarithm. The recognition that the two answers belong to one tensor, and that for stripes the tensor’s axes differ by exactly two, is from work in the 2000s on flow over superhydrophobic surfaces, by which time the surfaces could be made and the slip measured, and the gap between the clean prediction and the measurement had become the interesting part.

Still open: posts, and slip that depends on the flow #

Stripes give a logarithm because a stripe is a line: the no-slip solid extends without end in one direction and holds the liquid back along its whole length. Posts are points, and a liquid flowing past an array of them is held back far less, so the slip length of a post array grows as one over the square root of the solid fraction instead — a much stronger dependence, and a surface far more slippery at the same solid fraction. Computing it needs a genuinely two-dimensional pattern, where the substitution that gave the factor of two has no analogue.

Beside it is the question the slip-length idea hides in every case here: whether the liquid-gas surfaces stay clean, stay flat and stay in place as the flow runs over them. Slip at a molecular scale is a property of the gas’s own collisions and does not wash away; slip on a textured wall is a property of a few million menisci, and depends on every one of them surviving.