Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

The no-slip condition — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The air a wing carries along, and how little of it there is. The Blasius profile, in the wing's frame, with the free stream at one. No slip says the air at the surface is at rest relative to the surface, which in the ground's frame means it is moving with the wing — but only exactly at the wall. The deficit, integrated across the layer, is the displacement thickness: at a Reynolds number of 1e+6 and a metre of chord it is 1.72 millimetres of air moving at flight speed, which is the whole of what is 'carried'. The step drawn on the axis is that same deficit as a solid slab. A wing does not drag a blanket of air with it; it leaves a boundary layer behind it, and the layer is made of air that keeps being replaced.

    The air a wing does not carry

    No slip says the air touching a surface moves with it, and the usual reading is that a wing drags a blanket of air along. The blanket is 1.7 millimetres thick per metre of chord, it is different air every instant, and the drag it costs falls as it gets thicker.

    part 1 · misconceptions
  2. A force that depends on the molecular scale only through its logarithm. The force per unit length needed to move a 30° contact line of water at 1 mm/s, out to 1 mm, against the slip length on a logarithmic axis, from a picometre to a tenth of a millimetre. Each tenfold change in the slip length moves the force by the same fixed amount, so eight decades of the most uncertain length in the problem change the answer by a factor of about twenty — and at zero slip length the line keeps rising without end. The dashed curve takes the exact wedge's angle factor with a sharp cutoff; the solid one the thin-film wedge with Navier slip.

    The drop a no-slip wall would never let spread

    Liquid touching a solid moves with it, and nearly everywhere that is as close to exact as anything in fluid mechanics. At the edge of a spreading drop it cannot be: the stress in the corner rises as one over the distance from the edge, and the force needed to move the edge is infinite. Something slips over a nanometre, and because the answer depends on that length only through its logarithm, a drop spreads at almost the same rate whatever the something is.

    part 2 · misconceptions
  3. Where the inverse cube stops being an inverse cube. The force resisting a sphere's approach to a plane, against the gap, for four slip lengths. With no slip the force goes as one over the gap and has no limit. Any slip at all bends the curve over: below the slip length the force grows only as a logarithm. The curves separate where the gap falls through the slip length, which is exactly where a measurement has to be made and is why the apparatus has to close to nanometres rather than microns.

    A length you can only measure by squeezing

    No slip is a boundary condition rather than a law, and nothing in the equations requires it. The evidence that it is right for an ordinary liquid to within a nanometre comes from the one experiment whose small length is chosen rather than given — a squeeze film closed to tens of nanometres.

    part 3 · misconceptions
  4. Three dies, one melt, three flow curves. The apparent flow curve a capillary rheometer reports, for three dies whose radii differ by a factor of four. A material property cannot depend on the instrument, so the fact that these three do not lie on one another is a measurement of something other than a viscosity. The single curve drawn beneath them is what the same melt gives with the wall condition set to no slip.

    The wall a melt really does slip on

    For a simple liquid the slip length is under two nanometres, which is why no-slip is safe everywhere. For an entangled polymer melt it is a millimetre, which is larger than the die it is being pushed through — so the melt moves as a plug and the viscosity a rheometer reports is a property of the instrument.

    part 4 · misconceptions
  5. A slip law implies a slip length that changes with the rate. The slip length — slip velocity over the true wall shear rate — against the wall shear rate, for a melt whose slip follows a velocity law in stress with exponent 1.5, 2 or 3, all matched to slip at 2 mm/s at a stress of 100 kPa, and for a melt with a constant slip length, dashed. With the exponent of 2 the implied length falls from 0.22 mm to 0.08 mm across the range of rates; only a law whose exponent is exactly 1/n, 2.5 here, implies a constant length.

    A melt's slip length has an exponent it cannot choose

    A polymer melt slips at the wall, and there are two ways to say by how much: a slip length, fixed, so the slip velocity follows the wall's shear rate; or a slip law, the slip velocity as a power of the wall's stress. The literature moves between them as though they were the same, and for a melt they are not. A constant slip length forces the slip law's exponent to be exactly one over the melt's flow index — 2.5 for a typical melt — so a measured exponent of 2 is a slip length that changes with the rate. And temperature separates them outright: a tenfold change in viscosity moves the slip velocity tenfold under one description and three-hundredfold under the other.

    part 5 · misconceptions
  6. A layer read as slip gives a slip length that drifts. The slip length a drainage measurement would fit at each gap to the force of a 1-nanometre viscous layer, using the slip formula, for layers three and ten times as viscous and immobile. Far off it is the first-order value, −0.667 nm for κ = 3; closer, it shrinks towards zero — −0.597 nm at 10 nm and −0.525 at 5. A slip length that depends on the gap is the layer's signature, and the only one a drainage curve carries.

    A viscous nanometre reads as slip until the gap is ten of it

    A drainage experiment turns a molecular length at a wall into a measurable force, and reports it as a slip length. A layer of liquid a nanometre thick and several times as viscous as the bulk changes the same force by the same amount, to first order. Solved with the stratified viscosity, the two pictures part by a per cent only when the gap is eleven to seventeen layers wide, and the part of the difference a measurement can see is a slip length that drifts with the gap. A layer that does not move at all is exactly a wall in a different place.

    part 6 · misconceptions
  7. Elasticity adds a spring at first order and a slip only at second. The damping's deficit, one minus the damping ratio, and the in-phase force against Λ on logarithmic axes. The in-phase force rises in proportion to Λ — a fitted slope of 1.0000 — and the damping's deficit as its square, 2.0000. A slip length changes the damping at first order and adds no spring at all, so a soft wall and a slipping one are distinct in kind: the soft wall announces itself first in the phase of the force.

    A soft wall is read as slip only at second order

    A drainage force smaller than Taylor's has been read as the liquid slipping at the wall. A wall that gives under the drainage pressure lowers the force too, and it can be mistaken for slip. Solved together, the flow and the elastic wall say how: the softness first adds a spring to the force, in phase with the motion, and only at second order takes anything from the damping a slip length is read from. When it does, the slip it imitates grows as the square of the frequency and falls as the square of the gap.

    part 7 · misconceptions
  8. A mica sheet shields the glue's false slip only where the pressed region is narrower than the sheet bends. The slip length, in nanometres, that a dynamic drainage measurement at 100 Hz would read from a wall that does not slip, against the gap: a 3 mm sphere in water over bare glue of plane-strain modulus 1 GPa, and over mica sheets 1, 3 and 5 µm thick on the same glue. At 3 nm the bare glue reads 9.48 nm and the sheets 8.75, 2.68 and 1.06; at 30 nm, 0.1 against 0.099, 0.067 and 0.043.

    A mica sheet hides its glue only at the smallest gaps, and changes how the false slip falls

    A drainage measurement against a soft wall reads a slip that is not there, falling as the square of the gap. The wall in a surface-force apparatus is not a soft half-space but a stiff mica sheet on a soft glue, and the sheet spreads the drainage pressure over its own bending length before the glue feels it. Where the pressed region is wider than that length the sheet changes nothing; where it is narrower the sheet sinks as one piece, a spring in series with the liquid. That cuts the false slip only by the square of the bending length over the pressed width — threefold at three nanometres for three micrometres of mica — and it turns the slip's fall with the gap from the inverse square towards the inverse, which is the test an experimenter would have used to see it.

    part 8 · misconceptions

All series