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

Hele-shaw — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

    The exact theory, drawn by viscosity

    Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

    part 1 · inviscid
  2. The speed past the flank, in the cell and in the ideal flow it is meant to show. The gap-averaged speed along the top of a cylinder, against the distance from its surface in cylinder radii, for three cells — the cylinder's radius 10, 35 and 100 times the Brinkman length h/√12 — and for ideal flow. Ideal flow slides along the wall at twice the stream speed. The cell's flow must stop at the wall and does so in a layer a few Brinkman lengths thick, outside which it rejoins the ideal curve.

    The cell draws a larger cylinder

    A Hele-Shaw cell draws the streamlines of ideal flow past an obstacle and cannot obey the one rule ideal flow breaks: the fluid must stop at the obstacle's wall. It does stop there, in a layer a third of the gap thick, and far away the whole correction amounts to one thing — the cell is drawing ideal flow past a cylinder one layer-thickness too big, and the obstacle has exactly that cylinder's drag.

    part 2 · inviscid
  3. Long waves grow, short ones are held flat, and one in between grows fastest. The growth rate of a ripple on the interface where air displaces oil in a Hele-Shaw cell, against its wavenumber, at displacement speeds of 0.5, 1 and 2 mm/s. The viscosity contrast drives every wavelength at a rate proportional to its wavenumber; surface tension holds back short ones as the cube. At 1 mm/s the cut-off is 0.245 per mm and the fastest wavenumber 0.141 per mm, a wavelength of 44.4 mm, growing at 0.0942 per second.

    The fastest finger is set by the gap and one number

    Push air into oil between two glass plates and the interface breaks into fingers. The averaged law of the cell makes the instability exact in its linear stage: every wavelength is driven in proportion to its wavenumber, surface tension holds back the short ones as the cube, and the fastest finger is π gap widths divided by the square root of a capillary number. A narrow channel stays flat; a wide one chooses how many fingers to make; and a growing bubble of air makes more of them the larger it gets.

    part 3 · inviscid
  4. The meniscus shortens the fastest finger and its film lengthens it, and they cancel near Ca = 0.01. The fastest wavelength over the uncorrected π b/√Ca, against capillary number, with the in-plane tension reduced to π/4 alone, with the speed-dependent transverse jump alone, and with both. The reduced tension shortens every finger by √(π/4) = 0.8862; the film's jump lengthens it, more at higher capillary number; together they cancel at Ca = 0.0131, shorten the finger 5.8% at 10⁻⁴ and lengthen it 6.1% at 0.3.

    The meniscus slows the finger and hardly moves it

    Air pushing oil through a narrow gap fingers at a wavelength the gap and one number set, on the assumption that surface tension acts only on the interface's curve in the plane. The interface is really a meniscus across the gap, leaving a film behind, and Park and Homsy's correction for it does two things that nearly cancel in the wavelength: it cuts the tension to π/4 and adds a drag that grows with speed. At bench speeds the fastest finger barely moves; it grows a quarter more slowly, and a channel goes unstable at four-fifths of the speed.

    part 4 · inviscid

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