Concept

Darcy's law — where it appears

The proportionality between a porous medium's seepage velocity and the pressure gradient driving it, with the permeability as the constant. It holds while the pore Reynolds number is small, and what it reports is a flux per unit total area rather than the speed of anything.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

$Two velocities, 2.50 apart, and only one of them is anybody's. The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included: a speed no fluid particle ever has, since the fluid occupies only the fraction ε of that area. The speed the fluid actually averages is larger by exactly 1/ε — 2.50 times here — and it is the one that belongs in a residence time, in a pore Reynolds number and in any statement about when a tracer arrives. The grains are drawn to say that the pore-scale flow is not computed here: nothing above resolves it.$

A velocity nobody has

The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included — a speed no fluid particle in the bed ever has. Averaging buys a linear law and charges for it in exactly this coin, and the constant that comes with it is an area of about a square micron.

applied · Porous

Two terms, one with viscosity in it and one without. Ergun's two contributions to the pressure gradient, against the pore Reynolds number, on logarithmic axes. The viscous term rises with the first power of the velocity and the inertial one with the square, so on these axes they are straight lines of slope one and two and there is exactly one crossing. The second term contains no viscosity at all — it is the price of accelerating fluid into every pore and out again — which is why a linear resistance law has to fail eventually whatever the fluid is.

Where Darcy stops

A linear resistance law has to fail eventually, because pushing fluid into a pore and out again costs energy that has nothing to do with viscosity. Where it fails is one dimensionless number, and that number is 150/1.75 — read out of a correlation's own constants rather than measured.

applied · Porous

Two different moments of one distribution. The permeability and the specific surface of a log-normal bundle, against the width of its pore-size distribution at a fixed median. The permeability rises by five orders because it is a fourth moment and the widest tubes dominate it; the surface falls because it is a first moment and the narrowest tubes dominate that.

A permeability that is only the geometry

Kozeny–Carman says that a porous medium's permeability follows from its porosity and its specific surface. Both are real, both are exactly measurable, and they do not determine the answer: forty-nine tube bundles built with identical values of each span a factor of eighty-two in permeability.

applied · Porous

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.

inviscid · Hele-shaw

Time per litre rises in a straight line with the litres already filtered. The elapsed time divided by the filtrate collected, against the filtrate, for a constant-pressure filtration of a silica slurry through a cloth: integrated step by step, and Ruth's closed form. The line's slope is the cake's resistance, which grows with every litre, and its intercept is the cloth's, which does not. After a quarter of a cubic metre the cake is already the larger.

A filter is slowed by what it has caught

Push a slurry through a cloth and within minutes the cloth no longer matters: the particles it has caught form a cake, the cake is a packed bed, and every litre filtered makes the bed thicker for the next. The filtrate grows only as the square root of time. And if the cake is soft, it packs tighter against the cloth the harder it is pushed, until more pressure buys almost nothing.

applied · Porous

A washed filter's best cycle is shorter, and still has one. The filtrate a batch filter produces per hour, averaged over filtering, washing and twenty minutes of emptying and refilling, against the filtrate collected in each cycle, for a slurry of 400 kilograms of silica per cubic metre and wash ratios of 0, 1, 2 and 5 pore volumes. Each curve has a best cycle, marked, and the wash both lowers it and moves it to a thinner cake: from 0.307 to 0.217 m³ per m² of cloth.

A washed filter works as long as it rests

A batch filter that has to be emptied between runs has a best cycle, and the rule for it is exact: the time the cake is responsible for equals the time the filter stands idle. Washing the cake does not break the rule, because a wash through a finished cake costs time in proportion to the square of its thickness, just as filtering it did. What a wash changes is the price, and the price is set by the slurry's concentration and by how far the wash front smears, not by the cake's softness.

applied · Porous

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.

inviscid · Hele-shaw

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.

inviscid · Hele-shaw

Named alongside it

The objects these essays reach for when they reach for this one.

Model limit Packed bed Porosity Hele-shaw Permeability Scaling Capillary number Compressibility Correlation Creeping flow Ergun's equation Linear stability

All concepts