Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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 3 essays across one field — 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 anywhere: no figure on this site claims to resolve 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

Named alongside it

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

Packed bedCorrelationErgun's equationPermeabilityPore reynoldsPorosityAveragingCreeping flowDimensionless numberForchheimerInertiaKozeny carman

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