A number that is only the shape of the hole

For fully developed laminar flow in a straight duct of any cross-section, the pressure gradient, the flow rate and the geometry are related by one dimensionless number:

$$\mathrm{Po} = f\,Re = \frac{2\,A\,D_h^2}{\displaystyle\int \hat{w}\,dA}, \qquad \nabla^2\hat{w} = -1,$$

with $\hat{w}$ the velocity scaled on the pressure gradient and the viscosity, and $D_h = 4A/P$ the hydraulic diameter.

Look at what is in that expression: an area, a perimeter, and the mean of the solution of one Poisson equation with one boundary condition. No fluid, no size, no speed, no pressure. Everything physical has been scaled out before the equation is solved, so what is left can only be geometry.

The consequence is a quantity that is genuinely a property of a shape and nothing else, which is rare enough to be worth checking rather than believing — and rare enough that it is routinely assumed to be more general than it is.

Ten shapes, ten numbers #

Three of them have closed forms and are exact: 64 for a circle, 96 for parallel plates, and 160/3 for an equilateral triangle. The rectangles come from an exact Fourier series — the square is 56.9083, which is the 56.91 every handbook prints — and the ellipses from a closed form of their own.

The spread from the equilateral triangle to parallel plates is exactly $96/(160/3) = 1.8$.

Those are the shapes a single correlation treats as equivalent. Scaled to a common hydraulic diameter — which is the scaling the correlation asserts makes them the same — they carry, for the same pressure gradient and the same fluid, flow rates that differ by a factor of 1.8.

Why it separates at all #

The separation is worth deriving once, because it is what makes the whole quantity possible and it takes four lines.

Fully developed means nothing changes along the duct, so the axial momentum equation reduces to $\mu\nabla^2 w = dp/dx$ — a Poisson equation for the velocity in the cross-section, with $w = 0$ on the wall. Everything on the right is a constant, so scaling $w = (-dp/dx)\hat{w}/\mu$ removes it, and what is left is $\nabla^2\hat{w} = -1$ with no parameter in it whatever.

The mean velocity is then $(-dp/dx)/\mu$ times the mean of $\hat{w}$, so the friction factor — which is the pressure gradient non-dimensionalised on the dynamic pressure — carries a factor of $1/\bar{W}$, and the Reynolds number carries a factor of $\bar{W}$. Their product has neither, and the fluid and the speed have both cancelled.

That is the whole mechanism, and it is the reason the number exists rather than merely being convenient. It also says exactly what breaks it: anything that puts a second term in the momentum equation. Inertia does, in the entrance region; buoyancy does, in a heated pipe; a non-Newtonian viscosity does, because then $\mu$ cannot be taken out of the Laplacian — which is why a yield-stress fluid’s bracket is a function of a second parameter rather than a pure number.

It really does depend on nothing else #

Vary the radius over eight orders of magnitude and the number does not move — not to a part in $10^{15}$, exactly, because the radius cancels between the area, the hydraulic diameter and the mean velocity. That is what “depends only on the shape” means when it is computed rather than asserted.

The closed forms are checked against a Poisson solve on a masked grid. Two things went wrong there first and both are worth recording, because both produce a plausible answer.

A perimeter counted off a mask is a staircase. Summing the exposed cell faces of a discretised circle gives the Manhattan perimeter, which exceeds the true one by up to $4/\pi$ however fine the grid is — it does not converge. On the 2:1 ellipse it gave 12.00 against Ramanujan’s 9.69, and the Poiseuille number that came out was 43 against the closed form’s 67.3.

And a Dirichlet condition applied at grid points rather than at the boundary is first order. Every section came out about one per cent low, systematically, at every resolution: 63.39 for a circle where the answer is exactly 64. A single grid would have looked converged. Two grids extrapolate to 64.04.

One per cent is nothing for a picture. It is a great deal for an essay whose subject is that these numbers are exact, which is why the closed forms are the source here and the grid is the check.

Which kind of number this is #

It is worth placing the 64 among the four kinds of pure number this collection distinguishes, because the placement says how far to trust it.

It is an integral: the mean of a solved field. So it is worth as many digits as the solve is converged to — which for the three closed forms is all of them, and for the rectangles is however many terms of the series are kept. That is a better position than an optimum, whose location is worth only the square root of its objective’s precision, and about the same as a root.

The classification also explains why the grid solve needed extrapolating. An integral of a solution inherits the solution’s error, and a first-order boundary treatment gives a first-order error in the integral — so a single grid is wrong at first order and two grids extrapolate. Had the number been a root, the same grid error would have entered through a residual and been divided by a slope; had it been an optimum, it would have been square-rooted. The kind of number decides how a numerical error propagates into it, which is the practical content of where a pure number comes from.

What the shape is doing #

Flattening a rectangle takes it from 56.91 towards 96, reaching 95.35 at two hundred to one — approaching the parallel-plate limit from below and slowly, which is the signature of a Fourier series with a corner in its boundary.

Flattening an ellipse does not do the same thing. Its Poiseuille number rises past 96 and keeps going, because an ellipse tapers to a point at its ends where the flow is slow and the perimeter is still being counted. Two families of shapes, the same hydraulic diameter, and different answers — which is as direct a refutation of the hydraulic-diameter idea as the subject offers.

What the number is measuring is how much of the section is close to a wall. Plotting it against the perimeter over the square root of the area — a dimensionless shape factor with no flow in it — orders the sections almost perfectly. A slot puts nearly all of its fluid within a short distance of a wall and shears nearly all of it; a circle puts a core of fluid far from every wall and shears that core hardly at all. It is everything happening in a thin layer with the layer being the whole duct.

The three exact ones, and why they are exact #

Three of the ten have closed forms, and the reason each does is instructive.

The circle is exact because the Poisson equation separates in polar coordinates and the solution is a parabola: $\hat{w} = (a^2 - r^2)/4$, whose mean is $a^2/8$. That gives 64 immediately, and it is the parabola every pipe carries.

Parallel plates are exact for the same reason in one dimension, giving 96.

The equilateral triangle is exact for a reason that has nothing to do with either. Its three boundary lines can be written as three linear functions $L_1, L_2, L_3$ that each vanish on one side, and the product $L_1L_2L_3$ therefore vanishes on all three — so it is a candidate solution, and because the Laplacian of a cubic is linear, a multiple of it satisfies $\nabla^2\hat{w} = -1$ exactly. That trick works for exactly three polygons: the equilateral triangle, the 45–45–90 triangle and the 30–60–90 one. It does not work for a square, which is why the square needs a series.

And the ellipse is exact by the same device with a quadratic: $1 - x^2/a^2 - y^2/b^2$ vanishes on the boundary and has a constant Laplacian.

So “having a closed form” is not a property of being simple. It is a property of the boundary being the zero set of a low-order polynomial, which is a strong and rather arbitrary condition — and it is why a square, which is simpler to draw than an ellipse, is harder to solve.

What the hydraulic diameter costs #

Using 64 for parallel plates under-predicts the pressure drop by 33 per cent; for a twenty-to-one rectangle by 30; for a square by 12.

That is not scatter. It is a different number, known to four figures, and it is available in every handbook — the difficulty is that the same handbook also contains the hydraulic-diameter shortcut, and the shortcut is right in the turbulent regime that occupies most of the chapter.

The reason for the difference between the regimes is worth having. In turbulent flow the velocity profile is nearly uniform across most of the section and all the shear is in a thin wall layer, so the resistance really is set by the wetted perimeter and the area — which is what the hydraulic diameter encodes, and why it works. In laminar flow the shear fills the whole section, so the distribution of the area relative to the perimeter matters and one length cannot express it. The hydraulic diameter is not an approximation that degrades at low Reynolds number; it is an approximation whose justification is the wall layer, and in laminar flow there is no wall layer to speak of.

The same separation for heat, and a second number per shape #

The scaling argument is not about momentum. Run it on the energy equation for a fully developed laminar duct flow and the same thing happens: everything physical scales out, and what is left is a Nusselt number that is a pure property of the cross-section — an output, not an input, and tabulated once for all time.

With one complication the momentum problem does not have. The velocity has a single boundary condition, no slip, and there is nothing to choose. The temperature has a choice: a wall held at a constant temperature, or a wall supplying a constant heat flux. Those are different problems with different answers, so every shape has two Nusselt numbers rather than one — 3.66 and 4.36 for a circular pipe, 7.54 and 8.24 for parallel plates, and about 3.0 and 3.6 for a square.

Putting the two families side by side gives the compact-exchanger trade directly. Going from a circular passage to a slot at the same hydraulic diameter costs half again in pumping — 96 against 64 — and buys nearly double the heat transfer. Which of those is worth more is the whole design question, and neither number is 64.

Where it actually matters #

A factor of 1.8 in laminar resistance sounds academic, and it is not, because the places laminar duct flow occurs are places where the section is deliberately not circular.

Compact heat exchangers are built of narrow rectangular or triangular passages, precisely because a non-circular section puts more perimeter — more heat-transfer area — against a given flow area. The pressure drop those passages cost is what limits how far the idea can be taken, and it is set by the number in the table rather than by 64.

Microchannels are almost always rectangular, because that is what etching produces, and their aspect ratios are frequently ten to one or more. A design using the circular value there is out by about thirty per cent in pressure drop, which for a pump on a chip is the difference between working and not.

And the passages inside a turbine blade are irregular polygons at Reynolds numbers that can be laminar at altitude. The same arithmetic applies and the same shortcut is available to be misused.

In all three the honest procedure is short: identify the section, look up or compute its Poiseuille number, and use it. The number is exact, it is tabulated for every common shape, and computing a new one is a single Poisson solve. There is no reason to use 64 except habit — and the habit is inherited from the turbulent chapter, where it is right.

A better single length, if one is wanted #

The hydraulic diameter is one attempt to reduce a shape to a number, and it is not the only one. If a correlation with a single length is wanted for laminar flow, there is a better choice than $4A/P$.

The quantity $\sqrt{A}$ — the square root of the area — is the natural length of a cross-section in the sense that the Poisson problem cares about: the mean of $\hat{w}$ scales as $A$, so scaling on $\sqrt{A}$ removes the size and leaves only the aspect. A Poiseuille number defined on $\sqrt{A}$ rather than on $D_h$ collapses the ten sections here considerably better, which is the content of the shape-factor figure above.

That is not a new observation — it has been rediscovered several times in the compact-heat-exchanger literature — and it has not displaced the hydraulic diameter, for a reason worth naming. The hydraulic diameter is right in the turbulent regime and wrong in the laminar one; the square root of the area is the other way round. Neither is right in both, because the two regimes are governed by different geometry: turbulent resistance is a wall-layer phenomenon and cares about the perimeter, laminar resistance fills the section and cares about the area.

A single length that worked for both would have to know which regime it was in, which is to say it would have to be a function of the Reynolds number — at which point it is not a length, it is a correlation, and the honest thing is to use two.

One line to keep #

The Poiseuille number is one of very few quantities in this subject that is genuinely and only a property of a shape, and the reason is worth carrying: the governing equation has no parameter in it. Everything physical was scaled out before anything was solved, so what came back could only be geometry.

Whenever that happens — and it happens rarely — the resulting number is exact, is tabulated once for all time, and is a legitimate thing to look up. Whenever it does not, a number that looks like a shape factor is a shape factor at one set of conditions, and the conditions have to be quoted with it.

Telling the two apart takes one glance at the equation, and it is worth the glance.

What is not claimed #

Fully developed, laminar, straight, constant-section. The entrance region has its own behaviour and its own length, which is how far before a duct forgets; a curved duct develops secondary flow and a quite different resistance; and a duct whose section changes along it is not covered at all.

Constant properties. The viscosity is taken as uniform across the section, which fails as soon as the duct is heated or cooled: a temperature-dependent viscosity makes the Poisson equation nonlinear and the number a function of a heating parameter, which is the Brinkman number’s territory rather than this one’s.

Smooth walls and no slip. Roughness does nothing in laminar flow — which is the roughness a wall cannot feel, and is a separate exact statement — but slip at the wall, in a microchannel or with a rarefied gas, changes the number directly.

The rectangle’s series is truncated at two hundred terms, which is converged for the aspect ratios shown and is not a closed form. The 200:1 value of 95.35 is approaching 96 slowly enough that its own convergence is worth checking before it is quoted further.

The annulus and the sector are absent. Both have closed forms, both are common in practice — an annulus is what a double-pipe exchanger is — and both were left out because their Poiseuille numbers depend on a second parameter (the radius ratio, the included angle) and are therefore functions rather than numbers. That is not a difficulty for the physics; it is a difficulty for a table.

And the hydraulic diameter is not the only convention. Some references define the friction factor with a factor of four (the Fanning factor), which turns 64 into 16 and is the single commonest source of factor-of-four errors in this part of the subject — a version of the length in the number with the same cure, which is to say saying which one is meant.