An extra pipe can cut the flow

A number that is only the shape of the hole reduced a straight duct to one number: a resistance, set by its shape, in a Poiseuille flow. A loss with no viscosity in it found a different kind of resistance, a sudden expansion, whose head loss goes as the square of the flow whatever the viscosity. A pipe system is built from those two, and a network of them seems entirely straightforward. Kirchhoff’s two rules say that flows at a junction sum to zero and heads round a loop sum to zero, and Hardy Cross’s 1936 method solves any network on a sheet of paper.

So here is a question with an obvious answer. A network carries some total flow from a fixed head to a drain. Install a new pipe between two points in it, with a valve, and open the valve. What happens to the total flow?

The obvious answer has a good proof behind it, in the case of electrical resistors. A new path can only reduce the total resistance. Rayleigh proved it in 1899, and it’s not just a plausible rule. A proof is on the page below. It’s also false for a mixture of elements, and a small part of what follows is about how large the failure can be. The larger part is about the reason, which comes down to what a network is minimising.

The network #

Take two routes from a source to a drain, in opposite orders. One goes through a narrow pipe, with a laminar head loss $aq$, then past an element with a fixed head $c$ to clear, a lift or a weir, with a small loss $bq^2$ after that. The other route has the same two elements the other way round. By symmetry both routes carry the same flow, a bit less than $(H-c)/a$ each. Between the midpoints A and B, nothing moves: they stand at different heads, $H-aq$ and $aq$. So a pipe between them has something to do, and opening it looks like a free improvement.

Open it, with a low resistance. The midpoints now stand at the same head. By symmetry that head is $H/2$. And with $H<2c$, that head is below the lift’s height on both routes: A stands at $H/2<c$, so no water clears the lift from A to the drain; and the lift from the source to B has only $H-H/2=H/2<c$ available, so none clears that one either. Both lifts go dry. All the flow takes the only path left, through the first narrow pipe, across the bridge and through the second, in series, with

$$Q=\frac{H}{2a}.$$

With the bridge shut, the flow was twice $(H-c)/a$, with a small correction. The two are equal near $H=\frac43c$, and between there and about $2.3c$ the bridge has made things worse.

Just below $H=2c$, the flow falls from 1.82 to 0.995. Opening the valve has nearly halved the flow, and it’s lost nothing to any new resistance, since the bridge has none. Below the crossing, the bridge does what anyone would hope for. It lets flow start at a head too low to clear either lift: an open bridge gives flow from $H=0$ upward, while a shut one has none at all until $H=c$. And far above $2c$ it helps as well. Only in the band does it harm.

Why the water does not choose better #

The first thing to notice is that the bridged state is not optimal for anything. Close the valve and the flow is larger. Nothing forces the water to behave as if it’s a single entity trying to get as much of itself through as possible. Each parcel of water responds only to the local head difference, and the equations for that are Kirchhoff’s.

Road traffic works this way too. Braess’s paper of 1968 was about road networks, in which each driver chooses a route on their own, and the equilibrium is where no driver gains by changing route. A new road that seems to help each driver individually can leave them all in a state with longer journeys. Water parcels at a junction have no such thing as a choice, and yet the equations come out formally identical: head plays the part of journey time, and flow the part of traffic. Wardrop’s equilibrium of 1952, in which all used routes have equal cost, is the same statement as Kirchhoff’s second law.

Cohen and Horowitz built the version that has made the point best, in 1991. They made a network of two springs and two strings, hanging a weight. Cut a string that seems to hold things up, and the weight rises. A string plays the part of a constant-head element, with a fixed length, and a spring plays the part of a pipe with a resistance. A real microfluidic version, by Case and colleagues in 2019, used channels with a laminar regime and others with a non-laminar one, and they could switch flow directions by changing a bridge, a kind of logic gate.

What Rayleigh proved, and where it stops #

It’s worth seeing exactly why the theorem works, since it makes the exception obvious. A steady network with head laws $h_i(q_i)$ has a variational principle. Maxwell knew a version of it, and Millar gave the general form in 1951: the flows are the ones that minimise the total content,

$$G(q)=\sum_i\int_0^{q_i}h_i(q)\,dq-HQ,$$

over all flow patterns that conserve mass at each junction. It’s a convex minimisation, and it has one answer. Add a new pipe, and that minimisation has one more variable. Setting its flow to zero recovers the old network, so the new minimum of $G$ can’t be higher than the old. That’s all the new path does: it lowers the minimum of $G$.

Now, at the minimum, the energy identity says $\sum h_iq_i=HQ$. If every element has the same power law $h=kq^n$, then $\int_0^qh\,dq=hq/(n+1)$, and the sum simplifies:

$$G_{\min}=\frac{HQ}{n+1}-HQ=-\frac{n}{n+1}\,HQ .$$

So $G_{\min}$ is proportional to $-Q$. A lower minimum then means a larger $Q$. That’s Rayleigh’s monotonicity, for any single power law, and not only for electrical resistors. The bars below check it with both laws.

With a mixture, that proportionality goes, because an element’s contribution to $G$ per unit of $hq$ depends on its law. A laminar element counts half of its $hq$ in the integral. A quadratic one counts a third. A fixed lift, whose $h$ doesn’t vary at all, counts the whole of $hq$. So the minimisation is not of the flow but of a sum in which lifts are heavily weighted, and the principle will happily sacrifice flow to avoid paying for one. In the bridge case that’s exactly what happens. It drives both lifts’ flows to zero. The total head drop is the same and so is the driving force, but the system has found a pattern with less content by moving out of the lifts, which also has less flow. The network isn’t maximising its flow; it never was. Rayleigh’s theorem is what makes the two seem the same when a single law applies.

How large with ordinary pipes #

The lift gives the cleanest case. With only the two ordinary kinds of pipe, a laminar one and a quadratic loss, the content’s weights are a half and a third, a far smaller difference, and the effect should be much smaller. It is.

A fall of 2.4 per cent wouldn’t be noticed on most systems. It’s small, and it’s a genuine theorem-breaking fall. It exists only for a particular ratio of head to resistances, for which the laminar and quadratic losses are comparable, as the scaled axis says. It’s nearly lost in the turbulence of real components, and what makes the sign robust is that nothing else has to go right. In a hydraulic network with a head tank, a check valve or a siphon’s crest in one route, the lifts do appear, and the larger form of this applies.

Where would it matter? A constant-head element is common in practice: a spring-loaded non-return valve has a cracking pressure, and a weir or a siphon’s crest is a lift. Blood flow has its own version of this: a dilated collateral vessel near a stenosis can lead to reduced flow through a nominally improved route, which cardiologists call a steal. This isn’t Braess’s mechanism in any strict sense, and there’s more to it than a passive network, but it’s a warning of the same kind. A branching network like the airways or a river delta won’t be designed by adding paths until someone has checked. Road network planners have made a virtue of it. Seoul removed a six-lane road in 2005 and traffic did not get worse. Whether that was Braess’s effect at work has been argued about, and no study can run the same city twice.

The sign of the effect is therefore not a matter of how a pipe is sized. Nothing about the bridge in the figures is badly chosen; it could hardly be wider. It is the character of the elements on either side of it that decides.

Which networks can do this #

The bridge isn’t an accident of the construction. Milchtaich showed in 2006, for the same equilibrium in routing games, that a network can show this behaviour only if it contains a particular pattern: a Wheatstone bridge, in which a cross-link joins two routes partway along. A series-parallel network, built up entirely by combining elements end to end and side by side, is immune, and any element added in parallel at any stage only helps. Water running through a tree of pipes, and any manifold where branches simply divide, can’t have this at all. Murray’s branching is safe on that ground.

That’s a good design rule, with a clean statement. The dangerous pipe is one that crosses between two routes, and joins points at unequal heads, with those heads kept apart by elements of different characters. That’s exactly what a balancing valve or a cross-connection between two riser pipes in a building is. Whether such a link is safe needs to be checked with the laws that apply.

Less flow, and less power #

The energy account is worth reading as well. Nothing in a passive network produces anything: the head is fixed at both ends, and all of the power $HQ$ supplied is dissipated. At $H=1.99$ the shut network dissipates 3.61 units of power, and the open one 1.98. So opening a bridge has lowered the system’s dissipation by nearly half, along with its flow.

That’s a surprise against a familiar principle, and it looks at first like a confirmation of it. Helmholtz’s minimum-dissipation theorem, which applies to slow viscous flow, says the flow in a given geometry dissipates less than any other with the same boundary conditions, and it’s easy to read this bridge as another case. It’s not. Minimum dissipation holds a flow fixed and asks how it divides; here, a head is fixed, and for linear elements at fixed head a network dissipates more with each extra path, because it passes more flow. So for a resistor network a new path raises the dissipation, and the mixed network has done the reverse. What’s being minimised is $G$, and that isn’t the same as either the dissipation or the flow unless all of the laws agree.

On a bench #

The figures are in scaled units, and the experiment fits on a table. Take a pipe half a metre long and 1.5 millimetres in bore for each narrow element: its laminar resistance, in water, gives $a$ of about $4.1\times10^5$ seconds per square metre. For each lift, take a loop of wide tube that climbs 20 centimetres above its entrance and drains to the side. A reservoir 38 centimetres above the drain gives $H=1.9c$. With the bridge shut, the two routes together pass about 0.88 millilitres a second. Open a wide tube across the midpoints, and both lifts empty, and the flow drops to about 0.46. The pipe’s Reynolds number stays below four hundred throughout, so the laminar law applies, and the whole thing could be done with a bucket and a stopwatch. A person watching it would see the bridge begin to carry water, the loops drain, and the outflow visibly fall.

What the picture cannot show #

A real lift. The fixed-head element idealises a weir or spring valve. A real check valve’s cracking head varies with history and has its own hysteresis, and a real weir has a head loss that depends on its flow. The paradox needs only that the element’s loss not grow from zero in proportion to the flow, which is why a small quadratic term was kept: it doesn’t remove it.

Transient behaviour. A real valve opening doesn’t jump to the new steady state. A water column must decelerate first, and the transient may overshoot in a way a steady network can’t. The hammer a valve makes would be a larger concern than the steady state on a sudden opening.

Laminar-to-turbulent elements. A pipe with a flow near transition changes law as it carries more, which makes it a mixed element by itself and opens a range of cases. Two regimes in one pipe also create multiple steady solutions, of the kind Ledinegg found for boiling channels. The network here always has one solution, since the content is convex. A pipe whose head rises then falls with flow does not.

Reversal in the bridge. The bridge here takes whichever direction the heads ask. A valve that could only go one way would remove the paradox for one direction, and it would make the sign of the imbalance decisive. The figure’s symmetric arrangement hides that.

The convention #

Head is in units of the lift, $c$; flow in units of $c/a$; so a laminar element of resistance one drops one unit per unit flow. The lift’s small quadratic loss has $b=0.1$ in those units. In the mild figure, elements are $aq$ and $bq^2$, and the head is scaled by $a^2/b$, the head where the two would be equal at one flow. The law figure uses resistances 1 and 4 in the same pattern. Solutions are by nested bisection at the two nodes, whose monotonic balance guarantees the unique answer.

How it was checked #

The shut networks have closed forms, and the solution reproduces them to all figures. The free-bridge flow with lifts agrees with $H/2a$ to the three figures of the bridge’s own small resistance. The monotonic rise for each single law is the check on the theorem, and it was found at both exponents. The model refuses a head, a resistance, a lift or a bridge of zero or less.

Who found it, and when #

Kirchhoff’s rules are from 1845 and Rayleigh’s monotonicity from 1899. Hardy Cross’s method dates from 1936, and Wardrop’s equilibrium from 1952, with Millar’s content from 1951. Braess’s paper appeared in German in 1968. Cohen and Horowitz’s mechanical and electrical demonstration came in 1991, Calvert and Keady’s analysis of power-law networks in 1993, and Case and colleagues’ microfluidic form in 2019.

Still open: how often a random network has this #

A worst case is not a typical case. The next calculation would draw random networks of a few dozen pipes, with a realistic mix of laminar and quadratic elements and a few lifts, open each pipe in turn from a shut state, and count how often the total flow falls, and by how much. It would ask whether the share of paradoxical pipes grows or falls with the size of the network, whether it depends on the share of lifts or only on whether any exist, and so whether a working engineer should expect to meet this once in a career or in most systems, somewhere, by a tiny amount.