The only law that forbids it

Take the jump conditions of the previous rung and put a subsonic Mach number into them. Nothing complains. Out comes a discontinuity in which the gas accelerates, its pressure falls, its density falls, and its temperature falls: an expansion shock.

Substituting that solution back into mass, momentum and energy gives residuals of exactly zero, zero and zero. It is a solution.

It also does not happen, anywhere, ever.

The numbers, so the claim can be checked #

A jump from Mach 0.8, computed from the same relations that give the Mach 2 shock in the previous essay:

Mass balance residual: 0. Momentum: 0. Energy: 0. Every one exact to the last bit of a double, which is a stronger statement than the compression case manages.

The entropy change is $\Delta s/R = -0.01638$. It has gone down.

Every other check on this site would pass it #

The point of this essay is not that entropy forbids the expansion shock, which is in every textbook. It is what that fact does to a site built the way this one is.

This site’s discipline is that a claim is given a test it could fail, and that the test is a computation. Mass conservation is asserted on every field. Tangency at a wall is asserted. Circulation is checked against the Kutta condition, and lift by two independent routes. Every one of those would pass the expansion shock without a murmur, because every one of them tests a conservation law and the expansion shock conserves everything.

There is also nothing to see. The figure of an expansion shock would show a line, with gas on one side and different gas on the other, and every number on it correct. It would be indistinguishable from a figure of a real shock viewed in a mirror.

So the refusal has to be written down explicitly, as an assertion whose content is thermodynamic rather than mechanical, and assertShockRaisesEntropy is that assertion. Given a supersonic Mach number it requires the entropy rise to be positive and the total pressure ratio to be below one. Given a subsonic one it fails, with a message saying which laws are satisfied — because a check whose failure message reads “impossible” teaches nothing, and one that reads “mass, momentum and energy are all satisfied by this and the second law is not” teaches the whole lesson.

What entropy is doing here #

The second law is not a conservation law and does not behave like one. It is an inequality, and it selects among solutions rather than producing them.

For a perfect gas the entropy change between two states is

$$\frac{\Delta s}{R} = \frac{1}{\gamma - 1}\ln\frac{T_2}{T_1} - \ln\frac{\rho_2}{\rho_1} = -\ln\frac{p_{02}}{p_{01}}$$

and that second form is the useful one: entropy rise and total-pressure loss are the same statement. A shock that gained total pressure would be a device that lowered entropy, and running gas through it repeatedly would produce work from nothing.

That is why the expansion shock is not merely unobserved but impossible in the strong sense. It is not that the mechanism happens not to be available; it is that its existence would let anybody build a perpetual motion machine of the second kind out of a convergent duct and a supply of air.

The mechanical argument, and why it is not the reason #

There is a satisfying physical story about why compressions steepen and expansions spread, and it is in the previous rung: the back of a compression wave sits in hotter gas and gains on the front, while the back of an expansion wave sits in cooler gas and falls behind.

That story is correct and it is not the reason. It describes a mechanism by which a compression becomes discontinuous, and mechanisms of that kind explain how something forms rather than whether it is allowed. An expansion shock could in principle be produced by some other route — a piston withdrawn suddenly, a diaphragm burst, a clever arrangement of ducts — and the wave-steepening argument says nothing about those cases.

The entropy argument covers all of them at once, without knowing anything about how the jump was made. That is the characteristic strength of a thermodynamic argument and the characteristic reason it is unsatisfying: it rules out an enormous class of possibilities without describing any of them.

Where the refusal actually bites in practice #

It is tempting to file this under curiosities, since nobody sets out to build an expansion shock. The refusal earns its place for a different reason: it is the check that catches a sign error.

A solver that computes shocks has a direction convention in it somewhere, and getting it backwards is easy — the relations are symmetric in appearance and only asymmetric in what they mean. A shock solver with the two sides swapped returns states that satisfy every conservation law, produce smooth plausible curves, and are wrong.

That is why assertShockRaisesEntropy runs on every shock this site draws rather than only in the gate. It is not guarding against somebody deliberately inserting an impossible wave; it is guarding against the direction convention being wrong, which is the sort of error that has already happened once here with the sign of a circulation and produced an aerofoil that flew downwards while passing all its own checks.

The code that produces one anyway #

This essay’s central situation — a solution satisfying every conservation law, refused only by thermodynamics — has an exact counterpart in computation, and there the impossible solution is not a thought experiment. It gets computed.

The Euler equations are a hyperbolic system of conservation laws, and such systems admit weak solutions: fields with discontinuities in them, satisfying the conservation laws in an integral sense. Weak solutions are not unique. For a given initial state there are generally several, they all conserve mass, momentum and energy exactly, and the mathematics needs an extra condition to say which one nature takes. That condition is the entropy condition, and it is the same second law, written as a requirement that an admissible discontinuity have characteristics running into it rather than out of it.

A numerical scheme built on the conservation laws inherits the non-uniqueness. It is designed to conserve — that is what makes it capture the right shock speeds — and conservation is exactly the property the impossible solution also has. So unless something is added, the scheme has no reason to prefer one weak solution over another.

And some schemes duly compute the wrong one. Roe’s approximate Riemann solver, which is exactly conservative and is one of the most widely used methods in compressible aerodynamics, will admit a stationary expansion shock at a sonic point: a discontinuity through which the flow accelerates, sitting still in the mesh, perfectly converged, with residuals at machine zero. Every conservation check the code makes passes. The picture looks like a shock in the wrong place.

The remedy has a name — an entropy fix — and it consists of adding a small amount of dissipation precisely where the scheme’s own eigenvalue passes through zero, which is exactly where the ambiguity lives. It is not a numerical accuracy measure and it does not improve the resolution of anything. It is the second law, put into the code by hand, because the discretised conservation laws do not contain it.

That is this essay’s argument arriving as a line of source. The equations permit both branches, the arithmetic cannot tell them apart, every mechanical check passes either, and something outside the mechanics has to be added explicitly or the wrong one is returned. A solver written from the conservation laws alone is a solver that has forgotten which way time runs — and the correction is a few lines long, is switched off by default in some codes, and is one of the standard explanations when a supersonic calculation converges to something that looks nearly right and is not.

The exception that proves the rule is real #

A genuinely surprising footnote: in certain fluids, expansion shocks do exist.

The reason the ordinary argument works is the shape of the isentrope, and specifically the sign of $\partial^2 p / \partial v^2$ along it, which for a perfect gas is positive. Bethe and Zel’dovich noticed independently, in the 1940s, that in a fluid with a sufficiently large heat capacity near its critical point that second derivative can become negative — and in such a fluid an expansion discontinuity raises entropy while a compression one lowers it. Everything reverses.

These are called BZT fluids, they are typically heavy organic vapours, and rarefaction shocks in them were observed experimentally only in the twenty-first century. The delay is telling: the prediction was clear, and finding a fluid that could be held in the right region of its phase diagram for long enough was a fifty-year experimental problem.

The lesson for reading this site: “$\gamma = 1.4$, perfect gas” in a figure’s regime note is not boilerplate. It is the hypothesis under which the second law forbids what this essay says it forbids, and there exist real substances for which the conclusion is the other way round.

The direction of time, drawn #

The expansion shock is the closest thing in fluid mechanics to a clean laboratory example of time’s arrow, and it is worth putting the point plainly.

Film a compression shock and run the film backwards. What is on the screen is an expansion shock, and it satisfies every mechanical law in the subject: Newton’s laws are reversible, and mass, momentum and energy are conserved in both directions. Nothing a mechanic could measure distinguishes the film from reality.

Yet the reversed film is obviously wrong to anybody who watches it, and what makes it wrong is that disorder is decreasing. That is the whole of the second law’s content in this problem: it is the only statement in the description that is not indifferent to which way the film runs.

Creeping flow provides the counterpart elsewhere on this site — a flow that genuinely is reversible, in which the fore-and-aft asymmetry is measured at $10^{-16}$ and running it backwards returns every particle to where it started. The two essays together bracket the question: reversibility is a property some flows have and some do not, and the difference is a computable number rather than a philosophical position.

The weak shock, and the third power #

There is one more thing the entropy expression says, and it is quantitative rather than a matter of principle.

Expand $\Delta s$ for a shock of small strength. The leading term is not linear and not quadratic in $(M_1 - 1)$; it is cubic. A shock at Mach 1.01 produces an entropy rise of about $10^{-7}R$, which is an irreversibility so small that the process is isentropic for every practical purpose.

So the second law’s prohibition is absolute and its penalty is graded, and it goes to zero faster than the shock strength does. That is the whole design argument for supersonic intakes, and it appears again on the oblique-shock ladder where the turns are geometric rather than normal.

An entropy layer where nobody put one #

There is a consequence of the entropy rise that goes well beyond the shock itself, and it is the reason blunt-body supersonic flow is a computational problem rather than an analytical one.

A curved shock imposes a different normal Mach number on every streamline that crosses it, because each meets it at a different angle. So each streamline emerges with a different entropy rise — and therefore with a different total pressure, permanently.

The layer of high-entropy, low-total-pressure gas that flows back along the body from the nose region is called the entropy layer, and it does two awkward things. It carries vorticity — the entropy gradient generates it, by Crocco’s relation, in a fluid with no viscosity at all — so the flow behind a curved shock is not irrotational and no velocity potential exists for it. And it changes the state of the gas the boundary layer grows into, which changes the heat transfer to the wall.

A single number, computed from a jump condition, and the consequence is that a whole class of flows cannot be attacked with the machinery the rest of this site is built on.

What the solver computes, and how it is checked #

normalShock returns the jump at any Mach number, including subsonic ones, deliberately.

That decision is the interesting one. It would be easy to make the solver refuse a subsonic Mach number at source, and it would be worse: the whole content of this essay is that a solution exists and is refused, and a solver that cannot produce it cannot demonstrate that.

So normalShock computes it, rankineHugoniotResidual verifies that it satisfies the conservation laws, and assertShockRaisesEntropy refuses it. The gate exercises all three in sequence: it asserts that the residuals are below $10^{-12}$, asserts that $\Delta s < 0$ and $p_{02}/p_{01} > 1$, and then asserts that the refusal happens.

The rejection test for the other direction is the mirror image: a compression shock doctored to claim $\Delta s = -10^{-9}$ and a total pressure ratio slightly above one, which is what a subtly wrong implementation would produce, and which the check refuses.

Where the model stops #

A perfect gas with constant specific heats. The BZT exception above is precisely a failure of this assumption, and it reverses the conclusion rather than blurring it.

Equilibrium thermodynamics. Entropy is defined here for states in equilibrium, and the interior of a shock is not in equilibrium. That is not a problem for this argument, because the argument compares the two equilibrium states on either side and never mentions the interior — but it does mean the phrase “entropy inside a shock” has no meaning in this framework.

Single-species, non-reacting. Dissociation, ionisation and chemical reaction all contribute entropy of their own, and above about Mach 5 in air they dominate.

What the picture cannot show #

Everything, in this case, and that is why the essay exists.

Entropy has no picture. The figures here plot it as a curve — a number against a number — and that is as close as any of them get. There is no arrangement of streamlines, contours, colours or arrows that distinguishes a state of higher entropy from one of lower entropy, because entropy is not a local mechanical property of a fluid element in the way pressure and velocity are.

The consequence for this site is stated plainly in the figure rules: every gate here asks whether a label fits, contrasts and stays inside the viewBox, and none of them can ask whether the flow it labels is thermodynamically permitted. The expansion shock is the standing example of a figure that would pass every check and be false, and it is the reason the refusal lives in the solver where a build can stop on it, rather than in a caption.

Who found it, and when #

Rankine tried, in 1870, to rule out the expansion branch by an argument about heat conduction inside the shock, and it does not work: the argument concerns the interior, and the jump conditions were derived without reference to the interior.

The clean statement came from the thermodynamics rather than from the mechanics. Rayleigh in 1910 and G. I. Taylor in the same period settled it: the compression branch raises entropy and the expansion branch lowers it, and only the second law distinguishes them. Both were explicit that no mechanical argument would do.

The forty-year gap is the point. During it the relations were known, correct and used, and half of their solution set had no accepted justification for being discarded.

Where the ladder goes next #

The shock is allowed, in one direction, and it charges for the passage. What exactly it takes and what it leaves is the accounting that decides how every supersonic intake in existence is laid out — and the answer is that the heat all survives and the ability to use it does not.