The push that has no direction

The pressure that only knows depth establishes that the pressure in a still fluid depends on nothing but how far down the point is. That statement contains a hidden claim: that there is one number to state.

Force divided by area is not obviously one number. Take a small flat plate at a point in a fluid and measure the force on it; turn the plate through ninety degrees and measure again. Nothing in the definition says the two must agree, and in a solid they do not.

The wedge, and the cancellation in it #

The proof is due to Stevin and is nearly four hundred years old, and its structure is worth more than its conclusion.

Isolate a small wedge of fluid. It is at rest, so the forces on it sum to zero. The only forces are the pressures on its three faces and its own weight — a still fluid supports no shear, which is the one property being used and is what makes a fluid a fluid.

The horizontal balance is the clean half. The sloping face is longer than the upright one by $1/\cos\theta$, and the horizontal component of the force on it is smaller by $\cos\theta$. The two factors cancel exactly, so the balance reads $p_{\text{upright}} = p_{\text{slope}}$, at every angle, for every size, with no limit taken.

The geometry cancels. That is the whole argument, and it works because a pressure force is normal to the face and proportional to its area — which is a statement about fluids and not about geometry.

Why the limit is needed after all #

The vertical balance does not cancel, because the wedge has weight. The bottom face has to carry the pressure on the sloping face plus the weight of the fluid above it, and those two pressures are not equal.

The excess falls in exact proportion to the size of the wedge, and the figure verifies the exponent rather than quoting it. Halve the wedge and the surface forces quarter while the weight goes down by eight, so the ratio halves.

Surface effects beat body effects at small sizes, and this is one of the more consequential instances of a rule that runs through the whole subject. It is why surface tension dominates a small drop and gravity a large one, why an insect can stand on water, and why a scale model of a ship gets its waves right and its drag wrong.

Here it means the theorem is exact at a point and approximate anywhere else — and the approximation is extraordinarily good, because the ratio at any size a laboratory cares about is already a millionth.

What a motion does to it #

The theorem’s one assumption is that the fluid supports no shear, and that assumption is about a fluid at rest. A moving fluid supports shear — that is what viscosity is, and it is momentum going sideways — so the argument does not apply.

The picture of what replaces it is the Mohr circle, and the contrast is stark. A fluid at rest occupies a single point on the diagram: whichever way a plane is turned, the normal stress is the same and the shear is zero. A sheared fluid occupies a circle, whose radius is the viscous stress; now the normal stress depends on the orientation, and there are planes carrying shear.

There is still a natural candidate for “the pressure” — the centre of the circle, which is the average normal stress over all orientations — and it is what the symbol $p$ means in the Navier–Stokes equations. But it is now a definition rather than a theorem, and the difference matters: the stress at a point in a moving fluid needs six numbers, and the pressure is one combination of them chosen because it is the one that survives when the motion stops.

How much it matters in practice #

The second circle is the practically important case and it is nearly a point.

Water at a shear rate of five per second has a viscous stress of five millipascals. The pressure at ten metres is two hundred thousand times that. So the departure from isotropy in ordinary flowing water is a part in a hundred thousand, which is why hydraulics, meteorology and naval architecture all treat the pressure as one number without ever mentioning that they are entitled to.

The exceptions are where the viscosity is large or the pressure is small. In a polymer melt being extruded, the normal stress differences are comparable with the pressure and are the reason the extrudate swells when it leaves the die — the liquid that climbs the rod is the same effect made visible. In a rarefied gas the pressure can be low enough that viscous stresses compete. And in a granular material the whole notion fails, because the silo that does not weigh what it holds is a statement about a material that supports shear even at rest.

What the theorem buys #

It is worth being explicit about what would have to be carried around if the theorem were false, because the saving is enormous and is entirely invisible in a subject that has always had it.

Without isotropy, describing the state of a still fluid at a point would need six numbers rather than one, and a field of six numbers rather than a scalar field. Every statement about hydrostatics would become a statement about a tensor field. The condition for equilibrium would be that the divergence of that tensor balances the weight, which is three equations rather than one, and the pressure at a point would depend on the history of how the fluid got there.

With it, all of that collapses. One scalar field, one equation — the gradient of the pressure equals the weight per unit volume — and the answer depends only on depth. Force multiplied and nothing gained is a direct consequence: the pressure applied at one piston reaches the other undiminished precisely because there is one number and it is transmitted without regard to direction, which is Pascal’s principle and is a corollary of the theorem rather than an independent law.

A theorem that reduces six numbers to one is worth proving carefully, and the fact that the proof takes three sentences is the reason it usually gets none. The cost of the reduction is stated exactly: it holds at a point, in a fluid, at rest, and each of those three qualifications has a figure here showing what happens without it.

How small a point has to be #

The last figure closes a circle that is worth noticing, because it makes a contradiction dissolve.

Two things are said about pressure in a still fluid and they look incompatible. It is uniform in every direction at a point, so a body immersed in it feels a push that is the same all over. And it varies with depth, so the push on the bottom of a body exceeds the push on its top, which is exactly why the body floats.

Both are true and the reconciliation is scale. Across a cell the variation is a part in ten million and the fluid is uniform for every purpose. Across a fish it is a part in a few thousand — small, and it is the whole of the buoyancy, since the weight of the water that is not there is precisely that difference integrated over the surface.

A quantity can be negligible and load-bearing at once, and which it is depends on what else is being compared with it. The buoyancy is a small fraction of the pressure and it is not competing with the pressure; it is competing with the body’s weight, and against that it is everything.

The same argument in a solid, and how it fails #

Running the wedge argument on a solid is the quickest way to see which step is doing the work.

Isolate the same wedge inside a stretched steel bar. The forces on its faces are still surface forces, its weight is still a body force, and the scaling argument still applies — so in the limit the surface forces must balance alone. That much survives.

What does not survive is the step before it. In a fluid at rest the force on a face is normal to it, which is why the balance turned into a statement about one number per face. In a solid the force on a face has a shear component too, so each face carries two unknowns rather than one, the balance has more freedom, and the conclusion is much weaker: it says that the stress on a plane of any orientation is fixed by the stresses on three perpendicular planes, which is Cauchy’s theorem and is the definition of the stress tensor.

Isotropy is then the special case in which that tensor is a multiple of the identity — six independent numbers collapsing to one. A fluid at rest is not a material with a simple stress; it is a material whose stress happens to be the simplest possible one, and the reason is the definition of a fluid rather than anything about the geometry.

That also says exactly what a material has to fail to do for the theorem to hold: it has to be unable to hold a shear indefinitely. A solid can. A liquid cannot. A granular heap can, up to a limit, which is why it has an angle of repose and why a silo’s walls carry part of the load — a material somewhere between the two, and neither theorem applies to it cleanly.

Where the model stops #

The fluid is at rest and has no memory. A viscoelastic liquid that has recently been sheared carries residual stresses even when the motion has stopped, and the point on the diagram takes time to collapse. The liquid that remembers is that delay.

Surface tension is absent. At a free surface, or across a curved interface, there is a genuine stress that is not isotropic and does not vanish with the size of the region, because it lives on a surface rather than in a volume. Every argument here applies to the interior of a fluid.

Only mechanical equilibrium is assumed, not thermal. A fluid with a temperature gradient is still isotropic in this sense, but its pressure is not simply $\rho g h$, and separating the two is where atmospheric statics starts.

And the wedge argument assumes a continuum. At a scale approaching the mean free path the pressure is a statistical quantity with fluctuations, the fluctuations grow as the region shrinks, and the limit the proof takes cannot be taken all the way. In air at atmospheric pressure the trouble starts below about a micrometre.

The wedge is assumed to be in equilibrium. A fluid element that is accelerating is not, and the balance acquires a mass-times-acceleration term — which scales as the cube of the size, like the weight, and therefore drops out in the same limit. So the theorem survives acceleration and fails only for shear, which is a sharper statement than it first appears: pressure is isotropic in a fluid element that is falling, spinning, or being shaken, and stops being isotropic only when neighbouring elements are sliding past one another.

And nothing here is a statement about the pressure being positive. A liquid can be put under tension — the pressure becomes negative, the isotropy is untouched, and the fluid holds until it cavitates. The height a siphon cannot pass is the practical limit of that, and it is set by nucleation rather than by anything in this argument.

The measurement that has no orientation to choose #

A last consequence, and it is the one that makes hydrostatics usable in a laboratory.

A manometer is a tube of liquid whose two ends are at different pressures, and its reading is a height difference. Nothing about how the tube is bent, how it is oriented, or what shape its bore is enters the answer — only the vertical distance between the two free surfaces. That is a strong statement and it depends entirely on the theorem: if the pressure at a point had a direction, the reading would depend on which way the tube ran at each point along it, and the instrument would have to be calibrated for its own geometry.

The same is true of every hydraulic system. A brake line can take any route through a car and the pressure at the far end is the pressure at the near end plus $\rho g h$ for the net height change, and the route is irrelevant. Engineers rely on that without stating it, and it is a theorem about a wedge.

An instrument whose reading does not depend on its own shape is unusual, and where one exists there is generally a conservation law or a symmetry underneath it. Here it is the isotropy, and the isotropy is a scaling argument.

What the pictures cannot show #

The wedge figure draws two dimensions and the theorem is three-dimensional. Cauchy’s version uses a tetrahedron and shows that the stress on a plane of any orientation is fixed by the stresses on three perpendicular ones — a much stronger statement, from which the isotropy of a fluid is a corollary. The wedge shows the mechanism and understates the result.

The Mohr circle is a picture of a symmetric two-dimensional stress and the real object has six components. What the circle cannot show is that in three dimensions there are three principal stresses rather than two, so a fluid at rest is a point only because all three coincide — and a state with two equal and one different, which is what a uniaxial stress is, has no two-dimensional picture at all.

One more thing outside the figures is what an instrument actually reads. A pressure gauge has a diaphragm of finite area and finite stiffness, and what it returns is the average normal stress over that area on that orientation — which is the pressure only because the theorem says every orientation agrees. In a flowing fluid the gauge’s reading depends on how it is mounted: a tapping flush with a wall reads something close to the static pressure, and a tube facing upstream reads the static pressure plus the dynamic one. The difference between those two readings is how a Pitot tube measures speed, and it exists only because the isotropy that makes a still fluid’s pressure unambiguous has been broken by the motion.

That is worth carrying because it inverts the usual reading of the instrument. A Pitot tube is not measuring speed directly; it is measuring the amount by which the fluid has stopped being isotropic, and converting that into a speed through Bernoulli’s relation.

Where the ladder goes next #

The hydrostatics ladder began with the pressure that only knows depth, passed through force multiplied and nothing gained and the surface a spin decides, and reached the height a siphon cannot pass. This rung asks why there was one number to be uniform in the first place. The rungs after it: the stress tensor, of which pressure is the isotropic part; the bulk viscosity, which is the pressure’s own departure from the thermodynamic one during a compression; and the constitutive relation, which is where the assumption that a fluid supports no shear at rest becomes a definition of what kind of material is being described.

The habit worth carrying away is that a scaling argument can prove a theorem rather than merely estimate one. Two forces that scale differently cannot both matter in the limit, and identifying which one survives is often the whole of a proof — and always the whole of knowing when it stops applying.