Pressure and fluid statics
A fluid at rest cannot carry a shear stress, which sounds like a small restriction until you notice what it rules out. Every force this fluid exerts on anything — a wall, a gate, a submerged body, you — has to be a pressure, pushing square onto the surface it touches. Statics without shear turns out to be a subject of its own.
Everything in fluid properties led to one restriction: a static fluid supports no shear, at any angle, however small. What is left for it to carry is pressure, a normal force spread over a surface, and this article is entirely about how that pressure behaves — how it varies with position, how it is measured, and how it adds up through layers of different fluid. None of it needs the fluid to be moving. That is what makes it statics.
Pressure at a point
The first thing worth proving, rather than assuming, is that pressure at a point in a static fluid has no preferred direction — push a tiny probe into still water and it reads the same whichever way it faces. This is Pascal’s law, and the proof is a short piece of bookkeeping rather than a physical argument, which is why it is worth actually walking through once.
Isolate a small wedge-shaped element of fluid at rest, with three faces at some arbitrary angle to each other, and write down force balance in two directions. Each face carries a pressure force normal to itself — call them \(P_x\), \(P_y\) and \(P_s\) on the slanted face — and the element also carries its own weight, acting straight down. Resolve the geometry, and the weight term comes out proportional to the volume of the wedge, while every pressure term comes out proportional to an area.
That difference in scaling is the whole argument. As the wedge is shrunk toward a point, its linear size \(\ell\) falls, its areas fall as \(\ell^2\), and its volume falls as \(\ell^3\) — faster. The weight term becomes negligible next to the pressure terms long before the wedge has actually vanished, and what remains of the force balance says simply
however the wedge was oriented. Pressure at a point in a fluid at rest is therefore a single scalar, not a vector with a direction of its own — a fact used without comment in every calculation that follows, and one that stops holding the moment shear stresses reappear, which is the whole difference between statics and the rest of this course.
Isolate a wedge of fluid at rest, however small.
Pressure and depth
Pressure has no direction at a point, but it is certainly not the same at every point. Cut a thin horizontal slice out of a fluid at rest, of thickness \(dz\) and cross-sectional area \(A\), and ask what holds it up. The pressure on its underside pushes up, the pressure on its top pushes down, and the difference between the two has to support the slice’s own weight, because nothing else is available to do it — no shear exists to help. Writing that balance out and taking the limit as \(dz \to 0\) gives the hydrostatic equation:
with \(z\) measured upward. The minus sign says exactly what intuition already knows — go up, pressure falls; go down, pressure rises — but it is worth having it fall out of the algebra rather than being asserted. For an incompressible liquid, \(\rho\) is constant, and the equation integrates directly. Written in terms of depth \(h\) below a free surface at pressure \(P_0\),
Pressure increases linearly with depth, at a rate set entirely by \(\rho g\) — the same specific weight from the previous article, doing exactly the job its name suggests. The equation carries a second consequence that is easy to state and easy to forget under pressure: because \(P\) depends only on \(z\), any two points at the same depth in the same connected body of fluid are at the same pressure, however far apart they are horizontally, and however twisted the path of fluid between them. A U-shaped tube, an irregular tank, a submarine’s ballast line — none of the shape matters, only the vertical distance. Surfaces of constant pressure in a static fluid are horizontal planes, full stop, and that single fact is what makes a manometer readable at all.
One qualification earns its own sentence. The argument assumed a single, connected, constant-density fluid. Cross into a different fluid, or interrupt the connection with a solid partition, and the simple statement “same depth, same pressure” stops applying across that boundary — though, as the last section of this article shows, the hydrostatic equation itself still applies separately within each fluid, and the results still chain together in a perfectly systematic way.
In a single connected fluid, pressure depends on depth alone.
Absolute and gauge pressure
\(P = P_0 + \rho g h\) is only as good as the reference pressure \(P_0\) chosen at the top, and engineering practice uses two different zeros without always saying which. Absolute pressure is measured from a true, perfect vacuum — the lowest pressure that can physically exist — and it is the pressure that belongs in every equation of state and every thermodynamic relation. Gauge pressure is measured from local atmospheric pressure instead, because almost every pressure instrument you will ever hold is built with the atmosphere already pressing on one side of its sensing element, so that is what it reads zero against.
A gauge pressure can be negative — a pressure below atmospheric — and when it is, it is usually reported as a positive vacuum pressure instead, \(P_{\text{vac}} = P_{\text{atm}} - P_{\text{abs}}\), simply so the number itself does not carry a confusing sign. Mixing the two conventions inside one calculation is a standing source of factor-of-two and sign errors: a condenser running at “10 kPa” is almost certainly 10 kPa absolute, deep in vacuum, and treating that as a gauge reading of 10 kPa above atmosphere gets the physical situation backwards entirely.
Atmospheric pressure itself is not a fixed constant — it drifts with weather and falls steadily with altitude — and its value is measured with a barometer, an instrument that is really just the hydrostatic equation built out of glass and mercury. Fill a tube with mercury, seal one end, and invert it into an open reservoir of the same mercury. The liquid falls a short way, leaving behind a sealed vacuum at the closed top — mercury’s own vapour pressure there is negligible enough to ignore — and the column that remains is held up by nothing except the atmosphere pressing down on the open reservoir at the bottom.
Apply the hydrostatic equation between the reservoir surface, at atmospheric pressure, and the top of the column, at essentially zero, and the column height reads atmospheric pressure directly:
At sea level this comes out to a column about 760 mm tall, which is exactly why atmospheric pressure is so often quoted in millimetres of mercury out of sheer habit, long after the instrument itself has been replaced by something electronic. Mercury’s enormous density is not an accident of history — it is what keeps the column short enough to fit on a desk; water would need one over ten metres tall to do the same job.
Absolute pressure is measured up from a perfect vacuum.
Manometers
A barometer measures one pressure against a vacuum. A manometer measures one pressure against another, and it is the workhorse instrument of this whole subject precisely because it needs nothing but a bent tube, a working liquid, and the hydrostatic equation applied twice. Connect one leg of a U-tube to the unknown pressure and leave the other open to the atmosphere, or to a second unknown, and the liquid inside redistributes itself until the two columns balance whatever pressure difference is being applied.
Reading it is an exercise in walking the hydrostatic equation down one leg and back up the other, adding \(\rho g h\) for every drop in elevation and subtracting it for every rise, and insisting that the two routes agree at any point they share. For the simplest case — one fluid, open to atmosphere on one side — the bookkeeping collapses to
where \(\Delta h\) is the difference in height between the two liquid surfaces, not either one measured on its own. This is the detail that catches nearly everyone the first time: the absolute height of either column depends on how much liquid happens to be in the tube, which is not a property of the pressure being measured at all. Only the difference between the two legs is. A manometer with columns at 40 cm and 43 cm reads exactly the same pressure as one with columns at 4 cm and 7 cm, provided \(\Delta h = 3\) cm in both.
The working fluid is chosen to suit the pressure range. Mercury, dense and non-wetting, keeps the reading compact for large pressure differences and for absolute measurements like the barometer above; water or a light oil, far less dense, is used deliberately when the difference being measured is small, because it spreads the same pressure difference over a much longer, more legible column. Choosing a mercury manometer to resolve a pressure difference of a few pascals is a real mistake, not a stylistic one — the column movement would be smaller than you could read off the tube.
Open at both ends, a U-tube's two columns sit level.
Layered fluids
Nothing about the hydrostatic equation required there to be only one fluid. It requires only that within any given fluid, density is uniform, so that \(P = P_0 + \rho g h\) applies separately across each layer, with the pressure at the base of one layer becoming the reference pressure \(P_0\) for the layer beneath it. Immiscible liquids of different density settle into exactly such layers on their own, densest at the bottom, and a stack of oil over water over mercury is a genuine physical arrangement, not a textbook contrivance.
Finding the pressure at the base of such a stack is nothing more than chaining the single-layer result:
Work down from the top, one layer at a time, and each term depends only on that layer’s own density and thickness — the layers above it have already been accounted for in the running total you carry into it. Reverse the order, or try to apply a single average density across the whole stack, and the answer comes out wrong unless every layer happens to have the same density, which defeats the point of layering them in the first place.
The same chaining handles a manometer with two different liquids in it, or a tank with an immiscible layer floating on top of the process fluid, or the ballast and fuel layers in a ship’s tank — anywhere pressure has to be tracked through more than one density on its way to an answer. It is also the calculation that reappears, largely unchanged, once the surfaces involved stop being flat: finding the total force on a submerged gate or a dam wall, the subject of the next article, starts by integrating exactly this pressure distribution over an area instead of summing it down a column.