Pressure, Pascal's Law and Manometry

A fluid at rest can carry no shear, because a shear would mean it was deforming, which would mean it was moving. So the only stress left is normal to any surface you imagine through the fluid.

Take a small wedge of fluid and balance forces on it: the normal stress comes out the same on every face regardless of orientation. That is Pascal's law in its first sense — pressure at a point has a magnitude and no direction.

This is why the force on a submerged gate is perpendicular to the gate at any angle, and why you can talk about "the pressure at 3 m depth" without saying which way you are facing.

Balance the weight of a thin column of fluid against the pressure on its ends and you get dp/dz = −ρg. For constant density that integrates to p = ρgh below the free surface. Every metre of water adds 9.81 kPa.

Only depth appears. Not volume, not container shape, not the width of the tube. A one-centimetre pipe three metres tall pushes on its base exactly as hard as a swimming pool three metres deep — the hydrostatic paradox.

A pressure change applied to an enclosed fluid transmits undiminished to every point of it. Since force is pressure times area, a large piston at the same pressure delivers a proportionally larger force.

1. 1Push a 1 cm² piston with 10 N. The pressure rise is 100 kPa.
2. 2That 100 kPa appears throughout the connected fluid.
3. 3A 100 cm² piston at 100 kPa produces 1000 N.
4. 4A hundredfold force multiplication — but the small piston must travel a hundred times further.
5. 5Work in equals work out. It is a lever made of liquid, not a free lunch.

Every car brake, hydraulic digger and press in the world runs on this one sentence.

There are no special formulas per instrument. Start where the pressure is known, walk the tube to where you want it, and apply two rules with one permission.

• Going down through a fluid: add ρgh.
• Going up through a fluid: subtract ρgh.
• Pressure is constant along any horizontal line through a *single continuous* fluid — which is what lets you cross the U-bend.

Walk a U-tube from the open end and everything cancels except the height difference of the heavy fluid. That is why only h appears in the answer, and why the tube's shape between the ends is irrelevant.

Mercury is used because it is 13.6 times denser than water. A pressure that would need a 3.4 m water column shows up as 250 mm of mercury — readable on a bench.

A pressure gauge measures the difference from local atmospheric pressure, because the atmosphere acts on the back of its diaphragm too. p_abs = p_gauge + p_atm.

A tyre "at 220 kPa" holds 321 kPa absolute. And a perfect vacuum is −101 kPa gauge, which is the floor — a gauge cannot read lower.

p = ρgh assumes ρ is constant. For liquids that is excellent: water's density varies under 1% over kilometres of depth. For gases, density is proportional to pressure, so the integration becomes non-linear and pressure falls roughly exponentially with altitude, halving about every 5.5 km.

Inside a manometer's short gas limb, air's ρgh is worth a few pascals and is universally neglected. That is a judgement about scale, not a law — and the same neglect applied to a weather model would be nonsense.