Pressure in fluids

A fluid is anything that can flow: liquids and gases. A fluid cannot resist being sheared sideways, so it has no shape of its own and takes the shape of its container. What it can do is push, and it always pushes at right angles to any surface it touches.

Pressure is that push per unit area, P = F/A, in pascals (1 Pa = 1 N m⁻²). At a point in a fluid at rest the pressure is the same in every direction, so pressure has no direction: it is a scalar.

Some units you will meet: 1 atm = 1.013 × 10⁵ Pa (the air pressure at sea level); 1 bar = 10⁵ Pa; 1 torr = 1 mm of mercury ≈ 133 Pa. So 1 atm = 760 torr = 76 cm of mercury.

Pressure grows with depth. The liquid above any point has weight, and that weight presses down: P = P_a + ρgh, where P_a is the air pressure on the open surface.

• P is the absolute pressure; the extra part ρgh is the gauge pressure (what a tyre gauge shows).
• It depends only on the depth, not on the shape or width of the vessel. A tall thin tube and a wide tank give the same pressure at the same depth.
• Example: a diver 20 m down in the sea (ρ = 1030 kg m⁻³): P = 1.01 × 10⁵ + 1030 × 9.8 × 20 ≈ 3.0 × 10⁵ Pa, about 3 atm.

A car tyre filled to '200 kPa' means a gauge pressure of 200 kPa; the absolute pressure inside is 200 + 101 = 301 kPa. Drive up a hill where the air pressure is only 90 kPa, and the same air now shows 301 – 90 = 211 kPa on the gauge.

A mercury barometer measures air pressure: at sea level the air holds up a mercury column about 76 cm high. An open-tube manometer (a U-tube of liquid) measures the difference between a gas's pressure and the air's.

Pascal's law: a pressure applied to an enclosed fluid is passed on undiminished to every part of it and to the walls. In a hydraulic lift, a small force on a small piston gives a big force on a big piston: F₂ = (A₂/A₁)F₁. With pistons of 10 cm² and 500 cm², a push of 200 N lifts 10 000 N. Hydraulic brakes use the same idea: a light press on the pedal becomes a large, equal push on every wheel.

In a U-tube holding two liquids that do not mix, the pressures are equal at the same level in the lower liquid, so h₁ρ₁ = h₂ρ₂. Example: 10 cm of oil (ρ = 800) balances 8 cm of water (ρ = 1000) above the same level.