Fluid Statics & Pressure Measurement Notes

Last reviewed 16 Sept 2026 · 6 min read

Pressure

Pressure (intensity of pressure) is the normal force per unit area exerted by a fluid: $p = F / A$ (N/m² = pascal; 1 bar = 10⁵ Pa).

Pascal's law

At a point in a fluid at rest, the pressure is the same in all directions: $p_{x} = p_{y} = p_{z}$ . (Pressure acts normal to any surface; there is no shear in a static fluid.)

Applications: hydraulic press, hydraulic jack, hydraulic brakes — a small force on a small piston produces a large force on a large piston: $\frac{F _{1}}{A _{1}} = \frac{F _{2}}{A _{2}}$ .

Hydrostatic law

In a static fluid, pressure increases with depth:

\frac{d p}{d z} = - ρ g \Rightarrow p = ρ g h (gauge, below a free surface)

Pressure is the same at all points on a horizontal plane within the same continuous fluid.
The shape of the container does not matter (hydrostatic paradox) — only the vertical depth.
Pressure head $h = \frac{p}{ρ g}$ : the height of a column of the fluid that produces the pressure. Converting between liquids: $h_{1} S_{1} = h_{2} S_{2}$ .

Absolute, gauge and vacuum pressure

p_{ab s} = p_{a t m} + p_{g a ug e}, p_{v a c uu m} = p_{a t m} - p_{ab s}

Atmospheric pressure at sea level ≈ 101.325 kPa = 10.33 m of water = 760 mm of mercury.
Gauge pressure is measured relative to local atmosphere (positive or negative).
Vacuum (negative gauge) pressure — below atmospheric; absolute zero is a perfect vacuum.
Atmospheric pressure is measured by a barometer (mercury or aneroid).

Pressure measuring devices

Piezometer

A simple vertical tube connected to the pipe: $p = ρ g h$ . Limitations: cannot measure negative (vacuum) pressures (air enters), impractical for high pressures (very tall tube), unsuitable for gases.

Simple U-tube manometer

A U-tube containing a heavier manometric liquid (usually mercury, $S_{2}$ ) connected to the pipe carrying fluid of $S_{1}$ . Write pressures along the tube, equating pressures at the same level in the same liquid.

FormulaSimple U-tube manometer (pipe on the left, right limb open)

Positive pressure at A, fluid rising $h_{1}$ above the datum in the left limb, manometric liquid difference $h_{2}$ :

p_{A} + ρ_{1} g h_{1} = ρ_{2} g h_{2} \Rightarrow p_{A} = ρ_{2} g h_{2} - ρ_{1} g h_{1}

Negative (vacuum) pressure at A (manometric liquid higher in the left limb):

p_{A} = - ρ_{1} g h_{1} - ρ_{2} g h_{2}

Differential U-tube manometer

Measures the difference in pressure between two points A and B. For two pipes at the same level carrying the same fluid ( $S_{1}$ ) with mercury ( $S_{2}$ ) reading $x$ :

p_{A} - p_{B} = x g (ρ_{2} - ρ_{1}) \Rightarrow head difference = x (\frac{S _{2}}{S _{1}} - 1)

For water with mercury: $h = 12.6 x$ .

Inverted U-tube manometer

A lighter manometric fluid (air or oil) in an inverted U: used for small pressure differences in liquids: $p_{A} - p_{B} = x g (ρ_{1} - ρ_{l i g h t})$ (same-level points).

Micromanometers and inclined manometers

Inclined manometer — the limb is inclined at $θ$ , so a small vertical difference $h = L sin θ$ produces a longer reading $L$ — higher sensitivity.
Micromanometer — enlarged reservoir (well) on one limb so that only one limb is read; used for very small differences.

Mechanical gauges

Bourdon tube gauge — a curved elliptical tube tends to straighten under pressure; pointer motion; used for high pressures and vacuum.
Diaphragm gauge, bellows gauge, dead-weight gauge (used for calibration), and electronic pressure transducers.

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Fluid Statics & Pressure Measurement