Fluid statics

Why it matters

Dams, retaining walls, water tanks, ship hulls, manometers and hydraulic presses all depend on how pressure varies in a fluid at rest. Civil engineers size tank walls and gates from it, mechanical engineers use it in hydraulics, and chemists and materials scientists read manometers and pressure gauges every day. Fluid statics is also the gentlest entry into fluid mechanics: no flow, just weight and pressure.

The hydrostatic equation

In a fluid at rest, pressure increases with depth because each layer carries the weight of the fluid above it. For a fluid of constant density ρ:

P = P0 + ρ · g · h

• P0 is the pressure at the free surface (often atmospheric, about 101.3 kPa at sea level).
• ρ is density (water ≈ 1,000 kg/m³; seawater ≈ 1,025 kg/m³).
• g = 9.81 m/s².
• h is the depth below the surface, in metres.

Two consequences surprise beginners:

• Pressure depends only on depth, not on the shape or width of the container (the "hydrostatic paradox").
• At the same depth in a connected body of the same fluid, the pressure is the same. That is how manometers and hydraulic lifts work (Pascal's principle).

Gauge and absolute pressure

• Absolute pressure is measured from a perfect vacuum.
• Gauge pressure is measured relative to the local atmosphere: P_gauge = P_abs − P_atm.

Most tyre and process gauges read gauge pressure. Thermodynamic property tables need absolute pressure. Write "kPa(g)" or "kPa(a)" if there is any doubt.

Worked example 1: pressure at the bottom of a tank

A water tank is 3 m deep and open to the atmosphere.

P_gauge = ρ g h = 1000 × 9.81 × 3 = 29,430 Pa = 29.43 kPa(g)
P_abs   = 101.3 + 29.43 ≈ 130.7 kPa(a)

A useful rule of thumb: every 10 m of water adds about 1 atmosphere (98.1 kPa).

Force on a submerged plane surface

Pressure acts perpendicular to a surface, and it grows with depth, so the force on a wall or gate is the pressure averaged over its area. For a flat surface the resultant force is:

F = ρ · g · h_c · A

where h_c is the depth of the surface's centroid and A is its area (using gauge pressure, because the atmosphere usually acts on both sides).

The force does not act at the centroid. Because pressure is larger lower down, it acts at the centre of pressure, which is deeper. For a vertical surface whose top edge is at the free surface:

y_cp = y_c + I_c / (y_c · A)

with y_c the centroid depth and I_c the second moment of area about the centroid (b·d³/12 for a rectangle of width b and height d).

Worked example 2: a vertical gate

A rectangular gate 2 m wide and 3 m tall sits in a vertical wall, with its top edge at the water surface.

A    = 2 × 3 = 6 m²
h_c  = 1.5 m
F    = 1000 × 9.81 × 1.5 × 6 = 88,290 N ≈ 88.3 kN
I_c  = 2 × 3³ / 12 = 4.5 m⁴
y_cp = 1.5 + 4.5 / (1.5 × 6) = 1.5 + 0.5 = 2.0 m below the surface

The resultant acts two-thirds of the way down, which matches the triangular pressure distribution. A designer would use that location to check hinges, bolts and the moments on the gate.

Buoyancy in one line

Archimedes' principle: a submerged or floating body feels an upward force equal to the weight of fluid it displaces, F_B = ρ_fluid · g · V_displaced. A 0.5 m³ block fully submerged in water feels 1000 × 9.81 × 0.5 ≈ 4.9 kN of buoyancy.

Common mistakes

• Measuring h from the bottom instead of down from the free surface.
• Mixing gauge and absolute pressure in the same calculation.
• Putting the force at the centroid. It acts at the centre of pressure, below the centroid.
• Forgetting the density of the actual fluid. Oil, seawater, mercury and brine differ a lot from fresh water.

A note on real design

These are textbook idealisations. Real tanks, dams and gates are designed to codes with load factors, seismic and dynamic effects, corrosion allowances and inspection rules, and their designs must be checked and sealed by a licensed professional engineer.

Recap

• P = P0 + ρ g h, with h measured down from the surface.
• Gauge = absolute − atmospheric.
• Force on a flat surface: F = ρ g h_c A; it acts at the centre of pressure, below the centroid.
• Buoyancy equals the weight of displaced fluid.

Try it

On the Do tab, finish a Python function for hydrostatic gauge pressure and use it to find the force on the gate.