Fluid Properties, Pressure and Manometers

Fluid mechanics begins with the properties of fluids and with fluids at rest, where pressure depends only on depth.

Fluid properties

• Density ρ (kg/m³); specific weight γ = ρg; specific gravity = ρ/ρ_water (ρ_water = 1000 kg/m³).
• Viscosity: Newton's law of viscosity, τ = μ du/dy. μ is the dynamic viscosity (Pa·s); ν = μ/ρ is the kinematic viscosity (m²/s).
• Liquids get less viscous when heated; gases get more viscous.
• Non-Newtonian fluids: shear-thinning (paint), shear-thickening, Bingham plastic (toothpaste).
• Surface tension σ (N/m): pressure inside a droplet Δp = 4σ/d; inside a soap bubble (two surfaces) Δp = 8σ/d.
• Capillary rise in a tube of diameter d: h = 4σ cos θ/(ρ g d). Water rises (θ ≈ 0°); mercury falls (θ > 90°).
• Bulk modulus K = −dp/(dV/V); compressibility = 1/K.

Hydrostatic pressure

In a fluid at rest, pressure increases with depth:

p = p₀ + ρ g h

• Pressure at a point is the same in all directions (Pascal's law).
• Absolute pressure = gauge pressure + atmospheric pressure (about 101.3 kPa).
• Points at the same level in the same continuous fluid at rest have the same pressure.

Manometers

Start at one end, add ρgh going down and subtract going up, and set the sum equal to the pressure at the other end.

• Piezometer: p = ρgh (only for small positive gauge pressures of liquids).
• U-tube: p_A + ρ₁ g h₁ − ρ_m g h_m = p_atm.
• Differential manometer between two pipes A and B with the same fluid ρ and manometer fluid ρ_m, reading x: p_A − p_B = (ρ_m − ρ) g x (when A and B are at the same level).

Worked example

Water flows in two pipes at the same level connected by a mercury differential manometer (SG 13.6) with a reading of 100 mm. Find p_A − p_B.

• p_A − p_B = (13600 − 1000) × 9.81 × 0.1 = 12.36 kPa.
• In metres of water: 12.6 × 0.1 = 1.26 m.

Practice