Density: ρ = m/V (kg/m³)

Specific weight: γ = ρg (N/m³)

Specific gravity: SG = ρ/ρ_water (dimensionless)

Specific volume: v = 1/ρ (m³/kg)

Viscosity

Newton’s law: τ = μ(du/dy)

Dynamic viscosity: μ (Pa·s)

Kinematic viscosity: ν = μ/ρ (m²/s)

Surface Tension & Compressibility

Droplet pressure: ΔP = 2σ/r (sphere), ΔP = 4σ/r (bubble)

Capillary rise: h = 4σcosθ/(ρgd)

Bulk modulus: K = −ΔP/(ΔV/V) (Pa)

Pressure

Hydrostatic: P = P₀ + ρgh

Pressure head: h = P/(ρg)

Gauge pressure: P_gauge = P_abs − P_atm

Hydrostatic Force

Force on plane surface: F = ρgh̄A

Centre of pressure: h_cp = h̄ + I_G/(h̄A)

Curved surface: F_H = force on vertical projection; F_V = weight of fluid above

Buoyancy

Buoyant force: F_B = ρ_fluidgV_displaced

Floating body: Fraction submerged = ρ_body/ρ_fluid

Metacentric height: GM = BM − BG, where BM = I/V_displaced

Manometers

U-tube: Start at known pressure, add ρgh going down, subtract going up

Differential: P_A − P_B = (ρ_m − ρ_fluid)gh

General: ρ₁A₁V₁ = ρ₂A₂V₂ (mass conservation)

Incompressible: A₁V₁ = A₂V₂ = Q (volume conservation)

Volume flow rate: Q = AV (m³/s)

Mass flow rate: ṁ = ρAV = ρQ (kg/s)

Circular pipe area: A = πD²/4

Standard Form

P₁ + ½ρV₁² + ρgz₁ = P₂ + ½ρV₂² + ρgz₂

Head Form

P/(ρg) + V²/(2g) + z = H (total head)

Modified (with losses & pump)

P₁/(ρg) + V₁²/(2g) + z₁ + h_pump = P₂/(ρg) + V₂²/(2g) + z₂ + h_L

Applications

Pitot tube: V = √[2(P₀ − P)/ρ]

Torricelli: V = √(2gh)

Venturi: Q = C_dA₂√[2ΔP/(ρ(1 − (A₂/A₁)²))]

Stagnation pressure: P₀ = P + ½ρV²

Reynolds number: Re = ρVD/μ = VD/ν

Pipe flow: Re < 2,300 → Laminar; Re > 4,000 → Turbulent

Flat plate: Re_x,crit ≈ 5 × 10⁵

Hydraulic diameter: D_h = 4A/P_wetted

Hagen-Poiseuille (Laminar Pipe Flow)

Velocity profile: u(r) = (ΔP/4μL)(R² − r²)

V_max = 2V_avg

Flow rate: Q = πΔPD⁴/(128μL)

Pressure drop: ΔP = 32μLV/D²

Friction factor: f = 64/Re

Wall shear stress: τ_w = 8μV/D

Darcy-Weisbach (Major Losses)

h_f = f(L/D)(V²/2g)

ΔP = f(L/D)(ρV²/2)

Laminar: f = 64/Re. Turbulent: Moody chart or Colebrook equation.

Colebrook Equation (Turbulent)

1/√f = −2log₁₀(ε/(3.7D) + 2.51/(Re√f))

Minor Losses

h_m = KV²/(2g)

Sudden expansion: h = (V₁ − V₂)²/(2g)

Equivalent length: L_eq = KD/f

Series & Parallel Pipes

Series: Q same, h_L,total = Σh_Li

Parallel: h_L same, Q_total = ΣQ_i

Laminar BL (Blasius — Flat Plate)

δ = 5x/√Re_x

δ* = 1.72x/√Re_x

θ = 0.664x/√Re_x

C_f,x = 0.664/√Re_x

C̄_f = 1.328/√Re_L

Turbulent BL (1/7th Power Law)

δ = 0.37x/Re_x^0.2

C_f,x = 0.0592/Re_x^0.2

C̄_f = 0.074/Re_L^0.2

Drag

F_D = C_D × ½ρV²A

Flat plate drag: F_D = C̄_f × ½ρV² × A_wetted