Fluid mechanics formula sheet

The formulas of a first fluid mechanics course in one place, ready to print. Each row is a quantity and its formula, grouped by topic.

Fluid properties

Quantity	Formula
Density	$ρ = \frac{m}{V}$
Specific weight	$γ = ρ g$
Kinematic viscosity	$ν = \frac{μ}{ρ}$
Shear stress (Newtonian)	$τ = μ \frac{d u}{d y}$

Hydrostatics

Quantity	Formula
Hydrostatic pressure	$p = p_{0} + ρ g h$
Force on a plane surface	$F = ρ g h_{c} A$
Centre of pressure	$y_{c p} = y_{c} + \frac{I _{x c}}{y _{c} A}$
Buoyancy (Archimedes)	$F_{B} = ρ g V_{disp}$

Conservation laws

Quantity	Formula
Continuity	$ρ_{1} A_{1} V_{1} = ρ_{2} A_{2} V_{2}$
Volume flow rate	$Q = A V$
Bernoulli	$\frac{p}{ρ g} + \frac{V ^{2}}{2 g} + z = const$
Energy equation	$\frac{p _{1}}{ρ g} + \frac{V _{1}^{2}}{2 g} + z_{1} + h_{p} = \frac{p _{2}}{ρ g} + \frac{V _{2}^{2}}{2 g} + z_{2} + h_{t} + h_{L}$
Momentum (steady flow)	$\sum F = \overset{m}{˙} (V_{out} - V_{in})$
Pump power	$P = \frac{ρ g Q h _{p}}{η}$

Pipe flow

Quantity	Formula
Reynolds number	$R e = \frac{ρ V D}{μ} = \frac{V D}{ν}$
Laminar pipe flow	$R e < 2300$
Darcy–Weisbach	$h_{f} = f \frac{L}{D} \frac{V ^{2}}{2 g}$
Friction factor, laminar	$f = \frac{64}{R e}$
Colebrook	$\frac{1}{f} = - 2 lo g (\frac{ε / D}{3.7} + \frac{2.51}{R e f})$
Haaland (explicit)	$\frac{1}{f} = - 1.8 lo g [(\frac{ε / D}{3.7})^{1.11} + \frac{6.9}{R e}]$
Minor losses	$h_{m} = K \frac{V ^{2}}{2 g}$
Hydraulic diameter	$D_{h} = \frac{4 A}{P}$

External flow and dimensionless numbers

Quantity	Formula
Drag force	$F_{D} = \frac{1}{2} C_{D} ρ V^{2} A$
Lift force	$F_{L} = \frac{1}{2} C_{L} ρ V^{2} A$
Laminar boundary layer (Blasius)	$\frac{δ}{x} \approx \frac{5.0}{R e _{x}}$
Turbulent boundary layer	$\frac{δ}{x} \approx \frac{0.37}{R e _{x}^{1/5}}$
Froude number	$F r = \frac{V}{g L}$
Mach number	$M a = \frac{V}{c}$

Notes

Each Bernoulli term is a head in metres. It holds along a streamline for steady, incompressible, frictionless flow. With friction or a pump, use the energy equation.

The Re ≈ 2300 transition is for pipe flow. Over a flat plate, transition happens around Re_x ≈ 5×10⁵.

Before an exam, check your course syllabus for whether a formula sheet is allowed and which.