Fluid mechanics formulas with units and examples

Fluids are in almost every machine: oil in a hydraulic cylinder, coolant in an engine, air over a car. These ten formulas cover pressure, flow and losses, the quantities you need to size a cylinder, a pump or a duct. Water is taken as ρ = 1000 kg/m³ and air as ρ = 1.2 kg/m³, with g = 9.81 m/s².

1. Hydrostatic pressure

$$p = \rho\,g\,h$$

Assumptions: the fluid is at rest and incompressible. Add atmospheric pressure (101.3 kPa) for absolute pressure.

2. Pascal’s law and the hydraulic press

$$\frac{F_1}{A_1} = \frac{F_2}{A_2} \quad\Rightarrow\quad F_2 = F_1 \frac{A_2}{A_1}$$

Assumptions: a confined fluid at rest, pistons at the same height, no friction or leakage. The force is multiplied but the large piston moves less: $A_1 x_1 = A_2 x_2$.

3. Buoyancy (Archimedes’ principle)

$$F_B = \rho_f\,g\,V_{sub}$$

Assumptions: fluid at rest with uniform density. A body floats when the buoyant force equals its weight before it is fully submerged.

4. Continuity equation

$$Q = A_1 V_1 = A_2 V_2$$

Assumptions: steady flow of an incompressible fluid with no leaks. For gases with large density changes use $\rho_1 A_1 V_1 = \rho_2 A_2 V_2$.

5. Bernoulli’s equation

$$p_1 + \tfrac{1}{2}\rho V_1^2 + \rho g z_1 = p_2 + \tfrac{1}{2}\rho V_2^2 + \rho g z_2$$

Assumptions: steady, incompressible, frictionless flow along a streamline, with no pump or turbine between the two points. Real flow loses some pressure to friction; add a head loss term $\rho g h_L$ to the right-hand side.

A full walk-through with a diagram is in Bernoulli’s equation explained with a solved numerical.

6. Reynolds number

$$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$$

Assumptions: the limits below are for flow in a round, full bore. Below about 2300 the flow is laminar; above about 4000 it is turbulent; in between it can be either.

7. Friction loss (Darcy–Weisbach)

$$h_f = f\,\frac{L}{D}\,\frac{V^2}{2g}, \qquad \Delta p = \rho g h_f$$

Assumptions: fully developed flow in a straight, round, full bore. For turbulent flow, $f$ comes from the Moody chart or the Colebrook equation; it depends on Re and wall roughness. Bends and valves add further losses.

8. Laminar friction factor

$$f = \frac{64}{Re} \quad\Rightarrow\quad \Delta p = \frac{32\,\mu L V}{D^2}$$

Assumptions: laminar flow (Re below about 2300) of a Newtonian fluid in a round bore. This is the Hagen–Poiseuille result; it is exact for laminar flow and does not depend on roughness.

9. Pump power

$$P = \frac{\rho\,g\,Q\,H}{\eta}$$

Assumptions: steady flow of an incompressible liquid. $\rho g Q H$ alone is the hydraulic power given to the fluid.

10. Drag force

$$F_D = \tfrac{1}{2}\,\rho\,V^2\,C_D\,A$$

Assumptions: $C_D$ is roughly constant over the speed range, which holds for bluff bodies at high Reynolds number. The power to overcome drag is $F_D V$, so it rises with the cube of speed.