Fluid Dynamics — Euler's & Bernoulli's Equations

Forces in fluid flow, Euler's equation of motion along a streamline, Bernoulli's equation and its assumptions, energy heads and the total energy line and hydraulic gradient line, Bernoulli's equation for real fluids with head loss, kinetic energy correction factor, applications (tank draining, siphon, Pitot tube, flow through nozzles) and the limitations — with solved numericals.

Last reviewed 16 Sept 2026 · 6 min read

Forces acting on a flowing fluid

Gravity — weight of the fluid.
Pressure — from the surrounding fluid.
Viscous — internal friction.
Turbulent — Reynolds stresses in turbulent flow.
Compressibility and surface tension — usually negligible in hydraulic problems.

Newton's second law with gravity and pressure only gives Euler's equation; adding viscous forces gives the Navier–Stokes equations; adding turbulence terms gives the Reynolds equations.

Euler's equation of motion

For steady flow of an ideal fluid along a streamline:

\frac{d p}{ρ} + g d z + V d V = 0

Bernoulli's equation

Integrating Euler's equation for an incompressible fluid:

FormulaBernoulli's equation

\frac{p}{ρ g} + \frac{V ^{2}}{2 g} + z = constant (along a streamline)

$\frac{p}{ρ g}$ = pressure head (pressure energy per unit weight)
$\frac{V ^{2}}{2 g}$ = velocity (kinetic) head
$z$ = datum (potential) head

Total energy per unit weight (total head) is constant along a streamline.

Assumptions

Ideal fluid — non-viscous (no friction losses).
Steady flow.
Incompressible fluid.
Flow along a streamline (between two points on the same streamline; for irrotational flow, between any two points).
No energy added or removed (no pumps or turbines), no heat transfer.
Velocity uniform across the section (one-dimensional).

Real fluids

For real flow between sections 1 and 2 with head loss $h_{L}$ , pump head $H_{p}$ and turbine head $H_{t}$ :

\frac{p _{1}}{ρ g} + α_{1} \frac{V _{1}^{2}}{2 g} + z_{1} + H_{p} = \frac{p _{2}}{ρ g} + α_{2} \frac{V _{2}^{2}}{2 g} + z_{2} + H_{t} + h_{L}

Kinetic energy correction factor $α$ accounts for non-uniform velocity across the section:

α = \frac{1}{A V ^{3}} \int u^{3} d A

$α$ = 2.0 for laminar flow in pipes; about 1.01–1.1 for turbulent flow (often taken as 1). The momentum correction factor $β = \frac{1}{A V ^{2}} \int u^{2} d A$ is 4/3 for laminar pipe flow and about 1.01–1.04 for turbulent flow.

Energy line and hydraulic gradient line

Total energy line (TEL) — plots total head $\frac{p}{ρ g} + \frac{V ^{2}}{2 g} + z$ along the flow. It always slopes downward in the direction of flow (losses), except where a pump adds energy.
Hydraulic gradient line (HGL) — plots piezometric head $\frac{p}{ρ g} + z$ ; lies below the TEL by the velocity head.
Where the HGL falls below the pipe, pressure is negative (sub-atmospheric) — risk of air entry and cavitation.
At a sudden enlargement, the HGL may rise (velocity head converts to pressure) while the TEL drops by the loss.

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