Bernoulli's Equation

Equation

\[\boxed{\,p + \tfrac{1}{2}\rho v^2 + \rho g z = \text{constant along a streamline}\,}\]

Compressible variants (replace \(p/\rho\) by \(\int dp/\rho\)):

• Isothermal ideal gas: \(RT\ln p + \tfrac12 v^2 + gz = \text{const}\)
• Isentropic: \(\dfrac{\gamma}{\gamma-1}\dfrac{p}{\rho} + \tfrac12 v^2 + gz = \text{const}\)

Physical meaning

Energy conservation per unit volume for a frictionless fluid parcel: pressure work + kinetic energy + gravitational potential energy is a fixed budget. Speed up ⇒ pressure down; climb ⇒ pay from pressure or speed.

Variables

Symbol	Meaning	SI unit
\(p\)	static pressure	Pa
\(\rho\)	density	kg m⁻³
\(v\)	speed along the streamline	m s⁻¹
\(z\)	elevation	m
\(\gamma\)	ratio of specific heats	–

Assumptions

1. Steady flow · 2. Inviscid · 3. Incompressible (basic form) ·
2. Along a streamline (global constant only if also irrotational — potential flow) · 5. No shaft work or heat addition.

Derivation

Newton along a streamline (see the concept page for the full walk-through):

\[-dp - \rho g\,dz = \rho V\,dV \;\Rightarrow\; \frac{dp}{\rho} + \tfrac12 d(V^2) + g\,dz = 0\]

then integrate with the appropriate \(\rho(p)\) relation. Equivalently, integrate Euler's equation along \(d\mathbf{s} \parallel \mathbf{v}\).

Worked examples

• Venturi meter: \(h = \frac{v_1^2}{2g}\left(\frac{A_1^2}{A_2^2}-1\right)\)
• Torricelli draining & optimal hole: \(v = \sqrt{2g(H-h)}\)
• Necking stream: flow-rate measurement with a ruler
• Pitot tube: stagnation minus static pressure = \(\tfrac12\rho v^2\) → airspeed

Limitations

Fails in boundary layers and separated/viscous regions, across shocks and hydraulic jumps, through pumps/turbines (add work terms), and in strongly unsteady flow (an unsteady term \(\rho\,\partial\phi/\partial t\) can be retained — used in the interface instability derivation).

Related equations

• Euler's equation — parent
• Continuity equation — constant companion
• Interface dispersion relation — built on perturbed Bernoulli

Quiz

Q1 (computational). A pitot-static probe on an aircraft reads \(\Delta p = 4.5\) kPa in air of density 0.9 kg/m³. Airspeed?

Answer

\(v = \sqrt{2\Delta p/\rho} = \sqrt{2\times4500/0.9} = 100\) m/s.

Q2 (conceptual). Why can't Bernoulli's equation be used through a household fan, even though the flow before and after is fast and smooth?

Answer

The fan does shaft work on the fluid — the energy budget jumps between inlet and outlet streamlines. Bernoulli holds separately upstream and downstream, not across the energy source.