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Bernoulli's equation

Bernoulli's equation is an energy balance along a streamline for steady, frictionless flow: the sum of pressure energy, kinetic energy and potential energy is constant. Where the speed increases, the pressure must drop, and vice versa, at the same height.

p + \tfrac12\rho v^2 + \rho gz = \text{constant}

Bernoulli's equation along a streamline

Symbols

$p$	static pressure	Pa
$v$	speed	m/s
$z$	height	m

Example

Water flows horizontally: $v_1 = 2$ m/s at $p_1$ , and $v_2 = 5.56$ m/s where the pipe narrows. $p_1 - p_2 = \tfrac12\cdot 1000\cdot(5.56^2 - 2^2) \approx 13.4$ kPa.

Bernoulli only holds with no losses and no pump between the points – add such terms separately if they are present.

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Part of Fluid Mechanics: Continuity and Bernoulli.