Bernoulli's equation

See Bernoulli differential equation for an unrelated topic in ordinary differential equations.

In fluid dynamics, Bernoulli's equation, derived by Daniel Bernoulli, describes the behavior of a fluid moving along a streamline.

${v^2 \over 2}+gy+{P \over \rho}=constant$

v = fluid velocity along the streamline

g = acceleration due to gravity on Earth

y = height in the direction of gravity

P = pressure along the streamline

$ρ$ = fluid density

These assumptions must be met for the equation to apply:

Inviscid flow - Viscosity (internal friction) = 0
Steady flow
Incompressible flow - $ρ$ is constant. (There exists a second form of Bernoulli's equation that is applicable for compressible flow, which makes use of the thermodynamic enthalpy.)
The equation applies along a streamline. It applies throughout the flow field for irrotational flow.

The decrease in pressure simultaneous with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle.

The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler.

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The equation can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects:

the work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy

The work done by the forces;

$F_{1}\cdot s_{1}-F_{2}\cdot s_{2}=p_{1}\cdot A_{1}\cdot v_ {1}\cdot\Delta t-p_{2}\cdot A_{2}\cdot v_{2}\cdot\Delta t$

+ the decrease of potential energy:

$m\cdot g\cdot h_{1}-m\cdot g\cdot h_{2}=\rho\cdot g\cdot A _{1}\cdot v_{1}\cdot\Delta t\cdot h_{1}-\rho\cdot g\cdot A_{2}\cdot v_{2}\cdot \Delta t\cdot h_{2}$

= the increase in kinetic energy:

$\frac{1}{2}\cdot m\cdot v_{2}^{2}-\frac{1}{2}\cdot m\cdot v_{1}^{2}=\frac{1}{2}\cdot\rho\cdot A_{2}\cdot v_{2}\cdot\Delta t\cdot v_{2} ^{2}-\frac{1}{2}\cdot\rho\cdot A_{1}\cdot v_{1}\cdot\Delta t\cdot v_{1}^{2}$

gives;

$p_{1}\cdot A_{1}\cdot v_{1}\cdot\Delta t-p_{2}\cdot A_{2}\cdot v_{2}\cdot\Delta t+\rho\cdot g\cdot A_{1}\cdot v_{1}\cdot\Delta t\cdot h_{1}-\rho\cdot g\cdot A_{2}\cdot v_{2}\cdot\Delta t\cdot h_{2}=\frac{1}{2}\cdot\rho\cdot A_{2}\cdot v_{2}\cdot\Delta t\cdot v_{2}^{2}-\frac{1}{2}\cdot\rho\cdot A_{1}\cdot v_{1}\cdot\Delta t\cdot v_{1}^{2}$

$\frac{\rho\cdot A_{1}\cdot v_{1}\cdot\Delta t\cdot v_{1}^{ 2}}{2}+\rho\cdot g\cdot A_{1}\cdot v_{1}\cdot\Delta t\cdot h_{1}+p_{1}\cdot A_{1 }\cdot v_{1}\cdot\Delta t=\frac{\rho\cdot A_{2}\cdot v_{2}\cdot\Delta t\cdot v_{ 2}^{2}}{2}+\rho\cdot g\cdot A_{2}\cdot v_{2}\cdot\Delta t\cdot h_{2}+p_{2}\cdot A_{2}\cdot v_{2}\cdot\Delta t$

division by $Δ t$ , $ρ$ and $A_{1}\cdot v_{1}$ (= rate of fluid flow = $A_{2}\cdot v_{2}$ as the fluid is incompressible) gives;

$\frac{v_{1}^{2}}{2}+g\cdot h_{1}+\frac{p_{1}}{\rho}=\frac{v_{2}^{2}}{2}+g\cdot h_{2}+\frac{p_{2}}{\rho}$

or $\frac{v^{2}}{2}+g\cdot h+\frac{p}{\rho}=C$ (as stated in the first paragraph).

Further division by g gives;

$\frac{v^{2}}{2\cdot g}+h+\frac{p}{\rho\cdot g}=C$

A free falling mass from a height h will reach a velocity $v=\sqrt{\frac{h}{2\cdot g}}$ , or $h=\frac{v^{2}}{2\cdot g}$ . The term $\frac{v^2}{2\cdot g}$ is called the velocity head.
As the hydrostatic pressure or static head is defined as $p=\rho \cdot g \cdot h$ or $h=\frac{p}{\rho \cdot g}$ , the term $\frac{p}{\rho \cdot g}$ is also called the pressure head.

Bernoulli's equation

See also: Bernoulli's equation, Bernoulli's principle, Bernoulli differential equation, Conservation of energy, Daniel Bernoulli, Density, Earth, Enthalpy, Euler equations, Fluid