Categories: Fluid dynamics, Fluid mechanics, Physics.

# Bernoulli’s theorem

For inviscid fluids, Bernuilli’s theorem states that an increase in flow velocity $\va{v}$ is paired with a decrease in pressure $p$ and/or potential energy. For a qualitative argument, look no further than one of the Euler equations, with a material derivative:

\begin{aligned} \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} = \va{g} - \frac{\nabla p}{\rho} \end{aligned}

Assuming that $\va{v}$ is constant in $t$, it becomes clear that a higher $\va{v}$ requires a lower $p$.

## Simple form

For an incompressible fluid with a time-independent velocity field $\va{v}$ (i.e. steady flow), Bernoulli’s theorem formally states that the Bernoulli head $H$ is constant along a streamline:

\begin{aligned} \boxed{ H = \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} } \end{aligned}

Where $\Phi$ is the gravitational potential, such that $\va{g} = - \nabla \Phi$. To prove this theorem, we take the material derivative of $H$:

\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} &= \va{v} \cdot \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + \frac{\mathrm{D} \Phi}{\mathrm{D} t} + \frac{1}{\rho} \frac{\mathrm{D} p}{\mathrm{D} t} \end{aligned}

In the first term we insert the Euler equation, and in the other two we expand the derivatives:

\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} &= \va{v} \cdot \Big( \va{g} - \frac{\nabla p}{\rho} \Big) + \Big( \pdv{\Phi}{t} + (\va{v} \cdot \nabla) \Phi \Big) + \frac{1}{\rho} \Big( \pdv{p}{t} + (\va{v} \cdot \nabla) p \Big) \\ &= \pdv{\Phi}{t} + \frac{1}{\rho} \pdv{p}{t} + \va{v} \cdot \big( \va{g} + \nabla \Phi \big) + \va{v} \cdot \Big( \frac{\nabla p}{\rho} - \frac{\nabla p}{\rho} \Big) \end{aligned}

Using the fact that $\va{g} = - \nabla \Phi$, we are left with the following equation:

\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} &= \pdv{\Phi}{t} + \frac{1}{\rho} \pdv{p}{t} \end{aligned}

Assuming that the flow is steady, both derivatives vanish, leading us to the conclusion that $H$ is conserved along the streamline.

In fact, there exists Bernoulli’s stronger theorem, which states that $H$ is constant everywhere in regions with zero vorticity $\va{\omega} = 0$. For a proof, see the derivation of $\va{\omega}$’s equation of motion.

1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.