Bernoulli's equation and four applications

Bernoulli's equation from Euler's equation

Consider a small fluid element of length $ds$ and area $dA$ along a streamline, in steady flow of an ideal fluid. Newton's second law along $s$ (pressure and the weight component, no friction):

$$-\frac{\partial p}{\partial s}\,ds\,dA - \rho g\,ds\,dA\cos\theta = \rho\,ds\,dA\,V\frac{\partial V}{\partial s}$$

With $\cos\theta = dz/ds$, this gives Euler's equation:

$$\frac{dp}{\rho} + V\,dV + g\,dz = 0$$

For incompressible flow, integrate along the streamline:

$$\frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant}\quad\Rightarrow\quad\frac{p}{\gamma} + \frac{V^2}{2g} + z = \text{constant}$$

Assumptions: steady, incompressible, frictionless (inviscid) flow along a streamline, with no energy added or removed.

Four applications

1. Venturimeter: a pipe contraction raises the velocity and lowers the pressure. From the pressure difference, $Q = C_d\dfrac{A_1A_2}{\sqrt{A_1^2 - A_2^2}}\sqrt{2g\,h}$. It is used to measure discharge in pipelines.
2. Pitot tube: at the stagnation point the velocity head is converted into pressure; $V = \sqrt{2g\,h}$ gives the velocity in pipes and channels, and the airspeed in aircraft airspeed indicators.
3. Orifice / Torricelli's theorem: water issuing from a tank under head $H$ has velocity $V = \sqrt{2gH}$; this is used for orifices, sluice gates and discharge estimates.
4. Siphon: Bernoulli's equation gives the discharge velocity $\sqrt{2gh}$ and the pressure at the crest. The crest pressure must stay above the vapour pressure (maximum crest height about 7 to 8 m of water at sea level) to avoid cavitation and breaking of the flow.

Other uses include pump and turbine head calculations, flow nozzles and orifice meters, aerofoil lift and spray nozzles.