Deriving Bernoulli's equation from Euler's equation

Euler's equation of motion is Newton's second law applied to a fluid element moving along a streamline, considering only pressure and gravity forces. Integrating it gives Bernoulli's equation.

              (p + dp) dA
                 <---
            ____________
   p dA --> |  element  |  ds, along streamline
            |___________|   (inclined at theta)
                 |
                 v  weight = rho g dA ds

Forces on the element

Consider an element of area $dA$ and length $ds$ along a streamline inclined at $\theta$ to the horizontal.

1. Pressure force in the direction of flow: $p\,dA$.
2. Pressure force opposite to flow: $(p + dp)\,dA$.
3. Weight component along $s$: $-\rho g\, dA\, ds\sin\theta$, where $\sin\theta = dz/ds$.

Newton's second law

Mass × acceleration $= \rho\, dA\, ds\, a_s$, where for steady flow $a_s = V\dfrac{dV}{ds}$:

$$p\,dA - (p + dp)\,dA - \rho g\,dA\,ds\frac{dz}{ds} = \rho\,dA\,ds\, V\frac{dV}{ds}$$

This simplifies to $-dp\,dA - \rho g\,dA\,dz = \rho\,dA\,V\,dV$. Dividing by $\rho\, dA$:

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

This is Euler's equation of motion.

Integration (Bernoulli's equation)

For incompressible flow ($\rho$ constant):

$$\int \frac{dp}{\rho} + \int g\,dz + \int V\,dV = \text{constant}$$ $$\frac{p}{\rho} + gz + \frac{V^2}{2} = \text{constant}$$

Dividing by $g$:

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

This is Bernoulli's equation: pressure head + velocity head + datum head is constant along a streamline.

Assumptions: ideal (frictionless) fluid, steady, incompressible and irrotational flow along a streamline.