Consider a control volume fixes in space, bounded by flow streamlines. The length of the control volume is $ds.$ Flow across the bounding surfaces occurs only at the end sections, located at $s$ and $s+ds$ Assume the following: [1] inviscid [2] incompressible [3] along a streamline [4] steady ![[Pasted image 20260915192615.png|center|600]] $\rho\left(V-\frac{1}{2}dV\right)\left[-\left(A-\frac{1}{2}dA\right)\right]+\rho\left(V+\frac{1}{2}dV\right)\left(A+\frac{1}{2}dA\right)=0$ $\text{Neglecting higher order terms: } dA=-A\,dV \tag{1}$ $\text{Applying LME in }\hat{s} \text{ direction: }\left.(\vec{F}_B+\vec{F}_S)\right|_{\hat{s}}=\frac{d}{dt}\int_{CV}u_s\rho\,dV+\int_{CS}u_s\left(\rho\vec{u}_{rel}\cdot d\vec{A}\right)$ $=\left(V-\frac{1}{2}dV\right)\left[\rho\left(V-\frac{1}{2}dV\right)\left(-A+\frac{1}{2}dA\right)\right]+\left(V+\frac{1}{2}dV\right)\left[\rho\left(V+\frac{1}{2}dV\right)\left(A+\frac{1}{2}dA\right)\right]$ $\underbrace{ \rho A\,ds(-g\sin\theta) }_{ F_{B} }+\underbrace{ \left(p-\frac{1}{2}dp\right)\left(A-\frac{1}{2}dA\right)-\left(p+\frac{1}{2}dp\right)\left(A+\frac{1}{2}dA\right)+p\,dA }_{ F_{S} } \tag{2}$ $\text{Combining (1) and (2): }\boxed{ \frac{dp}{\rho}+V\,dV+g\,dz=0 } \text{ (differential form)}$ $\text{Integrating along streamline: }\boxed{ \frac{p}{\rho}+\frac{1}{2}V^2+gz=\text{const.} }$