Momentum Equation for compressible 1D flow

[Bohr1227]

Homework Statement

Derive the differential momentum equation: $$dp=-\rho udu$$.

Homework Equations

$$\oint_{ CS}^{ } \rho \vec{U}(\vec{U}\cdot \vec{n})dA=-\oint_{ CS}^{ } p\vec{n}dA$$ for steady state flow...

The Attempt at a Solution

I tried to solve it like in the attached picture, but can someone explain where the $$p+\frac{1}{2}dp$$ (marked with red line in the picture) comes from? Is it a linear interpolation or something?

Thanks!

 

[Chestermiller]

That is a term for representing the force that the solid boundary exerts (in the x direction) on the fluid passing through the control volume.