Momentum Equation for compressible 1D flow

AI Thread Summary
The discussion focuses on deriving the differential momentum equation for compressible 1D flow, specifically the equation dp = -ρu du. Participants reference the steady-state flow equation involving mass and momentum flux across a control surface. A question arises regarding the origin of the term p + (1/2)dp, with clarification that it represents the force exerted by the solid boundary on the fluid. The conversation emphasizes understanding the derivation process and the significance of each term in the equation. Overall, the thread aims to clarify the mathematical foundations of momentum in fluid dynamics.
Bohr1227
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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?
Untitled1.png


Thanks!
 

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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.
 
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