- #1
Bohr1227
- 13
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Homework Statement
Use a CV analysis to show that an element of fluid along a streamline gives
\[\partial p/\partial x=-\rho u\partial u/\partial x\]
Homework Equations
\[\sum F=\oint_{CS}^{ } \rho \overrightarrow{V}(\overrightarrow{V_{rel}}\cdot \overrightarrow{n})\]
The Attempt at a Solution
Using a CS with following conditions on left side: u and p.
Following conditions on right side of the element: \[u+\partial u/\partial x\Delta x] and \[p+\partial p/\partial x\Delta x]
Just looking at it per unit width inside the paper: (Neglecting small factors)
\[(p-(p+\partial p/\partial x\Delta x))\Delta y=\rho ((u+\frac{\partial u}{\partial x}\Delta x)^{2}-u^{2})\Delta y=\rho (u^{2}-u^{2}+2u\frac{\partial u}{\partial x}\Delta x))\Delta y\]
This gives:
\[\frac{\partial p}{\partial x}=-2\rho u\frac{\partial u}{\partial x}\]
I get the answer with a factor 2, which I am not supposed to get. What do I do wrong?