Pressure gradient along a streamline using CV analysis

[Bohr1227]

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?

 

[Chestermiller]

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?

Did you review this before you posted it to make sure that the LaTex displayed properly?

 

[Bohr1227]

Sorry about the first post, I didn't review the latex, and apparently it is not possible to edit it...so here is the edited version:

1. Homework Statement
Use a CV analysis to show that an element of fluid along a streamline gives
$$\frac{\partial p}{\partial x}=-\rho u\frac{\partial u}{\partial x}$$

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: $$u+\frac{\partial u}{\partial x}\Delta x$$ and $$p+\frac{\partial p}{\partial x}\Delta x$$

Just looking at it per unit width inside the paper: (Neglecting small factors because $$\Delta x\rightarrow 0$$)
$$(p-(p+\partial p/\partial x\Delta x))\Delta y=\rho ((u+\frac{\partial u}{\partial x}\Delta x)^{2}-u^{2})\Delta y=\rho (u^{2}-u^{2}+2u\frac{\partial u}{\partial x}\Delta x))\Delta y$$
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?

 

[Bohr1227]

Did you review this before you posted it to make sure that the LaTex displayed properly?

You are right, I did not review it. I'm sorry, but it is fixed now.

Thank you!

 

[Chestermiller]

Sorry about the first post, I didn't review the latex, and apparently it is not possible to edit it...so here is the edited version:

1. Homework Statement
Use a CV analysis to show that an element of fluid along a streamline gives
$$\frac{\partial p}{\partial x}=-\rho u\frac{\partial u}{\partial x}$$

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: $$u+\frac{\partial u}{\partial x}\Delta x$$ and $$p+\frac{\partial p}{\partial x}\Delta x$$

Just looking at it per unit width inside the paper: (Neglecting small factors because $$\Delta x\rightarrow 0$$)
$$(p-(p+\partial p/\partial x\Delta x))\Delta y=\rho ((u+\frac{\partial u}{\partial x}\Delta x)^{2}-u^{2})\Delta y=\rho (u^{2}-u^{2}+2u\frac{\partial u}{\partial x}\Delta x))\Delta y$$
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?

I don't see how you get the right hand side of this equation:
$$(p-(p+\partial p/\partial x\Delta x))\Delta y=\rho ((u+\frac{\partial u}{\partial x}\Delta x)^{2}-u^{2})\Delta y=\rho (u^{2}-u^{2}+2u\frac{\partial u}{\partial x}\Delta x))\Delta y$$