Pressure gradient along a streamline using CV analysis

In summary: The left hand side is just -\Delta p \Delta y, while the right hand side is -\rho u^2 \Delta y. Where does the factor of 2 come from?
  • #1
Bohr1227
13
0

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?
 
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  • #2
Bohr1227 said:

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?
 
  • #3
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
[tex]\frac{\partial p}{\partial x}=-\rho u\frac{\partial u}{\partial x}[/tex]

Homework Equations


[tex]\sum F=\oint_{CS}^{ } \rho \overrightarrow{V}(\overrightarrow{V_{rel}}\cdot \overrightarrow{n})[/tex]

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: [tex]u+\frac{\partial u}{\partial x}\Delta x[/tex] and [tex]p+\frac{\partial p}{\partial x}\Delta x[/tex]

Just looking at it per unit width inside the paper: (Neglecting small factors because [tex]\Delta x\rightarrow 0[/tex])
[tex](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[/tex]
This gives:
[tex]\frac{\partial p}{\partial x}=-2\rho u\frac{\partial u}{\partial x}[/tex]I get the answer with a factor 2, which I am not supposed to get. What do I do wrong?
 
  • #4
Chestermiller said:
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!
 
  • #5
Bohr1227 said:
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
[tex]\frac{\partial p}{\partial x}=-\rho u\frac{\partial u}{\partial x}[/tex]

Homework Equations


[tex]\sum F=\oint_{CS}^{ } \rho \overrightarrow{V}(\overrightarrow{V_{rel}}\cdot \overrightarrow{n})[/tex]

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: [tex]u+\frac{\partial u}{\partial x}\Delta x[/tex] and [tex]p+\frac{\partial p}{\partial x}\Delta x[/tex]

Just looking at it per unit width inside the paper: (Neglecting small factors because [tex]\Delta x\rightarrow 0[/tex])
[tex](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[/tex]
This gives:
[tex]\frac{\partial p}{\partial x}=-2\rho u\frac{\partial u}{\partial x}[/tex]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:
[tex](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[/tex]
 

1. What is pressure gradient along a streamline?

Pressure gradient along a streamline is a measure of the change in pressure along a flow line in a fluid. It represents the rate of change in pressure per unit distance in a specific direction.

2. How is pressure gradient calculated using CV analysis?

To calculate pressure gradient using CV analysis, the control volume (CV) method is used. This involves drawing a control volume around a fluid element and analyzing the forces acting on it. The pressure gradient is then determined using the Navier-Stokes equations.

3. What factors affect pressure gradient along a streamline?

The pressure gradient along a streamline is affected by several factors, including the fluid's viscosity, density, and velocity. It is also influenced by external forces such as gravity or pressure differences at different points along the streamline.

4. Why is pressure gradient along a streamline important in fluid dynamics?

Pressure gradient along a streamline is an essential concept in fluid dynamics as it helps determine the direction and magnitude of fluid flow. It is used in the analysis of various flow phenomena, such as turbulence, boundary layers, and separation of flow.

5. How does pressure gradient along a streamline affect the behavior of a fluid?

The pressure gradient along a streamline plays a crucial role in determining the behavior of a fluid. It is responsible for the acceleration or deceleration of fluid particles, which can result in changes in flow direction and the formation of vortices. It also affects the amount of drag and lift experienced by an object in the fluid.

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