Derivation of the Euler equation for a streamline

[greg_rack]

Hi guys, in the derivation of the Euler equation we apply Newton's 2nd law to a gas flowing through a streamline. To do so, we consider a "box" with sides $dx$ $dy$ and $dz$:

as such;

Here, with reference to the image, I can't understand where does that '$+dp$' comes from, and hence why the pressure would vary on the right side of the box compared to that acting on the left side.

 

[hutchphd]

If the pressure is not uniform along the x axis, then it will change with distance along x. For an infinitesimal change dx there will be an infinitesimal change in pressure. Given the function p(x), then the change of pressure is as described in the picture.

 

[greg_rack]

If the pressure is not uniform along the x axis, then it will change with distance along x. For an infinitesimal change dx there will be an infinitesimal change in pressure. Given the function p(x), then the change of pressure is as described in the picture.

And why would the pressure vary along x?

PS: I'm sorry for asking these probably silly questions, but I'm taking a really cool online course on aerodynamics for fun, without any expertise :)

 

[hutchphd]

For instance there could gravity in the x direction. There could be a Temperature difference. Or any dynamic situation (stuff in motion) can produce pressure differences. It might be zero but the theory requires us to be able to deal with such a difference.

 

[greg_rack]

For instance there could gravity in the x direction. There could be a Temperature difference. Or any dynamic situation (stuff in motion) can produce pressure differences. It might be zero but the theory requires us to be able to deal with such a difference.

Got it, thanks a lot for the brilliant clarification!

 

[boneh3ad]

Hi guys, in the derivation of the Euler equation we apply Newton's 2nd law to a gas flowing through a streamline. To do so, we consider a "box" with sides $dx$ $dy$ and $dz$:

View attachment 275385 as such;

Here, with reference to the image, I can't understand where does that '$+dp$' comes from, and hence why the pressure would vary on the right side of the box compared to that acting on the left side.

You have no reason to assume that it wouldn't vary, so you account for that in the derivation. If the variation isn't important to the final result, it would drop out. It doesn't, so obviously it's an important parameter. If you have a flow in which the pressure does not vary in a given direction, then the associated $\partial p/\partial ?$ term will just be zero.

And why would the pressure vary along x?

PS: I'm sorry for asking these probably silly questions, but I'm taking a really cool online course on aerodynamics for fun, without any expertise :)

To add to some of the things that @hutchphd suggested, it could also be due to any sort of flow acceleration as a result of interaction with an object (e.g. flow over an airfoil or into a corner).