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Old Aug 15th 2014, 06:05 AM   #1
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Eulerís Equation - Forces

please have a look at the attached pdf file....

I am interested in where the term circled RED comes from and how it translates to the term circled BLUE

many thanks
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File Type: pdf forces on an element of fluid.pdf (91.1 KB, 8 views)
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Old Aug 15th 2014, 08:35 AM   #2
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The term in red is taking the component of p_side that acts in the direction of the axis of the volume and including it in the balance of forces. It would have been clearer is the picture showed p_side acting perpendicular to the surface - I think it's a mistake to have shown it as they did, seemingly being perpendicular to the axis of the volume.

The term in blue is stating that p_side = p_1 plus some incremental amount kdp due to the change in depth of the side relative to the end - that value k is unknown (dependent on geometry), but in a later step is removed as being insignificant.
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Old Aug 15th 2014, 09:01 AM   #3
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Thank you very much for your input... I had already figured out the points that you mentioned... And I agree with you that pressure always acts perpendicular to the surface... They actually show it correctly on one side but not on the other side.....

What I am not sure about is how does Pside acting on (delta A) produces a force that is acting along ds or along the element axis.... As I see it, A and A+Delta A are areas of faces that are normal to ds or the element axis while Pside is acting at an angle to ds... Should there be a sin or cos of the angle that Pside makes with ds or the element axis?

Can you shed light on this please?

Many thanks
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Old Aug 15th 2014, 12:58 PM   #4
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Originally Posted by afaleeds View Post
Should there be a sin or cos of the angle that Pside makes with ds or the element axis?
Yes, you can do it that way. Let's call the length of the side dL. The component of force from P_side acting in the dS direction is p_side times sin(theta) times dL:

F= P_side sin(theta) dL

But sin(theta) = dA/dL, so you have

F= P_side (dA/dL) dL = p_side dA.
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Old Aug 16th 2014, 04:13 AM   #5
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Thank you very much for this... I was actually beginning to see this soon after my last post... I had previously somewhat misled myself by mis-visualising the element.... We are looking at an element that is parallel to the XY plane and then elevated at one end to give it a change in Z..... ie sloped in one direction only along ds....

Ok so Pside on either side work together to make the net effect into Pside Delta A.....

I guess the reason why they are not considering the effect of P on the top and on the bottom faces of the element is that they actually work in opposite directions along ds and hence cancel each other out....

As always, problem shared is a problem solved....

Many thanks....
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