Flow Rate of a Fluid Through Tubes

[WhiteCoatSyndrome]

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So maybe I'm digging too deep into this, but I figured I'd ask anyway. According to the continuity equation, flow rate for a given liquid in a pipe is constant regardless of changes in height or cross-sectional area. Poiseuille's equation also illustrates flow rate for an area of a pipe. According to Bernoulli's equation, as height/velocity increases, pressure decreases (This is known to be true), however I am confused how this is possible if flow rate is to be kept constant. For example, if a fluid gains velocity and loses pressure going through a narrower tube, then according to Poiseuille's equation flow rate should decrease too as both pressure and cross-sectional area decrease. I'm probably thinking too much into this, but does anyone have any insight?

 

[Chunkle]

So maybe I'm digging too deep into this, but I figured I'd ask anyway. According to the continuity equation, flow rate for a given liquid in a pipe is constant regardless of changes in height or cross-sectional area. Poiseuille's equation also illustrates flow rate for an area of a pipe. According to Bernoulli's equation, as height/velocity increases, pressure decreases (This is known to be true), however I am confused how this is possible if flow rate is to be kept constant. For example, if a fluid gains velocity and loses pressure going through a narrower tube, then according to Poiseuille's equation flow rate should decrease too as both pressure and cross-sectional area decrease. I'm probably thinking too much into this, but does anyone have any insight?

So if a fluid gains velocity, that means the pressure there is less according to Bernoulli. According to flow rate=vel * area, if the velocity is greater, the area is less, so it cancels out and the flow rate is constant.

 

[WhiteCoatSyndrome]

So if a fluid gains velocity, that means the pressure there is less according to Bernoulli. According to flow rate=vel * area, if the velocity is greater, the area is less, so it cancels out and the flow rate is constant.

Perhaps I wasn't clear enough. What you said all makes sense, I think I'm more confused as to why it doesn't cancel out in Poisueille's equation. For example, in Poisueille's if area decreases then pressure should decrease, both P and A are in the numerator and don't cancel.

 

[indianjatt]

One of the assumptions of the equation is : "through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. The equation is also known as the Hagen–Poiseuille law, Poiseuille law and Poiseuille equation." http://en.wikipedia.org/wiki/Hagen–Poiseuille_equation

So I think when there is a change in cross-sectional area, you cannot use the equation to compare the pressure between the two pipes of different cross sectional area. You can only compare the changes due to other factors like the radius.

 

[WhiteCoatSyndrome]

One of the assumptions of the equation is : "through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. The equation is also known as the Hagen–Poiseuille law, Poiseuille law and Poiseuille equation." http://en.wikipedia.org/wiki/Hagen–Poiseuille_equation

So I think when there is a change in cross-sectional area, you cannot use the equation to compare the pressure between the two pipes of different cross sectional area. You can only compare the changes due to other factors like the radius.

Ok, I figured it would be something along those lines. Essentially Poisueille's equation is best used for a singel location in a pipe and not for comparing flow rate/pressure at different points in a pipe. Bernoulli's equation and the continuity equation are best for that.