Why is resistance in fluid flow inversely related to r^4 instead of r^2?

[manohman]

So Volume Flow Rate (Q) = (P2-P1)/R where R is the total resistance of the system.

R is directly proportional to Length and inversely proportional to surface area, and the inherent resistance (viscosity) of the fluid. But R =nL/r^4. r^4 rather than r^2.

So there has to be another factor other than just surface area, that is dependent on radius^2, affecting resistance right? What am I missing?
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In terms of Volume Flow Rate Q, i can understand, since Q = Av and if you change r, you change not only the surface area but also the velocity since the pipe is larger so there is less resistance. A = pi*r^2 AND v is related to radius by a squared factor.

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[Hadi7183]

I thought my previous comment was a bit unclear, so here is another explanation.

The reason R is inversely proportional to r^4 is that, for a given pressure difference (P2-P1), the volumetric flow rate (Q) is actually proportional to A^2 (or r^4), where A is the cross-section area. In other words, R is inversely proportional to A^2 not A. Here is why:

In a pipe, the flow speed (v) and the viscous ******ing force (F) are directly proportional. Since F=(P2-P1)*A, v is directly proportional to A (again, for a given pressure difference). Now we know Q=v*A, therefore Q is proportional to A*A=A^2 (or r^4), and so R is proportional to A^-2 or r^-4.

 

[manohman]

I thought my previous comment was a bit unclear, so here is another explanation.

The reason R is inversely proportional to r^4 is that, for a given pressure difference (P2-P1), the volumetric flow rate (Q) is actually proportional to A^2 (or r^4), where A is the cross-section area. In other words, R is inversely proportional to A^2 not A. Here is why:

In a pipe is, the flow speed (v) and the viscous ******ing force (F) are directly proportional. Since F=(P2-P1)*A, v is directly proportional to A (again, for a given pressure difference). Now we know Q=v*A, therefore Q is proportional to A*A=A^2 (or r^4), and so R is proportional to A^-2 or r^-4.

Oh I see.

So since v is directly proportional to A, when it comes to the formula Q=vA, r^2 comes into play twice, with the velocity which is dependent on SA, AND the surface area component itself?

 

[Hadi7183]

Oh I see.

So since v is directly proportional to A, when it comes to the formula Q=vA, r^2 comes into play twice, with the velocity which is dependent on SA, AND the surface area component itself?

Correct.