Resistance of the pipe - exact solution

Classical Mechanics Level 4

Using the relation found in the last problem , connecting velocity and radius, we can integrate the velocity over all the fluid layers (from r = 0 r=0 to R R ) and find the expression for volume flow rate from our scaling solution, but now with the correct numerical coefficient.

Which of the expressions gives the correct expression for the hydraulic resistance of a viscous fluid in the stem?

R s t e m = π R 4 8 h η \mathcal{R}_{stem}=\frac{\pi R^4}{8h\eta} R s t e m = 3 π R 4 4 h η \mathcal{R}_{stem}=\frac{3\pi R^4}{4h\eta} R s t e m = R 4 4 π h η \mathcal{R}_{stem}=\frac{R^4}{4\pi h\eta} R s t e m = π R 4 4 h η \mathcal{R}_{stem}=\frac{\pi R^4}{4h\eta}

Josh Silverman by Josh Silverman

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