Fluid Mechanics: Rotating cylinder viscometer

[Feodalherren]

Homework Statement

Homework Equations

τ = μ (du/dy)

The Attempt at a Solution

For part B)

dM = τr dA

τ = μ (rΩ/Δr)

∫ (μrΩr2πr / Δr)dr = μR⁴πΩ / 2Δr

pretty sure that's correct.

I'm confused for part A though.

I want to set it up as

dM = τR dA = τR (2πRdL)

and

τ = μ (RΩ/L)

substitute and integrate over 0 to L. But this doesn't yield the correct result.

 

[Chestermiller]

Part A is done incorrectly. The velocity gradient in the gap is $\frac{ΩR}{Δr}$. So the shear stress is $μ\frac{ΩR}{Δr}$

Chet

 

[Feodalherren]

Ahh of course. It's so obvious now. Thank you, Sir.

Would that mean that part A becomes:

dM = τR dA = τR (2πRdL)

M = ∫ R(2πRdL) (ΩR/Δr) = 2πR³ΩLμ / Δr

correct?

 

[Chestermiller]

Ahh of course. It's so obvious now. Thank you, Sir.

Would that mean that part A becomes:

dM = τR dA = τR (2πRdL)

M = ∫ R(2πRdL) (ΩR/Δr) = 2πR³ΩLμ / Δr

correct?

Yes.

 

[Feodalherren]

Thanks!