Torque lost using Newton's law of viscosity

[felipe de carli]

Homework Statement

Two circular plates, as indicated in the figure, slide one over the other. Calculate the torque lost in this situation. The rotation w of the internal radius of the plates R1, the outer radius of the plates R2, the absolute viscosity of the lubricant between the plates (η) and the thickness of the lubricant film (e) are known. The speed profile is linear in the lubricant.

Homework Equations

Newton's law of viscosity:
τ = η ∂u / ∂z

The Attempt at a Solution

My first attempt was to get the angular velocity w from the formula v = ω R.
Substituting in the formula, I get τ = η w ∂R / ∂z.
I am stuck in here. How can I derive ∂R / ∂z ? I mean, this derivative should be equal 0, because they are not dependent. Is my approach correct?
Thank you in advance

 

[Chestermiller]

$$\frac{\partial u}{\partial z}=\frac{\omega R}{e}$$

 

[haruspex]

The rotation w of the internal radius

Some text missing in there?

I would start by finding u as a function of r and z.