# Fluid Mechanics: Cone-Plate viscometer

1. Jan 22, 2015

### Feodalherren

1. The problem statement, all variables and given/known data

2. Relevant equations
Fluid Mechanics

τ =μ (du/dy)
3. The attempt at a solution

I got far enough to write down

dM = μ (Ωr/tanθ) dA

from just substitutions, easy enough.

I get confused when I'm solving for the differential surface area. I somehow need the dA for a cone.
For a circle it's easy enough. A=πr^2 so then dA = 2πr dr.

But how do I go about getting this for a cone? the book lists it as (2πr/cosθ) dr, without any explanation, of course.

2. Jan 22, 2015

### erisedk

Take a strip element 2πxdr, where x is the radius of circle formed by that strip element. Try relating x to r using similarity of triangles.

3. Jan 22, 2015

### Nathanael

Well you should first understand how to find the surface area of a "truncated cone" (ignoring the faces)
It is the average circumference of the cone multiplied by the side length (not the height)

So the surface area of a truncated cone (ignoring the faces) is $2\pi r_{avg}L$ or you could say $2\pi L\frac{r_1+r_2}{2}$
(Side note, this also applies to normal cones; just treat the tip as r2=0)

So with this understanding of truncated cones, look at the following picture I made:

The differential surface area would be $2\pi r_{avg}L$ but as you can see from the picture, $L=\frac{dr}{\cos\theta}$ therefore the differential surface area is $\frac{2\pi R_{avg}}{\cos\theta}dr$

The two radii are r and (r+dr) but since dr is obviously infinitesimal, it suffices to say $r_{avg}=r$

Last edited: Jan 22, 2015
4. Jan 22, 2015

### erisedk

Does it really require all that? Isn't using similarity to relate x and R, considering a string element of radius x, just enough?

5. Jan 22, 2015

### Nathanael

No that is incorrect. The differential area element is $\frac{2\pi x}{\cos\theta}dx$

Me and the OP were using r not to represent the radius of the base of the cone, but to represent what you called "x"

6. Jan 22, 2015

### erisedk

Oh ok.

7. Jan 22, 2015

### Feodalherren

Excellent explanation. Thank you very much, Sir.