Fluid Mechanics question (shear stress)

[teknodude]

The problem and solutions is here

http://img206.imageshack.us/img206/6093/problem1rm4.th.png

Isn't the shear stress for both the inner and outer negative? When i take the derivative i get negative. The derivative is basically ln ( 1/x) since the rest are just constants.

Edit: I meant that the expression looks similar to ln (1/x).

Also I am clueless to how they got the values for that the velocity destribution graph. They picked a/b = 0.8 and i believe the # has to be less than 1 or the Vz equation is undefined.

 

[Astronuc]

Remeber ln (1/x) = ln (x^-1) = - ln x or

ln (a/b) = -ln (b/a).


Also, if one opens a spreadsheet, enters x = 0.8 . . . 1.0, by .01, and then plots 1/x vs x, one will see that it looks almost straight.

The solution indicates that the plot is almost linear. The radial dimension has been normalized to b, by using r/b for the abscissa.

 

[teknodude]

thanks Astronuc, but i still got a few questions.

Why did they do a/b? Is it because the problem wants to know the ratio of velocity distribution between the two cylinders?

Also where did r/b come from? The only way i see it is that they substituted ln (b/r) with -ln (r/b), but why?

 

[fire]

Either I'm very tired and can't think, or this solution has a serious problem with negative signes everywhere.

A ratio of a/b=0.8 was picked just as an example for a gap.

Using r/b is just a better choice when it comes to interpreting results. When the ratio is 0.8, you're at the inner cylinder (since the smaller cylinder is defined as a/b=0.8), when it's 1, you're at the outer.