Shear Stress Differential Equation

[Wildcat04]

Homework Statement

Equation solved from previous problem:

(g/$$\nu$$) = (1/r) d/dr [ r dv_z/dr]

$$T$$ = $$\mu$$ * dv_z/dr

Boundary condtions:

v_z (r=a) = v_z (r=b) = 0

The Attempt at a Solution

(d/dr)[d/dr r*v_z = g*r / $$\nu$$

(d/dr) r*v_z = gr²/2$$\nu$$ + A

v_z = gr²/6$$\nu$$ + A + B/r

I am not sure if this is correct or what I am supposed to do with the B.C.s. I know that I can have 2 equations with 2 unknowns ( v = 0 @ r = a and v = 0 @ r = b, unks = A, B)

0 = v_z (r=a) = ga²/6$$\nu$$ + A

A = -ga² / 6$$\nu$$

0 = v_z (r=b) = gb²/6$$\nu$$ - ga²/6$$\nu$$ + B/r

B = -[rg(b²-a²)] / 6$$\nu$$

v_z = gr/3$$\nu$$ - ga/3$$\nu$$ -[rg(b²-a²)] / 6$$\nu$$

I need to take the derivative of v_z to get the torque...


However the solution is supposed to be:

$$T$$=($$\mu$$/r) [ (g(b²-a²)/4$$\nu$$) / 2 ln(b/a) - gr²/2$$\nu$$ ]


Obviously I messed up somewhere in there. Could someone please point out my error so I can get this thing correct?

Thank you very much in advance!


I am sorry for it being a little sloppy...the $$\nu$$ kept shooting up to the top!

 

[Wildcat04]

Wow...I did butcher it. Went back and found my mistake.

Thanks!