The wall shear stress in a fully developed flow portion of a 12-in.-diameter pipe is 1.85 lb/ft^2.
determine the pressure gradient ∂p/∂x, wherexis the flow direction when (a) the pipe is horizontal, (b) the pipe is vertical with the flow up, and (c) the pipe is vertical with the flowdown.

a)

b)

c)

Explanation:

1) Notation

represent the shear stress  defined as "the external force acting on an object or surface parallel to the slope or plane in which it lies"

R represent the radial distance

L the longitude

since at the begin we have a horizontal pipe, but for parts b and c the angle would change.

D represent the diameter for the pipe

is the specific weight for the water  

2) Part a

For this case we can use the shear stress and the radial distance to find the pressure difference per unit of lenght, with the following formula

If we convert the difference's into differentials we have this:

We can replace and we have this:

Replacing the values given we have:

3) Part b

When the pipe is on vertical upward position the new angle would be , and replacing into the formula we got this:

4) Part c

When the pipe is on vertical downward position the new angle would be , and replacing into the formula we got this: