A viscous fluid with constant rho and μ flows through an inclined circular pipe with radius, R, and length, L. The pipe is inclined upwards and ϕ is the angle between the pipe and horizontal (ϕ>0 if inclined upwards). The gravity vector, g
​
=−g k
^
, acts downward in the vertical direction. The flow is steady, incompressible, fully developed, axisymmetric, laminar, and there is no swirl. Assume both an applied constant pressure gradient, ∂p/∂x, and gravity influences the flow. a. As done in class, perform differential analysis (begin with general equations (6.33) and (6.128)) and determine the velocity profile in the pipe. Show all steps. b. Derive expressions for the wall shear stress and volume flow rate (in terms of pressure drop, Δp ). c. Sketch the velocity and shear stress radial profiles (assume flow is upwards in inclined pipe)