Please include the steps needed for this problem! Thank you Some non-Newtonian f
ID: 1818062 • Letter: P
Question
Please include the steps needed for this problem! Thank you
Some non-Newtonian fluids behave as a Bingham plastic for which the shear stress can he expressed as r = ry + mu du/dr. For laminar flow of a Bingham plastic in a horizontal pipe of radius R, the velocity profile is given as u(r)= Delta P/4 mu L (r2-R2) + ry/mu (r-R), where delta P/L is the constant pressure drop along the pipe per unit length and r is the radial distance from the centerline. Determine the shear stress of at the pipe wall and the drag force acting on a pipe section of length L.Explanation / Answer
Given:
u(r) = (P/4L) (r2-R2) +( y/)(r-R)
du/dr = (P/4L)(2r) + ( y/)
At r= R
du/dr = [(P/4L)(2R) + ( y/)]
The shear stress is given by
= y + du/dr
= y + [(P/4L)(2R) + ( y/)]
= y + [(P/4L)(2R) + y]
= 2y + (P/4L)(2R)
The shear stress at the pipe wall is given by
= 2y + (P/4L)(2R)
Drag force on the pipe for a length L is
F= xlateral area of pipe
F= x 2RL
= [2y + (P/4L)(2R)]2RL
= [4yRL + (PR2)]
F= [4yRL + (PR2)]