Problem 8.61 [ Difficulty : 4] 8.61 Using Eq. 2.16, derive the velocity profile, flow rate, and average velocity of a non-Newtonian fluid in a circular tube. For a flow rate of Q = 1 ,L/min and R = 1 mm, with & having a value of unity in standard SI units, compare the required pressure gradients for # = 0.5, 1.0, and 15. Which fluid requires the smallest pump for the same pipe length? Given : Fully developed flow , Navier - Stokes equations ; Non - Newtonian fluid Find : Velocity profile , flow rate and average velocity Solution : According to equation (8.10 ), we can write the governing equation for Non -Newtonian fluid velocity in a circular tube du r Op ar 2 ax (1) However , as for the Newtonian fluid case , we must set c, = 0 as otherwise we'd have infinite stress at r = 0. Hence , equation (1) becomes r op 2 ax (2) The general solution for equation (2), obtained by integrating , is given by u = 1 ap 1 (3) 2k ax 1 + Apply the no slip boundary condition at r = R into equation (3), we get ( 1 op \\ " 1 C , = R (4) 2k ax (1 + 1 The fluid velocity then is given as u(r ) =- n ( 1 ap ) (n + 1) 2k 8x) F- R . (5) The volume flow rate is n ( 1 ap Q =[udA = = [ 27 (n + 1) 2k x) F- -R . ar (6) Hence