Problem 3. (40 points) An incompressible Newtonian liquid is confined between two concentric cylinders of infinite length-a solid inner cylinder of radius RA and a hollow outer cylinder of radius RB. The inner cylinder rotates at angular velocity o and the outer cylinder is stationary.The flow is steady, laminar, and two-dimensional in the r-0 plane. The flow is axi symmetric, meaning that nothing is a function of the coordinate 0. The flow is also circular so that u,=0everywhere from continuity equation, you do not need to derive this. a. Using the 0-momentum equation, generate an exact expression for the velocity component u,as a function of radius r. Ignore gravity. To simplify the solution method for the ODE, use the substitution of terms described in Figure 9-43 and Lesson I.4 b. Derive an expression for the wall shear stress on the inner rotating cylinder. For RA=3 cm, RB=6.0 cm, loi= 2.635 N:s/m?, and w=1500 rpm, plot the velocity profile ue(r)that you found in part a. Also plot a straight line between Ra to RB (the linear velocity profile for Couette flow between two flat plates). Your plot should be done on the computer using a spreadsheet or Matlab. Include a plot title and label the axes with the variable and dimensions. Include a legend identifying each curve. Use only lines, no symbols, to show your analytical solution. Your plot should only show the range from 3 cm to 6 cm. Is the linear velocity profile (planar Couette flow) accurate for this case? 1. For RA=3 cm, Rs=3.01 cm, loi= 2.635 N:s/m?, and w=1500 rpm, calculate the wall shear stress on the inner cylinder in two ways: first using the expression in part b and then by U.wall neglecting the wall curvature and using the planar Couette flow solution, T = !as youh did in HW1P3. (Show the substitution of numbers into each equation before solving.) What is the relative error when the wall curvature is neglected? Is it appropriate to use the planar Couette flow solution for thin lubricating flows as was done for Homework 1? Give your answer for the shear stress values to six digits of accuracy.

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Problem 3. (40 points) An incompressible Newtonian liquid is confined between two concentric cylinders of infinite length-a solid inner cylinder of radius RA and a hollow outer cylinder of radius RB. The inner cylinder rotates at angular velocity o and the outer cylinder is stationary.The flow is steady, laminar, and