Problem 1: (65 points) In class, we determined the torque on an inner cylinder of a Couette rheometer that rotated with an angular velocity 2. (a) (20 points) Consider the same setup; instead of solving this problem in dimensional form, use the inner cylinder radius a, velocity of the inner cylinder, and pressure scale of pal to non- dimensionalize distance r, velocity of the fluid, and pressure in the equations of motion (i.e., define new variables 7 =r/a, =v/(a12) =p/(pas??) and provide the boundary conditions for the ODEs in this framework). (b) (15 points) Determine the dimensionless velocity profile. (c) (20 points) Determine the dimensionless pressure distribution. (d) (10 points) Assume the outer cylinder is stationary. Plot the dimensionless velocity profile for E = 0.1,1.0, and 0.4. Can you approximate the shear rate near the inner wall for small values of E ? 121 o 122 2a 2a(1+E) Figure: Schematic of Couette flow rheometer. Problem 1: (65 points) In class, we determined the torque on an inner cylinder of a Couette rheometer that rotated with an angular velocity 2. (a) (20 points) Consider the same setup; instead of solving this problem in dimensional form, use the inner cylinder radius a, velocity of the inner cylinder, and pressure scale of pal to non- dimensionalize distance r, velocity of the fluid, and pressure in the equations of motion (i.e., define new variables 7 =r/a, =v/(a12) =p/(pas??) and provide the boundary conditions for the ODEs in this framework). (b) (15 points) Determine the dimensionless velocity profile. (c) (20 points) Determine the dimensionless pressure distribution. (d) (10 points) Assume the outer cylinder is stationary. Plot the dimensionless velocity profile for E = 0.1,1.0, and 0.4. Can you approximate the shear rate near the inner wall for small values of E ? 121 o 122 2a 2a(1+E) Figure: Schematic of Couette flow rheometer