Problem 2 You are a chemical engineer to design a new kind of electrically conducting polymer melt. To measure the viscosity of the polymer melt, you suggested using an annular-type cylindrical tank (tank radius: r₂ ) inside which is a rotating rod (rod radius: r₁, rotating angular velocity: ). Your assistant reported the typical value of the new polymer melt. You can assume that all the physical properties including the density () and viscosity () of the polymer melts are constants. You can also assume that the length of the rotating rod (H ) and the cylindrical tank is sufficiently long. Now, please address the following question.

2.1) From the continuity equation, please suggest the profile of the velocity component (i.e., v_r, v , v_z) of the fluid in the annular region of the tank in CCS as a function of (r, , z, t).

2.2) Using NS-equation in CCS, please suggest governing differential equation(s).

2.3) Your assistant suggested that you can use p = h(z) + f(r) + g() . From the simplified NS-equation in CCS and continuity equation, please derive algebraic or differential equation(s) on h(z), f(r), and g().

2.4) From the results in 2.3, please solve v(r). For this problem, you need to be very careful to check the physical unit.

2.5) Actually, you can suggest that g() = 0 even before you derive v(r). Why can you do this? Please comment on this.

2.6) Your assistant reported that p( r=r_2, =_, z=0 ). Please construct the complete profile of the pressure.

2.7) Now, you are ready to derive the torque measured at r = r₂. Your assistant reported that the absolute value of the torque per unit length of the cylindrical tank is equal to T. Please derive the viscosity of the new polymer melt in terms of , r₁, r₂, and T.

2.8) Your colleague suggested comparing your result in 2.7 by doing the experiment again to check the consistency of the measurement. In this case, your colleague suggested rotating the outer tank at an angular velocity of while fixing the inner rod. Please derive v(r) for this case.

2.9) For the case in 2.8, your assistant reported that the absolute value of the torque measured at the surface of the inner rod per unit length of the cylindrical tank is again equal to T. Is this report reliable? Yes or no, please comment on this.

Fluid Rotating cylinder 1. Figure 1. Polymer melt inside the annular region of the cylindrical tank with a rotating rod. Fluid Rotating cylinder 1. Figure 1. Polymer melt inside the annular region of the cylindrical tank with a rotating rod