$1)$ In a wire coating operation, a wire is pulled out of a cylindrical extruder die at speed $\backslash ($ v$\_\{0\}\backslash ).$ As the wire is pulled out of the die the polymer in the annular space coats the wire. In addition, product quality requires that the shear stress in the coating at the wire surface is zero; therefore, in addition to the motion of the wire, the coating is also forced around the wire with a pressure per unit length. The fluid is Newtonian. The situation is illustrated in the diagram below.

Instead of using $\backslash ($ R$\_\{$I$\}\backslash )$ and $\backslash ($ R$\_\{$O$\}\backslash ),$ you will probably find it more useful to use $\backslash ($ R $\backslash )$ and $\backslash ($ k R $\backslash )$ as in the diagram below.

a$)$ Simplify the appropriate form of the equation of motion to obtain a differential equation describing this system.

b$)$ Identify appropriate boundary conditions for this problem.

c$)$ Given a wire speed $\backslash ($ v$\_\{0\}\backslash ),$ solve the problem to obtain an expression for the pressure drop per unit length need to meet product requirements.