Problem $-1$: Rotating rod in a fluid

Figure $4$ shows a horizontal cross section of a long vertical cylinder of radius a that is rotated steadily counterclockwise with an angular velocity $$ in a very large volume of liquid of viscosity $.$ The liquid extends effectively to infinity, where it may be considered at rest. The axis of the cylinder coincides with the $z$ axis.

a$)$ What type of flow is involved? What coordinate system is appropriate?

b$)$ Write down the differential equation of mass and that one of the three general momentum balances that is most applicable to the determination of the velocity $v_{}.$

c$)$ Clearly stating your assumptions, simplify the situation so that you obtain an ordinary differential equation with $v_{}$ as the dependent variable and $r$ as the independent variable.

d$)$ Integrate this differential equation, and introduce any boundary condition$($s$),$ and prove that $v_{}=\frac{a^2}{r}.$

e$)$ Derive an expression for the shear stress ${}_r$ at the surface of the cylinder. Carefully explain the plus or minus sign in this expression.

f$)$ Derive an expression that gives the torque $T$ needed to rotate the cylinder, per unit length of the cylinder.

g$)$ Derive an expression for the vorticity component $(gradv{)}_z.$ Comment on your result.

Figure $4$: Rotating cylinder in a single liquid.