Problem $6$

An incompressible fluid $($density $$ and viscosity $)$ flows steadily in the annular region between two stationary, concentric cylinders under an applied pressure difference, as shown. The inner cylinder has radius $R_1$ and the outer cylinder has inner radius $R_2.$ The concentric cylinders have length $L.$ On the upstream side the pressure is $P_1$ and on the downstream side it is $P_2.$ Assume that the flow is laminar and fully developed over the entire length. Neglect gravitational forces.

Cross$-$section view

a$)$ Sketch the velocity profile, $v_z(r),$ and identify important features.

b$)$ Derive an expression for the velocity profile, $v_z(r),$ in terms of $P_1,P_2,R_1,R_2,L,,,$ and $r.$

c$)$ Calculate the average flow speed, $V,$ for the case where $P_1=2$kPa,$P_2=0Pa,R_1=3cm,R_2=6cm,L=10m,=1000k\frac{g}{m^3},$ and $=0\mathrm{.}001$Pas.