Find the velocity distribution for a steady state laminar flow of Newtonian fluid of viscosity $$ and

constant density $$ through a vertical annulus shown in the figure using shell momentum balances.

The flow occurs between two co$-$axial cylinders placed at radii $R$ and $R$ and is driven by a pressure

gradient $($ gradP where $P$ includes both thermodynamic and hydrostatic pressure, we also called it flow

potential$)$ in vertical direction.

a$)$ What are velocity assumptions in this case?

b$)$ What are the boundary conditions in this problem?

c$)$ Follow the procedure for flow in a vertical tube in terms of setting up the equations and

finding the velocity field. Useful tables for cylindrical system from BSL are given below

in Figure $1.$ The application of the boundary conditions changes the solution at the end.

d$)$ Plot the velocity and shear stress profiles vs$.$ radius in Python.

e$)$ Integrate the velocity field to obtain the volumetric flux and linearized flux $($average

velocity$).$

$$B$.1$ NEWTON'S LAW OF VISCOSITY $($continued$)$

${}_r=-[2\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{r}}]+(\frac{2}{3}-)(grad*v)$

${}_{}=-[2(\frac{1}{r}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+\frac{v_r}{r})]+(\frac{2}{3}-)(grad*v)$