A Stormer viscometer consists essentially of two concentrie cylinders, the inner of which rotates

while the outer is held stationary. Viscosity is determined by measuring the rate of rotation of the

inner cylinder under the application of a known torque, It is thus quite similar to the Couette

viscometer. Develop an expression for velocity distribution in this kind of apparatus, as a function

of applied torque, for laminar flow of a Newtonian fluid. Neglect end effects. The relevant equations

are:

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}+\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r{v}_r)+\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(v_{})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}(v_z)=0$

$(\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{t}}+v_r\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_{}}{r}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+v_z\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{z}}+\frac{v_r{v}_{}}{r})=\{\begin{array}{c}-\frac{1}{r}\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}}+g_{}\end{array}$

$-[\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r^2{}_r)+\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}{}_s+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}{}_z+\frac{{}_r-{}_{}}{r}]-(2)$

${}_r={}_r=-[r\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(\frac{v_{}}{r})+\frac{1}{r}\frac{del{v}_r}{\text{d}\text{e}\text{l}}]$

PLEASE SOLVE EACH STEP CLEARLY WITH COMPLETE DERIVATION BY HAND BY USING BIRD TRANSPORT PHENOMENA EDITION $2$ STYLE BOOK AND WITH COMPLETE LABELLED DIAGRAM $!$