This problem concerns the development of the analytical solution for the velocity and

pressure fields in a steady Poiseuille flow of an incompressible fluid of density $$ and

viscosity $$ in the thin gap between a pair of concentric cylindrical surfaces.

As shown in the figure above, the fluid enters through a narrow slit at the top and

leaves through another slit at the bottom. The $zz-$axis points into the plane of the

paper. Assume unit width in the $zz-$direction.

The outer and inner radii of the annular gap are $RR$ and $RR,$ respectively,

where

Assume that the flow between sections $1$ and $2$ is planar and unidirectional.

Assume gravity and other body forces are negligible.

Note that ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}xln(x)dx=\frac{x^2}{2}(ln(x)-\frac{1}{2})+C$

$1.$State the assumptions given in the problem description in a mathematical form. Use these

assumptions and the Continuity Equation to show that $v_{}$ is only a function of the radial

coordinate, $r.$

Use momentum conservation in the radial direction to show that the pressure $p$ can

depend on $r.$

Using momentum conservation in the $-$direction, show that $p=C_0+C_1(r),$ where $C_0$ is a

constant and $C_1$ is a function solely of $r.$

Use the $-$momentum equation along with appropriate boundary conditions to show that

v$\_$theta $=\backslash$left$(\backslash$frac$\{$R$(1-\backslash$kappa$^2)\}\{2\backslash$kappa$\}\backslash$right$)$ A$($r$)-\backslash$frac$\{$r$\}\{2\backslash$left$(\backslash$frac$\{$R$^2\}\{$r$^2\}-1\backslash$right$)\},$ where

$A(r)=\frac{r^{2}ln(r)-R^{2}ln(R)+\frac{R^2-r^2}{2}}{{}^2{R}^{2}ln(R)-R^{2}lnR+\frac{R^2-{}^2{R}^2}{2}}$