sider textbook Problem $2$B$\mathrm{.}7-$ Annular Flow with Inner Cylinder Moving Axially when ner cylinder is stationary and the outer cylinder moves axially with velocity $V_0.$ All other s of the original problem are unchanged.

etermine the velocity profile of the flowing fluid. $(25$ points$)$

etermine the viscous momentum flux in the fluid. $(25$ points$)$

Annular flow with inner cylinder moving axially $($see Fig. $2$B$\mathrm{.}7).$ A cylindrical rod $R$ moves axially with velocity $v_z=v_0$ along the axis of a cylindrical cavity of radius n the figure. The pressure at both ends of the cavity is the same, so that the flu hrough the annular region solely because of the rod motion.

Fig. $2$B$\mathrm{.}7$ Annular flow with the inner cylinder moving axiauy.

$($a$)$ Find the velocity distribution in the narrow annular region.

$($b$)$ Find the mass rate of flow through the annular region.

$($c$)$ Obtain the viscous force acting on the rod over the length $L.$

$($d$)$ Show that the result in $($c$)$ can be written as a "plane slit" formula multip ture correction." Problems of this kind arise in studying the performance of $w$ Answers: $($a$)\frac{v_z}{v_0}=\frac{ln(\frac{r}{R})}{ln}$

$($b$)w=\frac{R^2{v}_0}{2}[\frac{(1-{}^2)}{ln(\frac{1}{})}-2{}^2]$

$($c$)F_z=-2L\frac{v_0}{ln(\frac{1}{})}$

$($d$)F_z=\frac{-2L{v}_0}{}(1-\frac{1}{2}-\frac{1}{12}{}^2+$cdots$)$ where $=1-($see Prot