A liquid flows under a pressure gradient del$\frac{p}{d}$elz through the narrow annular space of a die, a cross section of which is shown in Fig$\mathrm{.}5.$ The coordinate $z$ is in the axial direction, normal to the plane of the diagram. The die consists of a solid inner cylinder with center P and radius b inside a hollow outer cylinder with center O and radius $a.$ The points O and P were intended to coincide but, due to an imperfection of assembly, are separated by a small distance $.$ By a simple geometrical argument based on the triangle OPQ, show that the gap width $$ between the two cylinders is given approximately by:

$=a-b-cos$

where the angle $$ is defined in the diagram. Now consider the radius arm $b$ swung through an angle $d,$ so that

Figure $5$: Off$-$center cylinder inside a die $($gap width exaggerated$)$: $($left$)$ complete cross section; $($right$)$ effect of incrementing $$

it traces an arc of length $bd.$ The flow rate $dQ$ through the shaded element in Fig. $5($right$)$ is approximately that between parallel plates of width $bd$ and separation $.$ Hence, prove that the flow rate through the die is given approximately by:

$Q=bc(2{}^3+3{}^2)$

$4$

in which $c=\frac{1}{12}(-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}})$ and $=a-b$ Assume that the flow rate per unit width between two flat plates separated by a distance $h$ is:

$\frac{h^3}{12}(-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}})$

What is the ratio of the flow rate if the two cylinders are touching at one point to the flow rate if they are concentric?