An incompressible fluid with density $\rho$ and viscosity $\mu$ is forced by a pressure difference $\Delta p$ through the narrow gap between two parallel circular cylinders of radius $a$ with axes $2 a+b$ apart. Explaining any approximations made, show that, provided $b \ll a$ and $\rho b^{3} \Delta p \ll \mu^{2} a$, the volume flux (per unit length of cylinder) is

$$\frac{2 b^{5 / 2} \Delta p}{9 \pi a^{1 / 2} \mu}$$

when the cylinders are stationary.

Show also that when the two cylinders rotate with angular velocities $\Omega$ and $-\Omega$ respectively, the change in the volume flux is

$$\frac{4}{3} b a \Omega .$$

For the case $\Delta p=0$, find and sketch the function $f(x)=u_{0}(x) /(a \Omega)$, where $u_{0}$ is the centreline velocity at position $x$ along the gap in the direction of flow. Comment on the values taken by $f$.