Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 39C (Fluid Dynamics II)

Steadiness, axial and rotational symmetry, and the absence of any imposed radial or azimuthal motion make the velocity independent of $z$ and $\theta$ and directed along the cylinders. Incompressibility is then automatic, so

The radial momentum equation makes $p$ independent of $r$. The axial Navier-Stokes equation reduces toThe no-slip boundary conditions are

The reduced axial equation shows that $dp/dz$ is constant, because its other terms depend only on $r$. Both open ends are at the same atmospheric pressure. Since a linear pressure cannot take the same value at the two ends unless its slope vanishes,

PutIntegrating the equation in part (b) and imposing both no-slip conditions givesFor $V=0$, gravity drives downward annular Poiseuille flow, positive in the interior and zero at both walls. Positive $V$ raises the inner-wall endpoint to $V$ and strengthens the downward flow; sufficiently negative $V$ creates a region of upward flow near the inner cylinder. In each case the graph is the displayed quadratic-plus-logarithmic profile.

The shear stress is ${\tau }_{rz}=\mu u^{\prime }(r)$, whereThus the axial forces exerted by the fluid per unit length on the inner and outer cylinders areTheir sum isthe weight per unit length of the fluid in the annulus. This is the expected overall vertical force balance.