Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 38C / Solution

The axial equation is $\mu r^{-1}(r{u}^{\prime }{)}^{\prime }=-G$. Regularity at zero, no slip $u_a(a)=0$, continuity of velocity and shear at $c$ giveThe annulus is controlled only by its own viscosity and the outer no-slip condition, explaining the absence of ${\mu }_c$. For the annular control volume, kinetic-energy rates vanish; pressure work plus interfacial traction work equals ${\int }_{V}2{\mu }_a{e}_{ij}{e}_{ij}dV$. Substituting $u_a^{\prime }=-Gr/(2{\mu }_a)$ evaluates both sides to the same boundary expression, verifying the mechanical-energy equation.