= Solution
The axial equation is $\mu r^{-1}(ru')'=-G$. Regularity at zero, no slip $u_a(a)=0$, continuity of <velocity> and shear at $c$ give
$$
u_a(r)=\frac{G(a^2-r^2)}{4\mu_a},\qquad u_c(r)=\frac{G(c^2-r^2)}{4\mu_c}+\frac{G(a^2-c^2)}{4\mu_a}.
$$
The 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 $\int_V2\mu_ae_{ij}e_{ij}dV$. Substituting $u_a'=-Gr/(2\mu_a)$ evaluates both sides to the same boundary expression, verifying the mechanical-energy equation.
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