We investigate viscous dissipation in linear flows driven by small-amplitude longitudinal librations in rotating fluid spheres focusing on the rapid rotation regime applicable to planets. Viscous coupling can resonate with inertial modes in the bulk of the fluid when the frequency of the forcing is within the range $(0,2\Omega _0)$, where $\Omega _0$ is the mean angular velocity of the sphere. We solve the linearised equations of motion with a semi-spectral numerical method and with an asymptotic expansion exploiting the small Ekman number, $E$, which quantifies the strength of viscous forces relative to the Coriolis force. Our results confirm that the dominant contribution to the dissipation occurs in the Ekman boundary layer with leading-order scaling $E^{1/2}$. When the forcing frequency coincides with that of an inertial mode, dissipation is reduced by as much as 9 % compared with boundary layer theory alone. The percentage-wise reduction is independent of $E$ and the frequency width of the reduction envelope scales as $E^{1/2}$. At non-resonant frequencies conic shear layers develop in the bulk interior and, together with the Ekman layer bulge at critical latitude, slightly enhance dissipation. We confirm critical latitude bulge and shear layer contributions to the overall dissipation scale as $E^{4/5}$ and $E^{6/5}$ respectively, becoming negligible compared with dissipation in the main boundary layer as $E\rightarrow 0$. The frequencies at which the dissipation enhancement from critical latitude effects is maximised are displaced from the inviscid limit periodic orbit frequencies by a factor that scales with $E^{0.23}$.

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