These three quantities — the skin depth, the viscous penetration depth, and the thermal penetration depth — share the same mathematical structure. Each describes how far an oscillatory disturbance can penetrate into a dissipative medium. In electromagnetics, the skin depth measures how deeply a time-harmonic field penetrates into a conductor.
In fluid mechanics, the viscous penetration depth gives the thickness of the oscillatory viscous layer near a vibrating boundary.
In heat transfer, the thermal penetration depth describes how far a periodic temperature variation can propagate into a medium. In all three cases, the penetration depth is determined by the competition between diffusivity and oscillation frequency, and scales as
《Physical Hydrodynamics》, §4.5.4 Oscillating flows in a viscous fluid
“ In Physical Hydrodynamics (§4.5.4), the oscillatory viscous boundary-layer problem is explicitly described as analogous to the skin effect in electrical conduction and to the penetration of seasonal temperature variations into the ground.”
