In general, a tangential harmonic oscillation of amplitude u0 and frequency
f applied to a wall at
z = 0 creates a viscous wave of the form
where, f is the frequency,
ρ0 is the static density, and
μ is the dynamic viscosity. The viscous shear waves are therefore dispersive with wavelength
and highly damped since their amplitude decays exponentially with distance from the boundary (see Ref. 3). In fact, in just one wavelength, the amplitude decreases to about
1/500 of its value at the boundary. Therefore, the viscous boundary layer thickness can for most purposes be considered to be less than
Lv. The length scale
δv is the so-called viscous penetration depth or viscous boundary layer thickness.
where Cp is the heat capacity at constant pressure and
k is the thermal conductivity. The wavelength is here
In air, this ratio is roughly 0.8, while in water, it is closer to
2.7. Thus, at least in these important cases, the viscous and thermal boundary layers are of the same order of magnitude. Therefore, if one effect is important for a particular geometry, so is probably the other.