Viscous diffusion time scale

For length $L>0$ and kinematic viscosity $\nu>0$ , the viscous diffusion time scale is

t_{\mathrm{diff}}=\frac{L^2}{\nu}.

Balancing $\partial_tu$ against $\nu\Delta u$ at spatial scale $L$ gives $U/t_{\mathrm{diff}}$ of size $\nu U/L^2$ , yielding this scale.

Interpretation

This is a characteristic scale, not an exact decay time for every solution. A Fourier mode of wave number magnitude $k$ under pure diffusion decays like $e^{-\nu k^2t}$ , with decay time $(\nu k^2)^{-1}$ ; a convention for wavelength introduces corresponding numerical factors.