The fluid’s kinematic viscosity, \( \nu = \mu / \rho \), specifies the propensity for vorticity to diffuse through a fluid. Consider \( \underbrace{\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega} \cdot \nabla)\mathbf{u} + \nu \nabla^2 \boldsymbol{\omega}}_{\text{the field equation governing vorticity}} \) for the \(z\)-component of vorticity in a two-dimensional flow confined to the \(x-y\) plane so that \( \boldsymbol{\omega} \cdot \mathbf{u} = 0 \)

\[D \omega_z / Dt = \nu \nabla^2 \omega_z\]

This equation states that the rate of change of \( \omega_z \) following a fluid particle is caused by diffusion of vorticity. Clearly, for the same initial vorticity distribution, a fluid with larger \( \nu \) will produce a larger diffusion term, \( \nu \nabla^2 \omega \), and more rapid changes in the vorticity. This equation is similar to the Boussinesq heat equation:

\[DT / Dt = \kappa \nabla^2 T\]

where \( \kappa \equiv k / \rho c_p \) is the thermal diffusivity, and this similarity suggests that vorticity diffuses in a manner analogous to heat

At a coarse level, this suggestion is correct since both \( \nu \) and \( \kappa \) arise from molecular processes in real fluids and both have the same units (length\(^2\)/time)

The similarity emphasizes that the diffusive effects are controlled by \( \nu \) and \( \kappa \), and not by \( \mu \) (viscosity) and \( k \) (thermal conductivity). In fact, the constant-density, constant-viscosity momentum equation:

\[D \mathbf{u} / Dt = - (1 / \rho) \nabla p + \nu \nabla^2 \mathbf{u}\]

also shows that the fluid particle acceleration due to viscous diffusion is proportional to \( \nu \)

Thus, at room temperature and pressure, air (\( \nu = 15 \times 10^{-6} \, \text{m}^2/\text{s} \)) is 15 times more diffusive than water
(\( \nu = 1 \times 10^{-6} \, \text{m}^2/\text{s} \)), although \( \mu \) for water is larger

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