Kinematic viscosity ($\nu =\mu /\rho$)

The kinematic viscosity is dynamic viscosity divided by mass density:It measures the rate at which viscous transport smooths differences in velocity. For constant $\mu$ and $\rho$ in incompressible flow, dividing the Navier-Stokes equation by mass density makes the viscous acceleration $\nu {\nabla }^2\mathbf{u}$, where ${\nabla }^2$ is the Laplacian. ^[ref]

For example, stipulate two Newtonian fluids with the same dynamic viscosity $0.10\mathrm{Pa}\mathrm{s}$ but different mass densities. In the same steady Couette flow as above, both have shear stress $2.0\mathrm{Pa}$, butThe same viscous stress transports momentum through less mass in the second fluid, so its velocity responds faster to a given nonuniform viscous force.

To see the diffusion equation directly, consider an unsteady parallel velocity field $\mathbf{u}=(u(y,t),0,0)$ with constant material coefficients and no streamwise pressure gradient or body force. Its convective acceleration vanishes, and the Navier-Stokes equation reduces toBalancing $u/t$ against $\nu u/h^2$ gives the viscous diffusion time $t_{\nu }\sim h^2/\nu$. For the $1.0\mathrm{mm}$ gap, the two fluids have scales $0.010\mathrm{s}$ and $0.0050\mathrm{s}$. This is a characteristic scale, not an exact settling time. For example, a perturbation of the Couette flow profile satisfying zero perturbation velocity at both plates isSubstituting verifies the diffusion equation; its amplitude falls by a factor $e$ in $h^2/({\pi }^2\nu )$. Kinematic viscosity is therefore useful for predicting how quickly a moving wall influences the interior, while dynamic viscosity gives the shear stress at a prescribed shear rate.

Viscous transfer of tangential momentum smooths gradients of velocity. For unidirectional incompressible flow without a pressure gradient, the Navier-Stokes equation reduces to the displayed diffusion equation, whose diffusion coefficient is the kinematic viscosity $\nu =\eta /\rho$.

For a vertically stratified accretion disk with surface density $\Sigma =\int \rho dz$, the density-weighted mean kinematic viscosity is $\bar{\nu }={\Sigma }^{-1}\int \rho \nu dz$. In an alpha disk with $\nu =\alpha p/(\rho \Omega )$, this becomes $\bar{\nu }=\alpha (\Sigma \Omega {)}^{-1}\int pdz$.

// Target: astrophysics.bigb

If angular velocity is independent of height, integrating the viscous stress $\rho \nu r{\Omega }^{\prime }$ gives $\Sigma \bar{\nu }r{\Omega }^{\prime }$. Thus the surface density weighted average, rather than an unweighted height average, enters vertically averaged viscous disk equations.