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At critical internal-wave reflection, the reflected group velocity is tangent to the boundary and the inviscid reflected wavenumber diverges. Kinematic viscosity, mass diffusivity, nonlinear steepening, and wave breaking regularize the ideal singularity.

With permeability $K$, depth $d$, kinematic viscosity $\nu$ and the relevant effective thermal diffusivity ${\kappa }_T$, the Darcy Rayleigh number compares buoyancy-driven Darcy advection to thermal diffusion. It differs from the clear-fluid Rayleigh number by replacing $d^3$ with $Kd$. Effective heat capacity and porosity factors may be absorbed into the definition of ${\kappa }_T$ and the thermal-time scale, and must be kept consistent with the chosen Darcy law convention.

An Ekman layer is a viscous boundary layer in a rotating fluid. For constant kinematic viscosity $\nu$ and Coriolis parameter $f$, its characteristic thickness is ${\delta }_E=\sqrt{2\nu /|f|}$, and its velocity turns with distance from the boundary.

The Hartmann number measures the strength of magnetic coupling relative to viscous and resistive effects in a channel:Here $L$ is the channel half-width, $\nu$ the kinematic viscosity, $\eta$ the magnetic diffusivity and ${\sigma }_c$ the electrical conductivity. In Hartmann flow, large $H$ produces thin magnetic-viscous wall layers and a flatter, slower core for fixed pressure gradient.

The linearized Boussinesq equations describe small velocity, pressure, and buoyancy perturbations about a motionless, stably stratified reference state. With kinematic viscosity $\nu$, mass diffusivity $\kappa$, and buoyancy frequency $N$, they are

For constant mass density $\rho$, kinematic viscosity $\nu$ and magnetic permeability ${\mu }_0$, the incompressible momentum equation isIt is the Navier-Stokes equation with Lorentz force density. The magnetic force decomposes into magnetic tension and a magnetic pressure gradient. A uniform streamwise pressure gradient need not make ordinary pressure independent of transverse coordinates.