Dean Number - The Dean Equations

The Dean Equations

The Dean number appears in the so-called Dean Equations. These are an approximation to the full Navier–Stokes equations for the steady axially uniform flow of a Newtonian fluid in a toroidal pipe, obtained by retaining just the leading order curvature effects (i.e. the leading-order equations for ).

We use orthogonal coordinates with corresponding unit vectors aligned with the centre-line of the pipe at each point. The axial direction is, with being the normal in the plane of the centre-line, and the binormal. For an axial flow driven by a pressure gradient, the axial velocity is scaled with . The cross-stream velocities are scaled with, and cross-stream pressures with . Lengths are scaled with the tube radius .

In terms of these non-dimensional variables and coordinates, the Dean equations are then

where

is the convective derivative.

The Dean number D is the only parameter left in the system, and encapsulates the leading order curvature effects. Higher-order approximations will involve additional parameters.

For weak curvature effects (small D), the Dean equations can be solved as a series expansion in D. The first correction to the leading-order axial Poiseuille flow is a pair of vortices in the cross-section carrying flow form the inside to the outrside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number (Dennis & Ng 1982). For larger D, there are multiple solutions, many of which are unstable.