text: The Dean number appears in the so-called Dean equations. These are an approximation to the full Navier&ndash;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 De 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 De), the Dean equations can be solved as a series expansion in De. The first correction to the leading-order axial Poiseuille flow is a pair of vortices in the cross-section carrying flow from the inside to the outside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number . For larger De, there are multiple solutions, many of which are unstable.