The Dean number (D) is a dimensionless group in fluid mechanics, which occurs in the study of flow in curved pipes and channels. It is named after the British scientist W. R. Dean, who studied such flows in the 1920s (Dean, 1927, 1928).
Definition
The Dean number is typically denoted by the symbol D. For a flow in a pipe or tube it is defined as:

\mathit{D} = \frac{\rho V\! d}{\mu} \left( \frac{d}{2 R} \right)^{1/2}
where

\rho is the density of the fluid

\mu is the dynamic viscosity

V is the axial velocity scale

d is the diameter (other shapes are represented by an equivalent diameter, see Reynolds number)

R is the radius of curvature of the path of the channel.
The Dean number is therefore the product of the Reynolds number (based on axial flow V through a pipe of diameter d) and the square root of the curvature ratio.
The Dean Equations
The Dean number appears in the socalled 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 leadingorder equations for a/r \ll 1).
We use orthogonal coordinates (x,y,z) with corresponding unit vectors (\hat{\boldsymbol{x}},\hat{\boldsymbol{y}},\hat{\boldsymbol{z}}) aligned with the centreline of the pipe at each point. The axial direction is \hat{\boldsymbol{z}}, with \hat{\boldsymbol{x}} being the normal in the plane of the centreline, and \hat{\boldsymbol{y}} the binormal. For an axial flow driven by a pressure gradient G, the axial velocity u_z is scaled with U=Ga^2/\mu. The crossstream velocities u_x, u_y are scaled with (a/R)^{1/2} U, and crossstream pressures with \rho a U^2/L. Lengths are scaled with the tube radius a.
In terms of these nondimensional variables and coordinates, the Dean equations are then

D \left( \frac{\mathrm{D} u_x}{\mathrm{D} t} + u_z^2 \right) = D \frac{\partial p}{\partial x} + \nabla^2 u_x

D \frac{\mathrm{D} u_y}{\mathrm{D} t} = D\frac{\partial p}{\partial y} + \nabla^2 u_y

D \frac{\mathrm{D} u_z}{\mathrm{D} t} = 1 + \nabla^2 u_z

\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0
where

\frac{\mathrm{D}}{\mathrm{D} t} = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y}
is the convective derivative.
The Dean number D is the only parameter left in the system, and encapsulates the leading order curvature effects. Higherorder 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 leadingorder axial Poiseuille flow is a pair of vortices in the crosssection carrying flow form 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 D_c \approx 956 (Dennis & Ng 1982). For larger D, there are multiple solutions, many of which are unstable.
References

Berger, S. A.; Talbot, L.; Yao, L. S. (1983). "Flow in Curved Pipes". Ann. Rev. Fluid Mech. 15: 461–512.

Dean, W. R. (1927). "Note on the motion of fluid in a curved pipe". Phil. Mag. 20: 208–223.

Dean, W. R. (1928). "The streamline motion of fluid in a curved pipe". Phil. Mag. (7) 5: 673–695.

Dennis, C. R.; Ng, M. (1982). "Dual solutions for steady laminarflow through a curved tube". Q. J. Mech. Appl. Math. 35: 305.

Mestel, J. Flow in curved pipes: The Dean equations, Lecture Handout for Course M4A33, Imperial College.
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