 Dean number

The Dean number 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 De. For flow in a pipe or tube it is defined as:
where
 ρ is the density of the fluid
 μ 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 ).
We use an orthogonal coordinates (x,y,z) with corresponding unit vectors aligned with the centreline of the pipe at each point. The axial direction is , with being the normal in the plane of the centreline, and the binormal. For an axial flow driven by a pressure gradient G, the axial velocity u_{z} is scaled with U = Ga^{2} / μ. The crossstream velocities u_{x},u_{y} are scaled with (a / R)^{1 / 2}U, and crossstream pressures with ρaU^{2} / L. Lengths are scaled with the tube radius a.
In terms of these nondimensional 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. 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 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.
References
 Berger, S. A.; Talbot, L.; Yao, L. S. (1983). "Flow in Curved Pipes". Ann. Rev. Fluid Mech. 15: 461–512. Bibcode 1983AnRFM..15..461B. doi:10.1146/annurev.fl.15.010183.002333.
 Dean, W. R. (1927). "Note on the motion of fluid in a curved pipe". Phil. Mag. 20: 208–223. http://www.informaworld.com/openurl?genre=article&issn=19415982&volume=4&issue=20&spage=208.
 Dean, W. R. (1928). "The streamline motion of fluid in a curved pipe". Phil. Mag. (7) 5: 673–695. http://www.informaworld.com/openurl?genre=article&issn=19415982&volume=5&issue=30&spage=673.
 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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