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The '''Dean number''' (''D'') is a [[dimensionless quantity|dimensionless group]] in [[fluid mechanics]], which occurs in the study of flow in curved pipes and channels. It is named after the [[Great Britain|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 flow in a pipe or tube it is defined as:

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

where
* <math>\rho</math> is the density of the fluid
* <math>\mu</math> is the dynamic viscosity
* <math>V</math> is the axial velocity scale
* <math>d</math> is the diameter (other shapes are represented by an equivalent diameter, see [[Reynolds number]])
* <math>R</math> 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 <math>V</math> through a pipe of diameter <math>d</math>) and the square root of the curvature ratio.

== The Dean Equations ==

The Dean number appears in the so-called '''Dean Equations'''. These are an approximation to the full [[Navier-Stokes equations|Navier&ndash;Stokes equations]] for the steady axially uniform flow of a [[Newtonian fluid]] in a [[torus|toroidal]] pipe, obtained by retaining just the [[leading-order|leading order]] curvature effects (i.e. the [[leading-order]] equations for <math>a/r \ll 1</math>).

We use [[orthogonal coordinates]] <math>(x,y,z)</math> with corresponding unit vectors <math>(\hat{\boldsymbol{x}},\hat{\boldsymbol{y}},\hat{\boldsymbol{z}})</math> aligned with the centre-line of the pipe at each point. The axial direction is <math>\hat{\boldsymbol{z}}</math>, with <math>\hat{\boldsymbol{x}}</math> being the normal in the plane of the centre-line, and <math>\hat{\boldsymbol{y}}</math> the [[binormal]]. For an axial flow driven by a pressure gradient <math>G</math>, the axial velocity <math>u_z</math> is scaled with <math>U=Ga^2/\mu</math>. The cross-stream velocities <math>u_x, u_y</math> are scaled with <math>(a/R)^{1/2} U</math>, and cross-stream pressures with <math>\rho a U^2/L</math>. Lengths are scaled with the tube radius <math>a</math>.

In terms of these non-dimensional variables and coordinates, the Dean equations are then
:<math>D \left( \frac{\mathrm{D} u_x}{\mathrm{D} t} + u_z^2 \right) = -D \frac{\partial p}{\partial x} + \nabla^2 u_x </math>
:<math>D \frac{\mathrm{D} u_y}{\mathrm{D} t} = -D\frac{\partial p}{\partial y} + \nabla^2 u_y</math>
:<math>D \frac{\mathrm{D} u_z}{\mathrm{D} t} = 1  + \nabla^2 u_z </math>
:<math>\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0</math>
where
:<math>\frac{\mathrm{D}}{\mathrm{D} t} = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y}</math>
is the [[convective derivative]].

The Dean number ''D'' is the only parameter left in the system, and encapsulates the [[leading-order|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 outside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number <math>D_c \approx 956</math> (Dennis &amp; Ng 1982). For larger ''D'', there are multiple solutions, many of which are unstable.

== References ==
* {{cite journal | last1 = Berger | first1 = S. A. | last2 = Talbot | first2 = L. | last3 = Yao | first3 = L. S. | year = 1983 | title = Flow in Curved Pipes | url = | journal = Ann. Rev. Fluid Mech. | volume = 15 | issue = | pages = 461–512 | doi = 10.1146/annurev.fl.15.010183.002333 |bibcode = 1983AnRFM..15..461B }}
* {{cite journal | last1 = Dean | first1 = W. R. | year = 1927 | title = Note on the motion of fluid in a curved pipe | url = http://www.informaworld.com/openurl?genre=article&issn=1941-5982&volume=4&issue=20&spage=208 | journal = Phil. Mag. | volume = 20 | issue = | pages = 208–223 }}
* {{cite journal | last1 = Dean | first1 = W. R. | year = 1928 | title = The streamline motion of fluid in a curved pipe | url = http://www.informaworld.com/openurl?genre=article&issn=1941-5982&volume=5&issue=30&spage=673 | journal = Phil. Mag. (7) | volume = 5 | issue = | pages = 673–695 }}
* {{cite journal | last1 = Dennis | first1 = C. R. | last2 = Ng | first2 = M. | year = 1982 | title = Dual solutions for steady laminar-flow through a curved tube | url = | journal = Q. J. Mech. Appl. Math. | volume = 35 | issue = | page = 305 }}
* Mestel, J. [http://www.ma.ic.ac.uk/~ajm8/M4A33/Dean.pdf Flow in curved pipes: The Dean equations], ''Lecture Handout for Course M4A33'', Imperial College.

{{NonDimFluMech}}

[[Category:Dimensionless numbers]]
[[Category:Fluid dynamics]]