Dean number
The Dean number ("De") 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 was the first to provide a theoretical solution of the fluid motion through curved pipes for laminar flow by using a perturbation procedure from a Poiseuille flow in a straight pipe to a flow in a pipe with very small curvature. Physical Context. If a fluid is moving along a straight pipe that after some point becomes curved, the centripetal forces at the bend  will cause the fluid particles to change their main direction of motion. There will be an adverse pressure gradient generated from the curvature with an increase in pressure, therefore a decrease in velocity close to the convex wall, and the contrary will occur towards the outer side of the pipe. This gives rise to a secondary motion superposed on the primary flow, with the fluid in the centre of the pipe being swept towards the outer side of the bend and the fluid near the pipe wall will return towards the inside of the bend. This secondary motion is expected  to appear as a pair of counter-rotating cells, which are called Dean vortices. Definition. The Dean number is typically denoted by "De" (or "Dn"). For a flow in a pipe or tube it is defined as: where The Dean number is therefore the product of the Reynolds number (based on axial flow formula_4 through a pipe of diameter formula_5) and the square root of the curvature ratio. Turbulence transition. The flow is completely unidirectional for low Dean numbers (De < 40~60). As the Dean number increases between 40~60 to 64~75, some wavy perturbations can be observed in the cross -section, which evidences some secondary flow. At higher Dean numbers than that (De > 64~75) the pair of Dean vortices becomes stable, indicating a primary dynamic instability. A secondary instability appears for De > 75~200, where the vortices present undulations, twisting, and eventually merging and pair splitting. Fully turbulent flow forms for De > 400. Transition from laminar to turbulent flow has also been examined in a number of studies, even though no universal solution exists since the parameter is highly dependent on the curvature ratio. Somewhat unexpectedly, laminar flow can be maintained for larger Reynolds numbers (even by a  factor of two for the highest curvature ratios studied) than for straight pipes, even though curvature is known to cause instability. 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 formula_10). We use orthogonal coordinates formula_11 with corresponding unit vectors formula_12 aligned with the centre-line of the pipe at each point. The axial direction is formula_13 , with formula_14 being the normal in the plane of the centre-line, and formula_15 the binormal. For an axial flow driven by a pressure gradient formula_16, the axial velocity formula_17 is scaled with formula_18. The cross-stream velocities formula_19 are scaled with formula_20, and cross-stream pressures with formula_21. Lengths are scaled with the tube radius formula_22. 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 formula_28. For larger "De", there are multiple solutions, many of which are unstable. Relation with Nusselt number wherein: