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Channel and fluid
A fluid in a curved channel feels a centrifugal push that sets up two counter-rotating vortices across the section. They grow with the Dean number, which grows with speed and falls with channel curvature radius.
Model checks
The science
What is modelled
Tracer particles move along a spiral channel with a parabolic axial speed, are carried across the section by a prescribed two-cell secondary flow whose strength scales with the Dean number, and diffuse randomly. A two-colour occupancy index measures how well the two inlet streams have mixed over the last part of the channel.
Equations
- Reynolds number
Re = ρ U Dh / μ, hydraulic diameterDh = 2 w h/(w + h). - Dean number
De = Re √(Dh/2R), with R the radius of curvature. - Axial profile
u = (9/4) U (1 - y²)(1 - z²), which has mean U over the square cross-section. - Diffusion step
σ = √(2 D dt)per axis.
Checks run in the page
- Re and De recomputed from the inputs.
- The mean of the axial profile.
- No flux of the secondary flow through any wall.
- Dye conservation: the outlet share of dye A equals the inlet share.
Honest simplifications
The secondary flow is a smooth prescribed streamfunction scaled by a weak-curvature rule of thumb, not a solution of the Navier-Stokes equations; the spiral radius is the median of an Archimedean spiral; the particle lift is a qualitative term. Real mixing and focusing depend on aspect ratio and particle size in ways this model only hints at.
What to try
- Press Fast flow, strong Dean: the cross-section fills with a swirled pattern and the mixing index rises.
- Press Viscous fluid and see that the lower Reynolds number weakens the vortices.
- Raise the diffusivity at low speed: mixing then comes from diffusion rather than from the vortices.