Math 2.0

Fluid mechanics Definition

Taylor dispersion

Builds on Advection–diffusion equationPoiseuille flowPéclet number

Definition

Consider steady Poiseuille flow in a straight circular pipe of radius aa with mean velocity UU. The flow carries a passive solute with molecular diffusivity D>0D > 0 according to the advection–diffusion equation, with no flux of solute through the wall. Let cˉ(x,t)\bar c(x, t) be the average of the concentration over the cross-section at axial position xx. Taylor dispersion is the regime in which, for times long compared with the radial diffusion time a2/Da^2 / D, the average cˉ\bar c satisfies the one-dimensional equation ∂cˉ∂t+U∂cˉ∂x=Deff∂2cˉ∂x2,Deff=D+a2U248D.\frac{\partial \bar c}{\partial t} + U \frac{\partial \bar c}{\partial x} = D_{\mathrm{eff}} \frac{\partial^2 \bar c}{\partial x^2}, \qquad D_{\mathrm{eff}} = D + \frac{a^2 U^2}{48 D}. The quantity DeffD_{\mathrm{eff}} is the Taylor dispersion coefficient (the formula including the term DD is often called the Taylor–Aris coefficient).

Remarks

With the Péclet number Pe=aU/D\mathrm{Pe} = a U / D, the coefficient is Deff=D(1+Pe2/48)D_{\mathrm{eff}} = D (1 + \mathrm{Pe}^2 / 48). So for large Pe\mathrm{Pe} the spreading along the pipe is much faster than molecular diffusion, and it grows as DD decreases.

Example: for a solute with D=10−5D = 10^{-5} cm2^2/s in a tube of radius a=0.01a = 0.01 cm with U=0.1U = 0.1 cm/s, Pe=100\mathrm{Pe} = 100 and Deff≈209DD_{\mathrm{eff}} \approx 209 D.

Non-example: at times short compared with a2/Da^2 / D, the solute has not yet sampled the cross-section. The band is then mainly stretched by the shear, and cˉ\bar c does not obey the effective equation.

Other geometries give the same form with a different constant; for flow between parallel plates at distance hh the term is h2U2/(210D)h^2 U^2 / (210 D).

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Advection–diffusion equation, concept, no evidence levelAdvection–diffusion e…conceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptNewtonian fluid, concept, no evidence levelNewtonian fluidconceptNo-slip condition, concept, no evidence levelNo-slip conditionconceptPéclet number, concept, no evidence levelPéclet numberconceptViscosity, concept, no evidence levelViscosityconceptPoiseuille flow, example, no evidence levelPoiseuille flowexampleTaylor dispersion, definition, no evidence levelTaylor dispersiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Advection–diffusion equation concept
  2. Navier–Stokes equations concept
  3. Newtonian fluid concept
  4. No-slip condition concept
  5. Péclet number concept
  6. Viscosity concept
  7. Poiseuille flow example
  8. Taylor dispersion definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Taylor_dispersion
About this page
Tag3RJQDCZ4MM
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Kinddefinition
WritingW0 stub
MSC76R50
DefinesTaylor dispersion
UsesAdvection–diffusion equation, Poiseuille flow, Péclet number
Changed3 November 2026

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