Infinitesimal generator (stochastic processes)

Taylor dispersion is an effect in fluid mechanics in which a shear flow can increase the effective diffusivity of a species. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it spreads in that direction. The effect is named after the British fluid dynamicist G. I. Taylor.

The canonical example is that of a simple diffusing species in uniform Poiseuille flow through a uniform circular pipe with no-flux boundary conditions.

Description

We use z as an axial coordinate and r as the radial coordinate, and assume axisymmetry. The pipe has radius a, and the fluid velocity is:

${\boldsymbol {u}}=w{\hat {\boldsymbol {z}}}=w_{0}(1-r^{2}/a^{2}){\hat {\boldsymbol {z}}}$

The concentration of the diffusing species is denoted c and its diffusivity is D. The concentration is assumed to be governed by the linear advection–diffusion equation:

${\frac {\partial c}{\partial t}}+{\boldsymbol {w}}\cdot {\boldsymbol {\nabla }}c=D\nabla ^{2}c$

The concentration and velocity are written as the sum of a cross-sectional average (indicated by an overbar) and a deviation (indicated by a prime), thus:

$w(r)={\bar {w}}+w'(r)$

$c(r,z)={\bar {c}}(z)+c'(r,z)$

Under some assumptions (see below), it is possible to derive an equation just involving the average quantities:

${\frac {\partial {\bar {c}}}{\partial t}}+{\bar {w}}{\frac {\partial {\bar {c}}}{\partial z}}=D\left(1+{\frac {a^{2}{\bar {w}}^{2}}{48D^{2}}}\right){\frac {\partial ^{2}{\bar {c}}}{\partial z^{2}}}$

Observe how the effective diffusivity multiplying the derivative on the right hand side is greater than the original value of diffusion coefficient, D. The effective diffusivity is often written as:

$D_{\mathrm {eff} }=D\left(1+{\frac {1}{192}}{\mathit {Pe}}_{d}^{2}\right)\,,$

where ${\mathit {Pe}}_{d}=d{\bar {w}}/D$ is the Péclet number, based on the channel diameter $d=2a$. The effect of Taylor dispersion is therefore more pronounced at higher Péclet numbers.

The assumption is that $c'\ll {\bar {c}}$ for given $z$, which is the case if the length scale in the $z$ direction is long enough to smoothen out the gradient in the $r$ direction. This can be translated into the requirement that the length scale $L$ in the $z$ direction satisfies:

$L\gg {\frac {a^{2}}{D}}{\bar {w}}={\frac {{\mathit {Pe}}_{d}\,d}{4}}$.

Dispersion is also a function of channel geometry. An interesting phenomena for example is that the dispersion of a flow between two infinite flat plates and a rectangular channel, which is infinitely thin, differs approximately 8.75 times. Here the very small side walls of the rectangular channel have an enormous influence on the dispersion.

While the exact formula will not hold in more general circumstances, the mechanism still applies, and the effect is stronger at higher Péclet numbers. Taylor dispersion is of particular relevance for flows in porous media modelled by Darcy's law.

References

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Other sources

• Aris, R. (1956) Template:Doi-inline, Proc. Roy. Soc. A., 235, 67–77.
• Frankel, I. & Brenner, H. (1989) Template:Doi-inline, J. Fluid Mech., 204, 97–119.
• Taylor, G. I. (1953) Template:Doi-inline, Proc. Roy. Soc. A., 219, 186–203.
• Taylor, G. I. (1954) Template:Doi-inline, Proc. Roy. Soc. A, 223, 446–468.
• Taylor, G. I. (1954) Template:Doi-inline, Proc. Roy. Soc. A., 225, 473–477.
• Brenner, H. (1980) Template:Doi-inline, Phil. Trans. Roy. Soc. Lon. A, 297, 81.
• Mestel. J. Taylor dispersion — shear augmented diffusion, Lecture Handout for Course M4A33, Imperial College.