Taylor dispersion
Encyclopedia
Taylor dispersion is an effect in fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

in which a shear flow
Shear flow
The term shear flow is used in solid mechanics as well as in fluid dynamics. Loosely speaking, shear flow is defined as:* the gradient of a shear stress force through the body ;...

can increase the effective diffusivity
Diffusivity
Diffusivity can refer to:*Diffusivity of heat*Diffusivity of mass:** Molecular diffusivity ** Eddy diffusivity*Momentum 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
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM was a British physicist, mathematician and expert on fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as "one of the most notable scientists of this century".-Biography:Taylor was born in St. John's Wood, London...

.

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

The concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

of the diffusing species is denoted c and its
diffusivity
Diffusivity
Diffusivity can refer to:*Diffusivity of heat*Diffusivity of mass:** Molecular diffusivity ** Eddy diffusivity*Momentum diffusivity...

is D. The concentration is assumed to be governed by

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:

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

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

where is the Péclet number
Péclet number
The Péclet number is a dimensionless number relevant in the study of transport phenomena in fluid flows. It is named after the French physicist Jean Claude Eugène Péclet. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same...

, based on the channel diameter . The effect of Taylor dispersion is therefore more pronounced at higher Péclet numbers.

The assumption is that for given , which is the case if the length scale in the direction is long enough to smoothen out the gradient in the direction. This can be translated into the requirement that the length scale in the direction satisfies
.

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
Porous medium
A porous medium is a material containing pores . The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid...

modelled by Darcy's law
Darcy's law
Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

.

## Other References

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