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 linear advection–diffusion equation:

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.