Taylor dispersion or Taylor diffusion or Shear-induced dispersion is an apparent or effective diffusion of some scalar field arising on the large scale due to the presence of a strong, confined, zero-mean shear flow on the small scale. 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, who described the shear-induced dispersion for large Peclet numbers. The analysis was later generalized by Rutherford Aris for arbitrary values of the Peclet number, and hence the process is sometimes also referred to as Taylor-Aris dispersion.
Article
Taylor dispersion or Taylor diffusion or Shear-induced dispersion is an apparent or effective diffusion of some scalar field arising on the large scale due to the presence of a strong, confined, zero-mean shear flow on the small scale. 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, who described the shear-induced dispersion for large Peclet numbers. The analysis was later generalized by Rutherford Aris for arbitrary values of the Peclet number, and hence the process is sometimes also referred to as Taylor-Aris dispersion.
The canonical example is that of a simple diffusing species in uniform Poiseuille flow through a uniform circular pipe with no-flux boundary conditions, but is relevant in many other contexts, including the spread of pollutants in rivers and of drugs in blood flow and rivulet flow.
Contents
Description
We use z as an axial coordinate and r as the radial coordinate, and assume axisymmetry. The pipe has radius a, and
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
)
=
w
¯
+
w
′
(
r
)
{\displaystyle w(r)={\bar {w}}+w'(r)}
c
(
r
,
z
)
=
c
¯
(
z
)
+
c
′
(
r
,
z
)
{\displaystyle 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:
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:
. The interesting result is that for large values of the Péclet number, the effective diffusivity is inversely proportional to the molecular diffusivity. The effect of Taylor dispersion is therefore more pronounced at higher Péclet numbers.
In a frame moving with the mean velocity, i.e., by introducing
ξ
=
z
−
w
¯
t
{\displaystyle \xi =z-{\bar {w}}t}
, the dispersion process becomes a purely diffusion process,
Dispersion is also a function of channel geometry. An interesting phenomenon 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.
Derivation
One may derive the Taylor equation using method of averages, first introduced by Aris. The result can also be derived from large-time asymptotics, which is more intuitively clear. In the dimensional coordinate system
(
x
′
,
r
′
,
θ
)
{\displaystyle (x',r',\theta )}
, consider the fully-developed Poiseuille flow
u
=
2
U
[
1
−
(
r
′
/
a
)
2
]
{\displaystyle u=2U[1-(r'/a)^{2}]}
flowing inside a pipe of radius
a
{\displaystyle a}
Higher order asymptotics
Integrating the equations obtained at the second order, we find
c
2
(
ξ
,
τ
)
=
c
2
a
(
ξ
,
τ
)
+
P
e
4
(
r
2
−
r
4
2
)
∂
c
1
a
∂
ξ
+
P
Generalization to other flows
The shear dispersion for oscillatory flows was studied by E. J. Watson and L. B. Progozhin. The concept of shear dispersion accounting for the effect of variable fluid properties such as density, viscosity and diffusivity is generalized recently. The concept of shear dispersion can be generalised to spatio-temporal periodic flows. Extension to Ostroumov flows is due to M. Emin Erdogan and Phillip C. Chatwin.
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
, where
U
{\displaystyle U}
is the average velocity of the fluid. A species of concentration
c
{\displaystyle c}
with some arbitrary distribution is to be released at somewhere inside the pipe at time
t
′
=
0
{\displaystyle t'=0}
. As long as this initial distribution is compact, for instance the species/solute is not released everywhere with finite concentration level, the species will be convected along the pipe with the mean velocity
U
{\displaystyle U}
. In a frame moving with the mean velocity and scaled with following non-dimensional scales
Substituting this series into the governing equation and collecting terms of different orders will lead to series of equations. At leading order, the equation obtained is
This equation can also be integrated with respect to
r
{\displaystyle r}
, but what is required is the solvability condition of the above equation. The solvability condition is obtained by multiplying the above equation by
2
r
d
r
{\displaystyle 2rdr}
and integrating the whole equation from
r
=
0
{\displaystyle r=0}
to
r
=
1
{\displaystyle r=1}
. This is also the same as averaging the above equation over the radial direction. Using the boundary conditions and results obtained in the previous two orders, the solvability condition leads to
