Taylor dispersion

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 ${\displaystyle (x^{\prime },r^{\prime },\theta )}$, consider the fully-developed Poiseuille flow ${\displaystyle u=2U[1-(r^{\prime }/a{)}^2]}$ flowing inside a pipe of radius ${\displaystyle a}$, where ${\displaystyle U}$ is the average velocity of the fluid. A species of concentration ${\displaystyle c}$ with some arbitrary distribution is to be released at somewhere inside the pipe at time ${\displaystyle t^{\prime }=0}$. 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 ${\displaystyle U}$. In a frame moving with the mean velocity and scaled with following non-dimensional scales

${\displaystyle t=\frac{t^{\prime }}{a^2/D},x=\frac{x^{\prime }-U{t}^{\prime }}{a},r=\frac{r^{\prime }}{a},Pe=\frac{Ua}{D}}$

where ${\displaystyle a^2/D}$ is the time required for the species to diffuse in the radial direction, ${\displaystyle D}$ is the diffusion coefficient of the species and ${\displaystyle Pe}$ is the Peclet number, the governing equations are given by

${\displaystyle \frac{\mathrm{\partial }c}{\mathrm{\partial }t}+Pe(1-2r^2)\frac{\mathrm{\partial }c}{\mathrm{\partial }x}=\frac{{\mathrm{\partial }}^{2}c}{\mathrm{\partial }{x}^2}+\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }c}{\mathrm{\partial }r}\right).}$

Thus in this moving frame, at times ${\displaystyle t\sim 1}$ (in dimensional variables, ${\displaystyle t^{\prime }\sim a^2/D}$), the species will diffuse radially. It is clear then that when ${\displaystyle t\gg 1}$ (in dimensional variables, ${\displaystyle t^{\prime }\gg a^2/D}$), diffusion in the radial direction will make the concentration uniform across the pipe, although however the species is still diffusing in the ${\displaystyle x}$ direction. Taylor dispersion quantifies this axial diffusion process for large ${\displaystyle t}$.

Suppose ${\displaystyle t\sim 1/ϵ\gg 1}$ (i.e., times large in comparison with the radial diffusion time ${\displaystyle a^2/D}$), where ${\displaystyle ϵ\ll 1}$ is a small number. Then at these times, the concentration would spread to an axial extent ${\displaystyle x\sim \sqrt{t}\sim \sqrt{1/ϵ}\gg 1}$. To quantify large-time behavior, the following rescalings

${\displaystyle \tau =ϵt,\xi =\sqrt{ϵ}x}$

can be introduced. The equation then becomes

${\displaystyle ϵ\frac{\mathrm{\partial }c}{\mathrm{\partial }\tau }+\sqrt{ϵ}Pe(1-2r^2)\frac{\mathrm{\partial }c}{\mathrm{\partial }\xi }=ϵ\frac{{\mathrm{\partial }}^{2}c}{\mathrm{\partial }{\xi }^2}+\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }c}{\mathrm{\partial }r}\right).}$

If pipe walls do not absorb or react with the species, then the boundary condition ${\displaystyle \mathrm{\partial }c/\mathrm{\partial }r=0}$ must be satisfied at ${\displaystyle r=1}$. Due to symmetry, ${\displaystyle \mathrm{\partial }c/\mathrm{\partial }r=0}$ at ${\displaystyle r=0}$.

Since ${\displaystyle ϵ\ll 1}$, the solution can be expanded in an asymptotic series, ${\displaystyle c=c_0+\sqrt{ϵ}{c}_1+ϵc_2+\cdots }$ 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

${\displaystyle \frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }r}\right)=0.}$

Integrating this equation with boundary conditions defined before, one finds ${\displaystyle c_0=c_0(\xi ,\tau )}$. At this order, ${\displaystyle c_0}$ is still an unknown function. This fact that ${\displaystyle c_0}$ is independent of ${\displaystyle r}$ is an expected result since as already said, at times ${\displaystyle t^{\prime }\gg a^2/D}$, the radial diffusion will dominate first and make the concentration uniform across the pipe.

Terms of order ${\displaystyle \sqrt{ϵ}}$ leads to the equation

${\displaystyle \frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }{c}_1}{\mathrm{\partial }r}\right)=Pe(1-2r^2)\frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }\xi }.}$

Integrating this equation with respect to ${\displaystyle r}$ using the boundary conditions leads to

${\displaystyle c_1(\xi ,r,\tau )=c_{1a}(\xi ,\tau )+\frac{Pe}{8}(2r^2-r^4)\frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }\xi }}$

where ${\displaystyle c_{1a}}$ is the value of ${\displaystyle c_1}$ at ${\displaystyle r=0}$, an unknown function at this order.

Terms of order ${\displaystyle ϵ}$ leads to the equation

${\displaystyle \frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }{c}_2}{\mathrm{\partial }r}\right)=Pe(1-2r^2)\frac{\mathrm{\partial }{c}_1}{\mathrm{\partial }\xi }+\frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }\tau }-\frac{{\mathrm{\partial }}^2{c}_0}{\mathrm{\partial }{\xi }^2}.}$

This equation can also be integrated with respect to ${\displaystyle r}$, but what is required is the solvability condition of the above equation. The solvability condition is obtained by multiplying the above equation by ${\displaystyle 2rdr}$ and integrating the whole equation from ${\displaystyle r=0}$ to ${\displaystyle r=1}$. 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

${\displaystyle \frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }\tau }=\left(1+\frac{P{e}^2}{48}\right)\frac{{\mathrm{\partial }}^2{c}_0}{\mathrm{\partial }{\xi }^2}\Rightarrow \frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }t}=\left(1+\frac{P{e}^2}{48}\right)\frac{{\mathrm{\partial }}^2{c}_0}{\mathrm{\partial }{x}^2}.}$

This is the required diffusion equation. Going back to the laboratory frame and dimensional variables, the equation becomes

${\displaystyle \frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }{t}^{\prime }}+U\frac{\mathrm{\partial }{c}_0}{\mathrm{\partial }{x}^{\prime }}=D\left(1+\frac{U^2{a}^2}{48D^2}\right)\frac{{\mathrm{\partial }}^2{c}_0}{\mathrm{\partial }{x}^{\prime 2}}.}$

By the way in which this equation is derived, it can be seen that this is valid for ${\displaystyle t^{\prime }\gg a^2/D}$ in which ${\displaystyle c_0}$ changes significantly over a length scale ${\displaystyle x^{\prime }\gg a}$ (or more precisely on a scale ${\displaystyle x\sim \sqrt{D{t}^{\prime }})}$. At the same time scale ${\displaystyle t^{\prime }\gg a^2/D}$, at any small length scale about some location that moves with the mean flow, say ${\displaystyle x^{\prime }-U{t}^{\prime }=x_s^{\prime }-U{t}^{\prime }}$, i.e., on the length scale ${\displaystyle x^{\prime }-x_s^{\prime }\sim a}$, the concentration is no longer independent of ${\displaystyle r}$, but is given by ${\displaystyle c=c_0+\sqrt{ϵ}{c}_1.}$

Higher order asymptotics

Integrating the equations obtained at the second order, we find

${\displaystyle c_2(\xi ,\tau )=c_{2a}(\xi ,\tau )+\frac{Pe}{4}\left(r^2-\frac{r^4}{2}\right)\frac{\mathrm{\partial }{c}_{1a}}{\mathrm{\partial }\xi }+\frac{P{e}^2}{32}\left(\frac{r^2}{6}+\frac{r^4}{2}-\frac{5r^6}{8}+\frac{r^8}{8}\right)\frac{{\mathrm{\partial }}^2{c}_0}{\mathrm{\partial }{\xi }^2}}$

where ${\displaystyle c_{2a}(\xi ,\tau )}$ is an unknown at this order.

Now collecting terms of order ${\displaystyle ϵ\sqrt{ϵ}}$, we find

${\displaystyle \frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }{c}_3}{\mathrm{\partial }r}\right)=Pe(1-2r^2)\frac{\mathrm{\partial }{c}_2}{\mathrm{\partial }\xi }+\frac{\mathrm{\partial }{c}_1}{\mathrm{\partial }\tau }-\frac{{\mathrm{\partial }}^2{c}_1}{\mathrm{\partial }{\xi }^2}.}$

The solvability condition of the above equation yields the governing equation for ${\displaystyle c_{1a}(\xi ,\tau )}$ as follows

${\displaystyle \frac{\mathrm{\partial }{c}_{1a}}{\mathrm{\partial }\tau }-\left(1+\frac{P{e}^2}{48}\right)\frac{{\mathrm{\partial }}^2{c}_{1a}}{\mathrm{\partial }{\xi }^2}=-\frac{P{e}^3}{2880}\frac{{\mathrm{\partial }}^3{c}_0}{\mathrm{\partial }{\xi }^3}.}$

Other sources

• Aris, R. (1956). "On the dispersion of a solute in a fluid flowing through a tube". Proceedings of the Royal Society of London, Series A. 235 (1200): 67–77. Bibcode:1956RSPSA.235...67A. doi:10.1098/rspa.1956.0065. S2CID 95229777.
• Frankel, I.; Brenner, H. (1989). "On the foundations of generalized Taylor dispersion theory". Journal of Fluid Mechanics. 204: 97. Bibcode:1989JFM...204...97F. doi:10.1017/S0022112089001679. S2CID 123719494.
• Taylor, Geoffrey (1953). "Dispersion of soluble matter in solvent flowing slowly through a tube". Proceedings of the Royal Society of London, Series A. 219 (1137): 186–203. Bibcode:1953RSPSA.219..186T. doi:10.1098/rspa.1953.0139. S2CID 97372019.
• Taylor, Geoffrey (1954). "The dispersion of matter in turbulent flow through a pipe". Proceedings of the Royal Society of London, Series A. 223 (1155): 446–468. Bibcode:1954RSPSA.223..446T. doi:10.1098/rspa.1954.0130. S2CID 96182418.
• Taylor, Geoffrey (1954). "Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion". Proceedings of the Royal Society of London, Series A. 225 (1163): 473–477. Bibcode:1954RSPSA.225..473T. doi:10.1098/rspa.1954.0216. S2CID 97701431.
• Brenner, H. (1980). "Dispersion resulting from flow through spatially periodic porous media". Philosophical Transactions of the Royal Society of London, Series A. 297 (1430): 81–133. Bibcode:1980RSPTA.297...81B. doi:10.1098/rsta.1980.0205. S2CID 121853893.
• Mestel. J. Taylor dispersion — shear augmented diffusion, Lecture Handout for Course M4A33, Imperial College.