Analysis of Enhanced Diffusion in Taylor Dispersion via a Model Problem

Abstract

We consider a simple model of the evolution of the concentration of a tracer, subject to a background shear flow by a fluid with viscosity ν ≪ 1 in an infinite channel. Taylor observed in the 1950s that, in such a setting, the tracer diffuses at a rate proportional to 1∕ν, rather than the expected rate proportional to ν. We provide a mathematical explanation for this enhanced diffusion using a combination of Fourier analysis and center manifold theory. More precisely, we show that, while the high modes of the concentration decay exponentially, the low modes decay algebraically, but at an enhanced rate. Moreover, the behavior of the low modes is governed by finite-dimensional dynamics on an appropriate center manifold, which corresponds exactly to diffusion by a fluid with viscosity proportional to 1∕ν.

Appendix: Convergence to the Center Manifold

The purpose of this appendix is to show that the center manifold constructed in Sect. 2 attracts all solutions. We know that we can construct the invariant manifold for the Eq. (21) (for a _k ′and b _k ′)globally since we have explicit formulas which hold for all values of a _k andη. In this note we show that any trajectory will converge toward the center manifold on a time scale of $\mathcal{O}(1/\nu )$.

Write $b_{k} = B_{k} + h_{k}(a_{k-2},\ldots,a_{0},\eta )$. We’ll prove that for any choice of initial conditions B _k goes to zero like $\sim e^{-\nu t}$.

First note that

$$\displaystyle\begin{array}{rcl} b_{k}'& =& B_{k}' +\sum _{ \ell=0,even}^{k-2}(\partial _{ a_{\ell}}h_{k})a_{\ell}' + (\partial _{\eta }h_{k})\eta ' {}\\ & =& B_{k}' -\sum _{\ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})\eta \frac{\ell} {2}a_{\ell} -\sum _{\ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})\eta b_{\ell-2} - (\partial _{\eta }h_{k})\eta ^{2} {}\\ & =& B_{k}' -\eta \sum _{\ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})B_{\ell-2} +\sum _{ \ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})(-\eta \frac{\ell} {2}a_{\ell} -\eta h_{\ell-2}) - (\partial _{\eta }h_{k})\eta ^{2}. {}\\ \end{array}$$

Thus, using the equation for b _k ′ in (21) we have

$$\displaystyle\begin{array}{rcl} & & B_{k}' -\eta \sum _{\ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})B_{\ell-2} + (\nu +\eta \frac{k} {2})B_{k} + \frac{2\eta } {\nu } B_{k-2} {}\\ & & \quad = -(\nu +\eta \frac{k} {2})h_{k} + \frac{\eta } {\nu ^{2}}a_{k-2} {}\\ & & \qquad \ -\sum _{\ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})(-\eta \frac{\ell} {2}a_{\ell} -\eta h_{\ell-2}) - (\partial _{\eta }h_{k})\eta ^{2} -\frac{2\eta } {\nu } h_{k-2}. {}\\ \end{array}$$

The key observation is that the terms on the right hand side precisely represent the invariance equation that defines h _k (and hence they all cancel), leaving the equation

$$\displaystyle\begin{array}{rcl} B_{k}'\quad = -(\nu +\eta \frac{k} {2})B_{k} -\frac{2\eta } {\nu } B_{k-2} +\eta \sum _{ \ell=2,even}^{k-2}(\partial _{ a_{\ell}}h_{k})B_{\ell-2}.& &{}\end{array}$$

(51)

We now see that the system of equations for B _k is homogeneous, linear, upper-triangular (but non-autonomous), and hence can be analyzed inductively. We’ll show that

$$\displaystyle\begin{array}{rcl} \vert B_{k}(t)\vert < \frac{C(N)} {\nu ^{k/2}} e^{-\nu t}& & {}\\ \end{array}$$

for $t > \frac{1} {\nu }$. We’ll only prove this for the even-indexed subsystem; the proof for the odd-subsystem is analogous. Notice that the base case, k = 0, holds since (51) for k = 0 reads B ₀′ = −ν B ₀, which implies $B_{0} \sim e^{-\nu t}$. Now let’s proceed with the induction argument: assume for j = 0, 2, … k, that

$$\displaystyle\begin{array}{rcl} \vert B_{j}(t)\vert < \frac{C(N)} {\nu ^{j/2}} e^{-\nu t}.& & {}\\ \end{array}$$

Next, we write the equation for B _k+2 from (51) with a key difference in the way the last term is written:

$$\displaystyle\begin{array}{rcl} B_{k+2}' = -\left (\nu B_{k+2} +\eta \frac{k + 2} {2} B_{k+2}\right ) -\frac{2\eta } {\nu } B_{k} + \frac{\eta } {\nu }\sum _{\ell=1}^{\frac{k} {2} }C_{k+2-2\ell}^{k+2} \frac{\eta ^{\ell}} {\nu ^{2\ell}}B_{k-2\ell}.& &{}\end{array}$$

(52)

This last sum is obtained from reindexing ℓ and using the fact that the formula for h _k in Proposition 1 implies that $(\partial _{a_{\ell}}h_{k})$ consists of a single term. Let’s proceed by noting that the equation

$$\displaystyle\begin{array}{rcl} y' = -\left (\nu y + a\eta y\right )& & {}\\ \end{array}$$

has the exact solution

$$\displaystyle\begin{array}{rcl} y = e^{-\nu (t-t_{0})}(1 + t)^{-a}(1 + t_{ 0})^{a}.& & {}\\ \end{array}$$

To derive this solution, it may help to recall that $\eta = \frac{1} {1+t}$. Applying this to (52) with $a = \frac{k+2} {2}$ and using Duhamel’s formula, we obtain

$$\displaystyle\begin{array}{rcl} B_{k+2}(t)& =& e^{-\nu (t-t_{0})}(1 + t)^{-\frac{k+2} {2} }(1 + t_{0})^{\frac{k+2} {2} }B_{k+2}(0) + D_{k}^{k+2}(t) \\ & & +\sum _{\ell=1}^{\frac{k+2} {2} -1}D_{k-2\ell}^{k+2}(t) {}\end{array}$$

(53)

where the Duhamel terms D _k−2ℓ ^k+2 satisfy

$$\displaystyle\begin{array}{rcl} D_{k-2\ell}^{k+2}(t)& & \\ & \sim & \frac{C} {\nu ^{2\ell+1}}\int _{t_{0}}^{t}e^{-\nu (t-s)}(1 + t)^{-\frac{k+2} {2} }(1 + s)^{\frac{k+2} {2} }(1 + s)^{-\ell-1} \frac{1} {\nu ^{\frac{k-2\ell} {2} }} e^{-\nu s}ds.{}\end{array}$$

(54)

Notice in the above Duhamel term, we have substituted, using the induction hypothesis $\vert B_{k-2\ell}(t)\vert \leq \frac{C} {\nu ^{\frac{k-2\ell} {2} }} e^{-\nu t}$ (we also assume $t > t_{0} > \frac{1} {\nu }$). These Duhamel terms are the most slowly decaying terms in the solution formula (53). Proceeding, we simplify (54) and obtain (for all ℓ),

$$\displaystyle\begin{array}{rcl} D_{k-2\ell}^{k+2}(t)& \sim & \frac{C} {\nu ^{\frac{k} {2} +\ell+1}}e^{-\nu t}(1 + t)^{-\left (\frac{k+2} {2} \right )}\left ((1 + t)^{\frac{k} {2} -\ell+1} - (1 + t_{0})^{\frac{k} {2} -\ell+1}\right ) {}\\ & =& \frac{C} {\nu ^{\frac{k} {2} +1}}e^{-\nu t}\left ( \frac{1} {\nu ^{\ell}(1 + t)^{\ell}} - \frac{1} {\nu ^{\ell}(1 + t_{0})^{\ell}} \frac{(1 + t_{0})^{\frac{k} {2} +1}} {(1 + t)^{\frac{k} {2} +1}} \right ). {}\\ \end{array}$$

Now since $t > t_{0} > \frac{1} {\nu }$, we obtain

$$\displaystyle\begin{array}{rcl} \vert D_{k-2\ell}^{k+2}(t)\vert \leq \frac{C} {\nu ^{\frac{k+2} {2} }} e^{-\nu t},& & {}\\ \end{array}$$

and subsequently we obtain, for $t > t_{0} > \frac{1} {\nu }$,

$$\displaystyle\begin{array}{rcl} \vert B_{k+2}(t)\vert \leq \frac{C} {\nu ^{\frac{k+2} {2} }} e^{-\nu t}& & {}\\ \end{array}$$

as desired. □