Dispersion of a solute by flow

Dispersion of a solute by flow combines molecular diffusion with spatially varying advection, often producing an enhanced large-scale longitudinal diffusivity.

A first-order irreversible transfer from fluid concentration $C$ to stored concentration $S$, measured per matching fluid storage volume, obeys the displayed balances. A sustained inlet can produce a stationary decaying fluid profile, but the stored amount then grows rather than becoming stationary. A finite pulse instead has $C\to 0$ and a finite final deposit. Decay, desorption or finite adsorption capacity is additional physics, not a consequence of the linear capture term alone.

Spreading of solute across the mean flow direction through diffusion and pore-scale velocity variations. In a reflecting layer of depth $H$, a constant transverse dispersion coefficient gives mixing time of order $H^2/D_T$. Repeated exchange between different longitudinal speeds produces Taylor dispersion on times long compared with this mixing time.

The Péclet number compares advection over length $L$ at speed $U$ with molecular diffusion of diffusivity $D$.

Taylor dispersion is enhanced longitudinal spreading produced when transverse molecular diffusion repeatedly moves solute between streamlines of different velocity.

For a passive scalar in a planar shear flow $u(y,t)$, let $m_j(y,t)={\int }_{\mathbb{R}}{x}^j\chi (x,y,t)dx$. Integrating the advection-diffusion equation by parts in $x$ gives the displayed hierarchy for $j\ge 0$, with negative-index terms omitted. Sufficient decay and finite moments justify the integrations. Insulating walls give Neumann boundary conditions for every moment. Solving the first two equations then determines the long-time growth of the second moment without solving the full scalar field.

For $u=U\cos (\omega t)\sin (\pi y/L)$ on $|y|<L/2$, positive diffusivity $\kappa$, nonzero frequency and insulating walls, a transversely uniform zeroth moment remains constant. Its first moment is proportional to $g(t)\sin (\pi y/L)$, where $g^{\prime }+ag=U\cos (\omega t)$. The periodic solution is $g=U[a\cos (\omega t)+\omega \sin (\omega t)]/(a^2+{\omega }^2)$. The longitudinal moment hierarchy for shear transport then gives $m_2=2Kt{m}_0+O(1)$ after both transverse and periodic transients. The spatial average of ${\sin }^2(\pi y/L)$ is $1/2$ and the time average of ${\cos }^2(\omega t)$ is $1/2$, producing the factor $1/4$. At exactly zero frequency the steady enhancement is instead $U^2/(2a)$; averaging indefinitely over slow cycles does not commute with setting the frequency to zero.

For a passive scalar transported by $U(t)\sin (my)$ with transverse period $2\pi /m$, positive diffusivity $\kappa$ and periodic $U$, let $g$ be the unique periodic solution of $g^{\prime }+\kappa m^{2}g=U$. A multiple-scale expansion has first corrector $-g(t)\sin (my){\partial }_X\overline{\chi }$. Averaging the next-order equation over the space-time cell gives the displayed longitudinal diffusion coefficient. Averaging $g{g}^{\prime }+\kappa m^2{g}^2=Ug$ proves the second form. This is a long-time, long-wavelength effective equation, after transverse and periodic transients have decayed, rather than an exact finite-time closure for arbitrary concentrations.

When velocity consists of impulses $U_1$ spaced by $T$, the periodic corrector jumps by $U_1$ and decays between kicks. With $a=\kappa m^2$, $g(t)=U_1{e}^{-at}/(1-e^{-aT})$ for $0<t<T$. Integrating $g^2$ in the effective diffusivity of a periodic sinusoidal shear formula yields the displayed result. For $aT\gg 1$, independent transverse phases give longitudinal variance $U_1^2/2$ per kick and enhancement $U_1^2/(4T)$. For $aT\ll 1$, the enhancement is $U_1^2/(2\kappa m^2{T}^2)$ to leading order, the steady-shear result for average speed $U_1/T$. The zero-diffusivity limit does not commute with the long-time limit: without transverse diffusion the displacement is ballistic.

For interstitial velocity $u=6U\xi (1-\xi )$, $\xi =z/h$, reflecting transverse walls and molecular diffusion $D$, solve $D{\chi }^{\prime \prime }=u-U$ with zero mean and zero normal derivative. One solution is $\chi =(U{h}^2/D)({\xi }^3-{\xi }^4/2-{\xi }^2/2+1/60)$. Averaging the advection-diffusion equation gives enhancement $-\overline{(u-U)\chi }=D\overline{({\chi }^{\prime }{)}^2}=U^2{h}^2/(210D)$. The approximation requires transverse equilibration and slowly varying longitudinal concentration.

For pore velocity $u(y)=2Uy/H$ on $0<y<H$, with reflecting transverse boundaries, solve $D_T{\chi }^{\prime \prime }=u-U$, ${\chi }^{\prime }(0)={\chi }^{\prime }(H)=0$, $\overline{\chi }=0$. The Taylor dispersion enhancement is $D_T\overline{({\chi }^{\prime }{)}^2}=U^2{H}^2/(30D_T)$. Here $U$ is the mean advective pore velocity, not the Darcy velocity, and $D_L$ is the original longitudinal dispersion. The averaged equation applies after transverse equilibration and on suitably long longitudinal scales.

For $u(y)=Uy/h$ between reflecting walls at $y=\pm h$, the flow-induced diffusivity is