Taylor dispersion in the presence of cross flow and interfacial mass transfer

Transverse velocity gradients can enhance the effective diffusion coefficient of a scalar in the primary flow direction, a phenomenon known colloquially as Taylor dispersion. In this work, we perform Taylor dispersion analysis on a canonical pressure-driven flow in a channel with a cross flow, using both perturbation theory and Brownian dynamics simulations. Moreover, we illustrate how mass transfer at the wall affects the evolution of the scalar. By writing a one-dimensional advection-diffusion-mass-transfer equation for the cross-sectionally averaged concentration, we elucidate how the effective diffusion coefficients, effective advective velocities, and effective mass-transfer rates depend on the strength of the cross flow and the wall transfer coefficient. We perform an asymptotic analysis to investigate the limit of strong cross flow and demonstrate that, in this limit, dispersion rapidly decreases with increasing cross-flow velocity $V$ as $V^{-4}$ and scales with the Brownian diffusivity $D$ as $D^3$. Additionally, we discuss the effect of the Schmidt number, the ratio of momentum to mass diffusion, which effectively controls the extent to which the cross flow affects the axial velocity profile. Finally, we describe how our results can be extended to include transverse velocities and diffusivities that are spatially nonuniform.

Figure 1

(a) Schematic of the problem considered. A pressure gradient drives flow through the channel in the $x$ direction (${\mathrm{Re}}_v=1$ shown), and a constant, uniform cross-flow velocity $V$ is supplied in the $-y$ direction. No flux is permitted at the top wall, and the scalar is transferred with a dimensional mass-transfer coefficient $k$, or dimensionless Sherwood number $\mathrm{Sh}$, at the bottom wall. (b) Cross-sectionally averaged concentration $\langle C\rangle$ is shown when ${\mathrm{Pe}}_u=10$ for ${\mathrm{Pe}}_v=0$ (top) and ${\mathrm{Pe}}_v=10$ (bottom) at several times, in increments of $\mathrm{\Delta }t=2$, with a delta function initial condition. Numerically determined histograms are shown in gray, and best-fit Gaussians are shown in black.

Figure 2

Concentration profiles $C/\langle C\rangle$ when ${\mathrm{Pe}}_u=10$ and ${\mathrm{Re}}_v=1$ are shown for (a) $\mathrm{Sh}=0, {\mathrm{Pe}}_v=\{0,5,10\}$, and (b) ${\mathrm{Pe}}_v=0.1, \mathrm{Sh}=\{0,0.2,0.4\}$. In BD simulations, concentration profiles at the center of the traveling Gaussian are averaged over time.

Figure 3

Taylor dispersion coefficients $D_{\mathrm{TD}}/D$ in the absence of mass transfer, $\mathrm{Sh}=0$, and when ${\mathrm{Re}}_v=1$ are shown as functions of (a) ${\mathrm{Pe}}_u$, for ${\mathrm{Pe}}_v=\{5,10,15,20\}$, and (b) ${\mathrm{Pe}}_v$, for ${\mathrm{Pe}}_u=10$. The inset shows the asymptotic behavior of the dispersion coefficient when ${\mathrm{Pe}}_v\gg 1$.

Figure 4

(a) The effective advective velocity ${\langle u\rangle }_{\mathrm{eff}}$ when ${\mathrm{Re}}_v=1$ is shown as function of ${\mathrm{Pe}}_v$. The inset shows the asymptotic behavior of the velocity when ${\mathrm{Pe}}_v\gg 1$. (b) Schematic describing the asymptotic analysis performed when ${\mathrm{Pe}}_v\gg 1$. The velocity profile is approximated as a simple shear flow.

Figure 5

Effective Sherwood numbers ${\mathrm{Sh}}_{\mathrm{eff}}$ when ${\mathrm{Pe}}_u=10$ and ${\mathrm{Re}}_v=1$ are shown as functions of (a) $\mathrm{Sh}$, for ${\mathrm{Pe}}_v=\{0,1,2\}$, and (b) ${\mathrm{Pe}}_v$, for $\mathrm{Sh}=\{0.2,0.4,0.6,0.8,1.0\}$.

Figure 6

(a) Taylor dispersion coefficients $D_{\mathrm{TD}}/D$ in the absence of mass transfer, $\mathrm{Sh}=0$, are shown as functions of ${\mathrm{Pe}}_v$, for ${\mathrm{Pe}}_u=10$ and $\mathrm{Sc}=\{0.1,0.3,1,3,10,30,100\}$. (b) The maximum $D_{\mathrm{TD}}/D=D_{\mathrm{TD}}^{*}/D$ and (c) corresponding ${\mathrm{Pe}}_v={\mathrm{Pe}}_v^{*}$ are plotted as a function of $\mathrm{Sc}$.

Figure 7

Concentration profiles $C/\langle C\rangle$ are shown for ${\mathrm{Pe}}_v=\{5,25,45\}$ when the transverse particle velocity is (a) outwards, $v_{\mathrm{p}}\left(y\right)=-1+2y$, and (b) inwards, $v_{\mathrm{p}}\left(y\right)=1-2y$.

Figure 8

(a) Taylor dispersion coefficients $D_{\mathrm{TD}}/D$ as a function of ${\mathrm{Pe}}_v$ for ${\mathrm{Pe}}_u=10$ for both the outward and inward particle velocities. The inset shows the asymptotic behavior of the dispersion coefficient when ${\mathrm{Pe}}_v\gg 1$ for the inward particle velocity. (b) Effective velocity ${\langle u\rangle }_{\mathrm{eff}}$ as a function of ${\mathrm{Pe}}_v$ for both the outward and inward particle velocities.