Active control of dispersion within a channel with flow and pulsating walls

Channels are fundamental building blocks from biophysics to soft robotics, often used to transport or separate solutes. As solute particles inevitably transverse between streamlines along the channel by molecular diffusion, the effective diffusion of the solute along the channel is enhanced, an effect known as Taylor dispersion. Here we investigate how the Taylor dispersion effect can be suppressed or enhanced in different settings. Specifically, we study the impact of flow profile and active or pulsating channel walls on Taylor dispersion. We derive closed analytic expressions for the effective dispersion equation in all considered scenarios providing hands-on effective dispersion parameters for a multitude of applications. In particular, we find that active channel walls may lead to three regimes of dispersion: either dispersion decrease by entropic slow down at small Péclet number, or by dispersion increase at large Péclet number dominated either by shuttle dispersion, or by Taylor dispersion. This improves our understanding of solute transport, e.g., in biological active systems such as blood flow, and opens a number of possibilities to control solute transport in artificial systems such as soft robotics.

Figure 1

Taylor dispersion in long slender channels. Schematic illustrating how Taylor dispersion influences transport of solutes. (a) The initially radially homogeneous concentration profile is displaced by a Poiseuille flow profile. (b) This results in a strong radial and axial solute concentration imbalance, which induces radial diffusion gradients at the rear and at the front of the concentration profile. (c) After some time, the cloud of solute particles is dispersed longitudinally more than it would have by pure molecular diffusion in the absence of the flow profile.

Figure 2

Setting for Taylor dispersion in a straight channel with solute absorption at the channel wall. Straight channel schematic with solute (light blue particles) dispersing in a straight channel (light gray) and advected by a Poiseuille flow (black arrows). The diffusing particles are adsorbed with rate $\gamma$ at the channel wall.

Figure 3

Coefficients of zeroth to second spatial derivatives resulting from different methods for absorption at the channel wall. Exact coefficients (LM 1982) from Ref. [21] (black), heuristic approach (MA 2018) from Ref. [60] (gray, dotted), and invariant manifold (IM) approach (blue, green, yellow, dashed) for expansion to second, third, and fourth order, respectively.

Figure 4

Setting for Taylor dispersion in a straight channel with different flow profiles. Straight channel schematic with a solute (light blue particles) dispersing in a straight channel (light gray) and advected by a Poiseuille flow with slip at the boundaries (black arrows), with slip length $b$ (orange), or advected by an arbitrary flow profile (dark blue arrows on the right-hand side).

Figure 5

Optimal flow profile for Taylor dispersion in comparison with Poiseuille flow. (Right) Poiseuille profile (black) and optimal profile $u_{\mathrm{opt}}\left(r\right)$ (blue). (Left) Both profiles are rescaled by the cross-sectionally averaged flow velocity $U$ to highlight their differences as a function of the radial coordinate $r$.

Figure 6

Setting for Taylor dispersion in a space-time contracting channel. Channel schematic with a solute (light blue particles) dispersing in a channel (light gray) and advected by a Poiseuille like-flow (black arrows). The flow also has a radial component. The borders of the channel are actively contracting (orange arrows) and therefore moving with time.

Figure 7

The different regimes of dispersion enhancement in a peristaltic channel. (a) Dispersion enhancement defined ${\kappa }_{\mathrm{eff}}-\kappa$ with respect to the Péclet of the wave ${\text{Pe}}_{\mathrm{wave}}=\omega /\left(\sqrt{2}\kappa k^2\right)$ for different values of the wave number $k$ and (b) associated phase map in the $(k{a}_0,{\text{Pe}}_{\mathrm{wave}})$ plane of the dispersion enhancement. The color scale corresponds to the colorbar of (a) saturated at 10 for readability. The line separating the entropic slow down, and the shuttle dispersion regions correspond to ${\text{Pe}}_{\mathrm{wave}}^2=1/6$, and the line separating the Taylor dispersion region to ${\left(k{a}_0\right)}^2=3\times 48/(2{\text{Pe}}_{\mathrm{wave}}^2+1)$ or nearly $k{a}_0\simeq 8/{\text{Pe}}_{\mathrm{wave}}$. The dotted yellow line of constant drift corresponds $v_{\mathrm{eff}}$ constant, i.e., to ${\text{Pe}}_{\mathrm{wave}}=(0.1/\sqrt{2})\times 1/k{a}_0$. All lines of constant drift are parallel to this one, with increasing velocities going towards the right.

Figure 8

Setting for Taylor dispersion in a straight channel. Straight channel schematic with a solute (light blue particles) dispersing in a straight channel (light gray) and advected by a Poiseuille flow (black arrows).