Taylor$\unicode{x2013}$Aris dispersion of active particles in oscillatory channel flow

TL;DR

This work investigates the Taylor–Aris dispersion of active particles in oscillatory channel flow using two complementary analytical frameworks: a two-time-variable expansion to capture transient dispersion of orientation-position coupled swimmers, and a generalized Taylor dispersion theory (GTD) to describe long-time, phase-resolved periodic dispersion. By applying these methods to swimmers governed by Jeffery dynamics and possible gyrotaxis/elongation, the authors show that motility can either enhance or suppress diffusivity relative to passive solutes depending on the oscillation frequency, with cross-streamline migration disrupted by shear. They reveal non-monotonic effects when a steady component is superimposed and demonstrate that gyrotaxis significantly modulates drift and dispersivity, while elongation yields weaker, more nuanced changes. The combined transient and asymptotic analyses, validated by Brownian dynamics simulations, provide a foundation for understanding periodic active dispersion and offer a pathway to explore more complex periodic variations in swimmer behavior or flow.

Abstract

Mass dispersion in oscillatory flows is intimately linked to various environmental and biological processes, offering a distinct contrast to dispersion in steady flows due to the periodic expansion and contraction of particle patches. In this study, we investigate the Taylor$\unicode{x2013}$Aris dispersion of active particles in laminar oscillatory flows between parallel plates. Two complementary approaches are employed: a two-time-variable expansion of the Smoluchowski equation is used to facilitate Aris' method of moments for the preasymptotic dispersion, while the generalised Taylor dispersion theory is extended to capture phase-dependent periodic drift and dispersivity in the long-time asymptotic limit. Applying both frameworks, we find that spherical non-gyrotactic swimmers can exhibit greater or lesser diffusivity than passive solutes in purely oscillatory flows, depending on the oscillation frequency. This behaviour arise primarily from the disruption of cross-streamline migration governed by Jeffery orbits. When a steady component is superimposed, oscillation induces a non-monotonic dual effect on diffusivity. We further examine two well-studied shear-related accumulation mechanisms, arising from gyrotaxis and elongation. Although these accumulation effects are less pronounced than in steady flows due to flow unsteadiness, gyrotactic swimmers respond more effectively to the unsteady shear profile, significantly altering their drift and dispersivity. This work offers new insights into the dispersion of active particles in oscillatory flows and also provides a foundation for studying periodic active dispersion beyond the oscillatory flow, such as periodic variations in shape and swimming speed.

Figures (12)

• Figure 1: Schematic illustration of swimmer dispersion in a vertical oscillatory channel flow. ($a$): A gyrotactic swimmer in the oscillatory channel flow experiences a viscous torque and a gravitational torque, in addition to rotational diffusion. ($b$): Time evolution of oscillatory velocity profile for the case with $P_0^{\ast} = 0$, $Q_0^{\ast} = 8 \nu^{\ast} D_r^{\ast}/W^{\ast}$, $\delta^{\ast} = 0.86 W^{\ast}$, and $\omega^{\ast} = D_r^{\ast}$.
• Figure 2: Comparison of the results obtained from moments equations and BD simulations over the first two oscillation periods. ($a$): First-order total moment $\left\langle P_1 \right\rangle_{z,\theta}$. ($b$): Mean square displacement of the cross-section-averaged concentration $\sigma^2$. Note that quantities obtained with moments equations are expressed with the single original time variable $t$ using the substitutions $t_0 \to t$ and $t_1 \to \omega t$, and the hat symbol are simultaneously removed. Parameters: $\hbox{Pe}_s=0.1$, $\hbox{Pe}_f^s=0$, $\hbox{Pe}_f^o=1$, $\alpha_0=0$, $\lambda=2.19$, $\omega=1$, $\hbox{Wo}=1.72$, $I_{ini} =1/(2 { \math@atom{\pi}{ \hbox{$\m@th\pi$}} })$.
• Figure 3: Comparison of the results obtained from the moments equations, GTD, and BD simulations over an oscillation period long after the initial release ($t \in [14T,15T]$). ($a$): Drift $U_d$. ($b$): Dispersivity $D_T$. Note that quantities obtained with moment equations and GTD are expressed with the single original time variable $t$ using the substitutions $t_0 \to t$ and $t_1 \to \omega t$. The parameters are consistent with those used in \ref{['fig:transient validation']}.
• Figure 4: ($a$,$b$): Transient drift $U_d$ and dispersivity $D_T$ of solute and spherical non-gyrotactic swimmers (SNS) over the first three periods following a uniform line release for several oscillatory flow Péclet numbers $\hbox{Pe}_f^o$. ($c$,$d$): Long-time asymptotic periodic drift $U_d^{\infty}$ and dispersivity $D_T^{\infty}$ of solute and SNS over one period for several oscillatory flow Péclet numbers $\hbox{Pe}_f^o$. Parameters for flow: $\hbox{Pe}_f^s=0$, $\omega=1$, $\hbox{Wo}=1.72$. Parameters for solute: $\hbox{Pe}_s=0$, $\alpha_0=0$, $\lambda=0$, $D_t=0.005$. Parameters for SNS: $\hbox{Pe}_s=0.1$, $\alpha_0=0$, $\lambda=0$, $D_t=0$.
• Figure 5: Long-time asymptotic period-averaged dispersivity $\overline{D_T^{\infty}}$ as functions of ($a$) the swimming Péclet number $\hbox{Pe}_s$ and ($b$) the oscillatory flow Péclet number $\hbox{Pe}_f^o$. Parameters for flow: $\hbox{Pe}_f^s=0$, $\omega=1$, $\hbox{Wo}=1.72$. Parameters in ($a$): $\alpha_0=0$, $\lambda=0$, $D_t=0$. Parameters in ($b$): $\hbox{Pe}_s=0$, $\alpha_0=0$, $\lambda=0$, $D_t=0.005$ (solute); $\hbox{Pe}_s=0.1$, $\alpha_0=0$, $\lambda=0$, $D_t=0$ (SNS).