The spread of a pulse of solute in a pressure-driven channel flow is well described for a wide range of Newtonian flows for which the viscosity and diffusivity are constants. Over many decades, various extensions have been suggested for the dispersion in pressure-driven non-Newtonian channel flows. While many theoretical studies have examined the effect of shear-rate-dependent viscosity on dispersion for a variety of non-Newtonian constitutive models, the solute diffusivity has invariably been treated as a constant. This assumption, however, is in contrast to the expectation that the diffusivity of a colloidal particle is inversely related to the viscosity, e.g. recall the Stokes–Einstein relation. We account for this coupling of transport coefficients – viscosity and diffusivity – by assuming a generalised form of the Stokes–Einstein equation, inspired by the recognition that the viscosity is now a field, although only transport transverse to the main flow direction is relevant because of the common assumptions of Taylor–Aris dispersion. Thus, we derive a general formula for axial dispersion in steady, pressure-driven shear-rate-dependent flows in uniform channels. In particular, we apply our general relation to calculate the Taylor–Aris dispersion coefficient for steady flows of a shear-thinning Carreau fluid and a viscoelastic Phan-Thien–Tanner fluid. Finally, we highlight new theoretical questions raised by this transport situation, where the underlying diffusivity is also a (tensorial) field related to variations in viscosity.

We derive expressions for the long-time effective dispersion coefficients of a solute in a slender annular channel with a spatiotemporally pulsating inner boundary. The problem is motivated by transport in perivascular spaces (PVSs) that are subjected to pulsations induced by travelling waves in the brain. Compared with steady flow, pulsations enhance the effective diffusivity and solute drift which scale quadratically with the wave amplitude, and depend on the ratio of wave-induced to bulk-flow-induced Péclet numbers, $ \textit{Pe}_c/ \textit{Pe}_b$. The mean enhancement in diffusivity can be decomposed into a bulk-flow-induced contribution and two wave-induced corrections: entropic slowdown, which reduces diffusivity due to solute lodging in the constrictions, for $ \textit{Pe}_c/ \textit{Pe}_b \lesssim \mathcal{O}(1)$, and shuttle dispersion which enhances diffusivity due to oscillatory solute transport for $ \textit{Pe}_c/ \textit{Pe}_b \gtrsim \mathcal{O}(1)$. The pulsations also induce an effective solute drift, that scales quadratically with the wave amplitude and linearly with $ \textit{Pe}_c/ \textit{Pe}_b$. The effective dispersion coefficients are sensitive to the annular cross-sectional area ratio with narrower geometries yielding stronger enhancements. For a representative murine PVS geometry subjected to pulsations under delta wave parameters, the mean enhancement of diffusivity, normalised by its value in steady flow, is $\mathcal{O}(10^{3})$ and the mean enhancement in effective solute drift is $-\mathcal{O}(1)$. Physiological mechanisms such as frequency-dependent wave amplitude, and approximately constant wave velocity across brain travelling waves, may diminish the enhancement magnitudes. The research presents a generalised framework for quantifying dispersion in spatiotemporally varying annular conduits and improves our understanding of perivascular solute transport.

Predicting and controlling the transport of colloids in porous media is essential for applications ranging from contaminant remediation to drug delivery. In these complex environments, solute gradients are ubiquitous and could drive diffusiophoretic particle migration, yet their impact on macroscopic colloid dispersion remains poorly understood. Here we combine experiments and simulations to quantify how diffusiophoresis alters the spreading of a colloidal blob in a two-dimensional ordered/disordered porous medium. A joint blob of colloids and salt at high concentration is introduced into a medium filled with salt at low concentration and advected by a background flow. Intuition suggests that when colloids are attracted towards or repelled from the solute-rich blob, dispersion should be suppressed or enhanced, respectively. Instead, we observe the opposite trend: longitudinal dispersion is enhanced in the attractive case, whereas dispersion is suppressed in the repulsive case. Numerical simulations reveal that this striking reversal arises from diffusiophoretic exchange of particles between slow and fast streamlines, which we capture using a minimal two-layer model of coupled fast and slow plug flows. Finally, we probe how geometric disorder in the medium modulates this mechanism. Our results demonstrate that diffusiophoresis can strongly modulate macroscopic dispersion of colloids in porous media with implications for transport in subsurface and biological environments.

Active particles exhibit complex transport dynamics in flows through confined geometries such as channels or pores. In this work, we employ a generalised Taylor dispersion (GTD) theory to study the long-time dispersion behaviour of active Brownian particles in an oscillatory Poiseuille flow within a planar channel. We quantify the time-averaged longitudinal dispersion coefficient as a function of the flow speed, flow oscillation frequency and particle activity. In the weak-activity limit, asymptotic analysis shows that activity can either enhance or hinder the dispersion compared with the passive case. For arbitrary activity levels, we numerically solve the GTD equations and validate the results with Brownian dynamics simulations. We show that the dispersion coefficient can vary non-monotonically with both the flow speed and particle activity. Furthermore, the dispersion coefficient shows an oscillatory behaviour as a function of the flow oscillation frequency, exhibiting distinct minima and maxima at different frequencies. The observed oscillatory dispersion results from the interplay between self-propulsion and oscillatory flow advection – a coupling absent in passive or steady systems. Our results show that time-dependent flows can be used to tune the dispersion of active particles in confinement.

Dispersion in turbulent flows is of broad interest in engineering and environmental processes, particularly for rivers, lakes and oceanic water bodies. Based on our streamwise dispersion model grounded in a Lagrangian perspective of convection–diffusion dynamics (Guan & Chen, 2024, J. Fluid Mech., vol. 980, A33), this work presents a comprehensive solution that consistently unifies dispersion across the Reynolds number spectrum, bridging laminar and turbulent regimes. The streamwise dispersion mechanism is general across time scales, yet its statistical behaviour cannot be fully described using conventional coarse-grained moments averaged over cross-sections. While classical drift–diffusion models that are effective for long-time asymptotics fail to capture the turbulent dynamics of the pre-asymptotic phase, our analytical model enables a complete spatio-temporal characterisation of concentration, and reveals how local statistics evolve towards their asymptotic, coarse-grained limits. Through asymptotic expansions and eigenfunction analysis, we quantify the time-dependent behaviour of phenomenological dispersion coefficients, and distinguish between local and mean statistics, which diverge significantly during the pre-asymptotic phase. The early regime exhibits robust features, including an overshoot in local dispersivity, asymmetric long tails in mean concentration, and island-shaped solute accumulation near the free surface. Three regimes are identified in the evolution of the local concentration: (i) an initially uniform line source, (ii) a transitional logarithmic profile shaped by vertical shear, and (iii) an emergent Gaussian dispersion regime approaching vertical uniformity. Comparisons of both local and mean concentration demonstrate quantitative agreement with finite difference and Monte Carlo simulations across all regimes. These findings clarify the interplay between shear and turbulent diffusion, laying a foundation for addressing more intricate and physically significant transport problems.

Because of the high dimensionality and geometric complexity of the circular-pipe problem, formulating and implementing boundary conditions are challenging, and most existing theoretical studies either neglect boundary effects or impose purely specular-reflection boundary conditions. To address this gap, we devise and explore an analytical model for microswimmer dispersion in a cylindrical pipe flow under a diffuse-reflection boundary condition, extending our earlier studies (Jiang & Chen, J. Fluid Mech., vol. 899, 2020, A18; Zeng et al., J. Fluid Mech., 1018, 2025, A27). We derive a well-posed Laplacian eigenvalue problem under diffuse reflection and obtain a complete basis formed by products of Bessel functions and spherical harmonics. The moment equations are solved by the Galerkin spectral method, and the computation is simplified by decomposing the operator and basis functions, together with an analytical treatment of the orientational integrals. The study follows the entire transport process by examining the local and radial distributions, the drift velocity and the dispersivity, and we assess the effects of key parameters with comparisons to the specular reflection conditions. Our results show that diffuse reflection drives microswimmers away from the wall more efficiently and promotes downstream alignment and cross-stream migration. When swimming is strong, non-gyrotactic microswimmers can develop centre accumulation, whereas gyrotaxis promotes near-wall accumulation that counteracts the effect of diffuse reflection, in contrast to classical behaviour. Distinct mechanisms dominate different stages of the transient evolution, leading to different temporal trends in the radial distribution and dispersivity. Overall, diffuse reflection yields a larger drift velocity and a smaller dispersivity, while both gyrotaxis and elongation increase dispersivity.

This study investigates the influence of wind tunnel ground conditions (stationary/moving) on flow topology and passive scalar dispersion in the wake of the Ahmed body with rear slant angles, $\phi$ = 25$^\circ$ and 40$^\circ$. We implement field measurements of both velocity and scalar concentrations in the wake, for both the ground conditions, within the same experimental set-up, allowing for structural correlation between wake topology and scalar dispersion. Particle image velocimetry measurements reveal the existence of a third spanwise vortex (vortex G) near the stationary wind tunnel ground, due to the floor boundary layer, for both of the Ahmed bodies ($\phi$ = 25$^\circ$, 40$^\circ$). Concentration field measurements performed using quantitative smoke visualisation show higher scalar dispersion in the wake of both Ahmed bodies for the stationary ground condition. Comparing the velocity and concentration fields further identifies vortex G as the primary physical driver for the enhanced vertical dispersion of the scalar, observed in stationary ground conditions. To quantify the dispersion and characterise these effects, we introduce dispersion parameters, such as non-dimensional dispersion ($\mathscr{D}$) and dispersion length scales ($\mathscr{L}_y, \mathscr{L}_z$). These parameters confirm that, while lateral dispersion remains relatively insensitive to wind tunnel ground conditions, the presence of vortex G in stationary ground conditions leads to an overestimation of vertical dispersion by up to $\approx$29 % ($\phi$ = 25$^\circ$) and $\approx$49 % ($\phi$ = 40$^\circ$). This study quantifies the overestimated dispersion, identifies the vortical structures responsible for scalar redistribution, provides physical insight into the wake dispersion phenomenon and highlights the importance of correct wind tunnel ground conditions in the vehicle wake dispersion studies.

The superlinear scaling relationship between the hydrodynamic dispersion coefficient and the Péclet number in porous media has been widely acknowledged. Nevertheless, the mechanisms driving this behaviour remain inadequately understood. In this work, we investigate the mechanism responsible for this superlinear scaling using a Lagrangian framework that combines a statistical model, which links the global probability density function of tracer transition time to flow variability in porous media, with a continuous time random walk framework. Our analysis reveals that the intra-pore and inter-pore flow variabilities are the primary sources responsible for the superlinear scaling, with their relative significance characterised by a structure-specific parameter, $\chi$. Specifically, the inter-pore flow variability dominates when $\chi \gt 1$, while the intra-pore variability prevails for $0\lt \chi \lt 1$. The parameter $\chi$ is derived exclusively from the statistical distributions of pore-throat radius, length and orientation angle, which can be readily obtained from structural characterisation techniques such as X-ray computed tomography imaging. These theoretical predictions are validated through extensive numerical simulations on tube networks with substantial structural variation. This study resolves discrepancies in previous studies regarding the mechanisms of superlinear scaling in hydrodynamic dispersion and offers valuable insights into modulate dispersion and mixing in porous media.

The dispersion phenomenon of mass and heat transport in oscillatory flows has wide applications in environmental, physiological and microfluidic flows. The method of concentration moments is a powerful theoretical framework for analysing transport characteristics and is well developed for steady flows: general solution expressions of moments have been profoundly derived by Barton (J. Fluid Mech. 126, 1983, 205–218). However, it was thought that these expressions could not be directly applied to unsteady flows. Prior studies needed to re-solve the governing equations of moments from scratch, encountering the complication induced by the time-periodic velocity, leaving higher-order statistics like skewness and kurtosis analytically intractable except for specific cases. This work proposes a novel approach based on a two-time-variable extension to tackle these challenges. By introducing an auxiliary time variable, referred to as oscillation time to characterise the inherent oscillation in the dispersion due to the oscillating flow, the transport problem is extended to a two-time-variable system with a ‘steady’ flow term. This enables the direct use of Barton’s expressions and thus avoids the prior complication. This approach not only offers an intuitive physical perspective for the influence of the velocity oscillation, but also clarifies the solution structure of concentration moments. As a preliminary verification, we examine the transport problem in an oscillatory Couette flow. The analytical solution agrees well with the numerical result by Brownian dynamics simulations. The effects of the point-source release and the phase shift of velocity on the transport characteristics are investigated. By extending the classic steady-flow solution to the time-dependent flows, this work provides a versatile framework for transient dispersion analysis, enhancing predictions in oscillatory transport problems.

Predicting and controlling the transport of colloids in porous media is essential for a broad range of applications, from drug delivery to contaminant remediation. Chemical gradients are ubiquitous in these environments, arising from reactions, precipitation/dissolution or salinity contrasts, and can drive particle motion via diffusiophoresis. Yet our current understanding mostly comes from idealised settings with sharply imposed solute gradients, whereas in porous media, flow disorder enhances solute dispersion, and leads to diffuse solute fronts. This raises a central question: Does front dispersion suppress diffusiophoretic migration of colloids in dead-end pores, rendering the effect negligible at larger scales? We address this question using an idealised one-dimensional dead-end geometry. We derive an analytical model for the spatio-temporal evolution of colloids subjected to slowly varying solute fronts and validate it with numerical simulations and microfluidic experiments. Counterintuitively, we find that diffuseness of the solute front enhances removal from dead-end pores: although smoothing reduces instantaneous gradient magnitude, it extends the temporal extent of phoretic forcing, yielding a larger cumulative drift and higher clearance efficiency than sharp fronts. Our results highlight that solute dispersion does not weaken the phoretic migration of colloids from dead-end pores, pointing to the potential relevance of diffusiophoresis at larger scales, with implications for filtration, remediation and targeted delivery in porous media.

Non-Newtonian fluid flow in porous media results in spatially varying viscosity, driven by flow–pore–geometry interactions, potentially leading to non-monotonic dispersion. In this work, using high-resolution micro-particle image velocimetry, we present a direct experimental observation of shear-viscosity-distribution-dependent transport with non-Newtonian fluid flows in porous media. We experimentally investigate dispersion in porous media in a microfluidic chip featuring a physical rock geometry, comparing a shear-thinning, non-Newtonian fluid with its Newtonian analogue at various Péclet numbers. We demonstrate that, in the absence of advective fluxes driven by elastic instabilities, non-Newtonian fluid flows at either extreme of the shear-dependent viscosity ($\eta _0,\eta _{\infty }$) converge to the Newtonian analogue. In contrast, for flows between these extremes, the non-Newtonian velocity fields are broadly distributed along the streamline curvature, leading to a larger enhancement in dispersion.

We introduce a description of passive scalar transport based on a (deterministic and hyperbolic) Liouville master equation. Defining a noise term based on time-independent random coefficients, instead of time-dependent stochastic processes, we circumvent the use of stochastic calculus to capture the one-point space–time statistics of solute particles in Lagrangian form deterministically. To find the proper noise term, we solve a closure problem for the first two moments locally in a streamline coordinate system, such that averaging the Liouville equation over the coefficients leads to the Fokker–Planck equation of solute particle locations. This description can be used to trace solute plumes of arbitrary shape, for any Péclet number, and in arbitrarily defined grids, thanks to the time reversibility of hyperbolic systems. In addition to grid flexibility, this approach offers some computational advantages as compared with particle tracking algorithms and grid-based partial differential equation solvers, including reduced computational cost, no Monte-Carlo-type sampling and unconditional stability. We reproduce known analytical results for the case of simple shear flow and extend the description of mixing in a vortex model to consider diffusion radially and nonlinearities in the flow, which govern the long time decay of the maximum concentration. Finally, we validate our formulation by comparing it with Monte Carlo particle tracking simulations in a heterogeneous flow field at the Darcy (continuum) scale.

Lagrangian transit times on basin to planetary scales are controlled by the interplay of multiscale processes. The primary advective time scale is set by throughflow currents, such as interhemispheric western boundary currents. Dispersion by mesoscale eddies introduces fluctuations that erase memory and enhance dispersion, widening the transit-time distribution. The tortuous paths of Lagrangian parcels, particularly within ocean gyres, significantly enhance dispersion beyond the levels attributed to mesoscale eddies alone. Additionally, trapping by ocean gyres leads to multimodal distributions of Lagrangian transit times. These processes are illustrated in three complementary contexts: eddy-permitting ocean state estimates, simplified spatially extended three-dimensional flows and diffusively coupled two-dimensional pipe models.

The dispersion of solutes has been extensively studied due to its important applications in microfluidic devices for mixing, separation and other related processes. Solute dispersion in fluids can be analysed over multiple time scales; however, Taylor dispersion specifically addresses long-term behaviour, which is primarily influenced by advective dispersion. This study investigates Taylor–Aris dispersion in a viscoelastic fluid flowing through axisymmetric channels of arbitrary shape. The fluid’s rheology is described using the simplified Phan-Thien–Tanner (sPTT) model. Although the channel walls are axisymmetric, they can adopt any geometry, provided they maintain small axial slopes. Drawing inspiration from the work of Chang & Santiago (2023 J. Fluid Mech. vol. 976, p. A30) on Newtonian fluids, we have developed a governing equation for solute dynamics that accounts for the combined effects of fluid viscoelasticity, molecular diffusivity and channel geometry. This equation is expressed using key dimensionless parameters: the Weissenberg number, the Péclet number and a shape-dependent dimensionless function. Solving this model allows us to analyse the temporal evolution of the solute distribution, including its mean and variance. Our analysis shows that viscoelasticity significantly decreases the effective solute diffusivity compared with that observed in a Newtonian fluid. Additionally, we have identified a specific combination of parameters that results in zero or negative transient growth of the variance. This finding is illustrated in a phase diagram and provides a means for transient control over dispersion. We validated our results against Brownian dynamics simulations and previous literature, highlighting potential applications for the design and optimisation of microfluidic devices.

Statistical regularities can be acquired from usage. To examine language speakers’ statistical metacognition about multiword expressions (MWEs), we collected ratings for frequency, dispersion, and directional association strength of English binomials from L1, advanced and intermediate L2 speakers. Mixed-effects modeling showed all speakers had limited speaker-to-corpus consistency but significant sensitivity to statistical regularities of language, supporting usage-based (Gries & Ellis, 2015) and statistical learning theories (Christiansen, 2019). Their statistical metacognition was also shaped by word-level cues, consistent with dual-route model (Carrol & Conklin, 2014). Despite similarities, frequency metacognition showed the strongest speaker-to-corpus consistency, while dispersion metacognition was the hardest to develop. Advanced L2 speakers showed the greatest speaker-to-corpus consistency and sensitivity, while lower-proficiency speakers relied more on word-level cues in metacognitive judgments, supporting the shallow-structure hypothesis (Clahsen & Felser, 2006). Overall, L1 and L2 speakers develop diverse statistical metacognition, with L2 speakers not necessarily inferior, suggesting that statistical metacognition is not solely shaped by usage-based experience.

This work investigates the long-time asymptotic behaviour of a diffusing passive scalar advected by fluid flow in a straight channel with a periodically varying cross-section. The goal is to derive an asymptotic expansion for the scalar field and estimate the time scale over which this expansion remains valid, thereby generalising Taylor dispersion theory to periodically modulated channels. By reformulating the eigenvalue problem for the advection–diffusion operator on a unit cell using a Floquet–Bloch-type eigenfunction expansion, we extend the classical Fourier integral of the flat channel problem to a periodic setting, yielding an integral representation of the scalar field. This representation reveals a slow manifold that governs the algebraically decaying dynamics, while the difference between the scalar field and the slow manifold decays exponentially in time. Building on this, we derive a long-time asymptotic expansion of the scalar field. We show that the validity time scale of the expansion is determined by the real part of the eigenvalues of a modified advection–diffusion operator, which depends solely on the flow and geometry within a single unit cell. This framework offers a rigorous and systematic method for estimating mixing time scales in channels with complex geometries. We show that non-flat channel boundaries tend to increase the time scale, while transverse velocity components tend to decrease it. The approach developed here is broadly applicable and can be extended to derive long-time asymptotics for other systems with periodic coefficients or periodic microstructures.

In this paper, we study experimentally the dispersion of colloids in a two-dimensional, time-independent, Rayleigh–Bénard flow in the presence of salt gradients. Due to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra velocity $\boldsymbol{v}_{{dp}} = D_{{dp}}\, \boldsymbol{\nabla }\log C_s$, where $D_{{dp}}$ is the phoretic constant, and $C_s$ is the salt concentration. Such a configuration is motivated by the theoretical work by Volk et al. (2022, J. Fluid Mech., vol. 948, A42), which predicted enhanced transport or blockage in a stationary cellular flow depending on the value of a blockage coefficient. By means of high dynamical range light-induced fluorescence, we study the evolution of the colloids concentration field at large Péclet number. We find good agreement with the theoretical work, although a number of hypotheses are not satisfied, as the experiment is non-homogeneous in space, and intrinsically transient. In particular, we observe enhanced transport when salt and colloids are injected at both ends of the Rayleigh–Bénard chamber, and blockage when colloids and salt are injected together and phoretic effects are strong enough.

The integration of electro-osmotic effect to the underlying flow enhances solute dispersion precision in microfluidic systems, which is crucial for applications such as drug delivery and on-chip fluidic functionalities. We investigate, in this study, the solute dispersion characteristics of couple-stress fluids in a two-dimensional microchannel configuration under the combined effects of electro-osmotic actuation and applied pressure gradients. We consider both homogeneous and heterogeneous reactions in the present analysis. Couple-stress fluids, which account for additional stresses due to the presence of the microstructures in the fluids, offer a more accurate model to describe the rheological behaviour of biofluids. While previous studies have addressed longitudinal Gaussianity and transverse uniformity of solute distribution, we focus uniquely in this endeavour on longitudinal uniformity. Using Mei’s multiscale homogenisation technique, we solve a two-dimensional convection–diffusion model, extending it to third-order approximation to analyse the dispersion coefficient, concentration profiles, and variation rates of concentration within microchannel flow. Results show that forcing and couple-stress parameters enhance the gradients of the longitudinal variation rate, while boundary absorption reduces this variation rate near the walls. The couple-stress parameter exhibits dual behaviour: initially, it enhances solute dispersion, but beyond a certain value of couple-stress parameter $B_{cr}$ (which depends on forcing comparison and the Debye–Hückel parameter), it reduces dispersion. In the absence of pressure, solute distribution remains longitudinally uniform. However, as the pressure gradient increases, concentration levels drop sharply, and the distribution shifts to a parabolic profile, underscoring the significant influence of pressure on flow behaviour in electro-osmotic flow.

Mass dispersion in oscillatory flows is closely tied to various environmental and biological processes, differing markedly from dispersion in steady flows due to the periodic expansion and contraction of particle patches. In this study, we investigate the Taylor–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 pre-asymptotic 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 arises 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 strongly to the unsteady shear profile, significantly modifying 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.