We study stirring and dispersion in steady flow through two-dimensional heterogeneous porous media. A fluid is stirred by the porous medium structure that leads to spatial flow variability and the deformation of fluid elements. While dispersion measures the extension of a solute distribution, stretching underpins solute mixing, chemical reactions and biological activity. Despite this, much less is known about fluid stretching in porous media. Its relation to the dispersion process on the one hand and the flow and medium properties on the other remain open questions. To close this gap, we probe the heterogeneity mechanisms that are at the origin of stretching and dispersion of fluid elements, and quantify the upscaled laws that govern their dynamics. Thus we perform detailed numerical simulations of Darcy flow in two-dimensional porous media characterised by a spatially varying hydraulic conductivity. Dispersion is analysed in terms of first-passage time distributions of fluid elements and the temporal evolution of their displacement mean and variance, while deformation is studied in terms of the probability density function and average elongation of fluid elements. We find that the dispersion and stretching behaviours are closely related. The exponents that govern the decay of the first-passage time distributions and evolution of the displacement variances also determine the evolution of the mean elongation. This dynamics is quantified in a single upscaled theoretical framework that bridges between Eulerian flow structure, large-scale dispersion and stretching of fluid elements.

Since Taylor’s seminal work on the dispersion of soluble matter, the modelling and control of solute dispersion have received considerable attention. This study investigates the solute dispersion in polymer solutions driven by unsteady magnetohydrodynamic (MHD)-oscillatory electro-osmotic flow within a microtube, aiming to reveal the transport mechanisms in complex rheological media through microscale flow. Firstly, based on the Debye–Hückel approximation, analytical solutions for the electric potential and velocity distributions of mixed MHD-electrohydrodynamic flows are rigorously derived. Subsequently, using Sankarasubramanian & Gill’s generalised dispersion model (1970 Proc. R. Soc. Lond. A, vol. 316, pp. 341–350), mathematical models are developed to systematically investigate the temporal evolution of hydrodynamic dispersion. By investigating the synergistic modulation of electrokinetic and magnetic fields, we identify three distinct flow regimes, namely the electric-field-dominant, competitive transition and magnetic-field-limited regimes, and quantitatively define their boundaries and evolution with the oscillatory Reynolds number $\textit{Re}$ and Deborah number $De$, thus allowing accurate flow regime classification. Moreover, a non-monotonic critical response governing solute dispersion is observed over a wide range of oscillatory Reynolds numbers, which provides a novel control paradigm for microfluidic mixing and separation. Additionally, through a comparative analysis of fractional Maxwell, classical Maxwell and Newtonian fluids, we show that microscopic structural relaxation plays a crucial role in macroscopic solute transport under multi-field coupling. Our findings help to reveal flow mechanisms for precise mass transport control in electromagnetically coupled systems, advancing the development of microfluidic devices and biomedical detection technologies.

Advective dispersion of neutral solutes in flow is pertinent to the design of a host of physical systems, including microfluidic devices and chemical separation systems. The cross-sectional shapes of the channels within these devices are often complex and may vary (in shape or area) in the flow direction. However, previous analyses of Taylor–Aris dispersion in channels of varying geometry typically focus on either periodic channels or channels with simple cross-sections. To our knowledge, no reduced-order theory exists to predict dispersion in channels of arbitrary cross-section whose shape or size may vary slowly in the axial direction. In the current study, we develop a theoretical framework for a reduced-order model applicable to the analysis of dispersion in channels with such shapes given steady or time-dependent flow conditions. We first derive a one-dimensional partial differential equation for the area-averaged solute concentration and associated expressions for the full, three-dimensional tracer distribution. Next, we formulate two coupled ordinary differential equations that govern the solute axial mean position and variance. Additionally, we derive an ordinary differential equation to engineer channels to produce a priori specified solute variance distributions. We apply our model to the example geometries of axisymmetric channels of varying radii, channels with a varying ‘dumbbell’ cross-section and rectangular channels of varying width. We benchmark our model’s performance against Brownian dynamics simulations for all tested geometries. Overall, our framework is straightforward to apply to different geometries and flow conditions, and may yield critical insights for the design and optimisation of microfluidic systems.

We experimentally investigated the turbulent flow in a water channel developing past a two-dimensional array of photovoltaic panels, with varying incidence angle and spacing. Particle image velocimetry revealed two distinct regimes, depending on the incidence angle, while the dependence on inter-panel spacing was weaker. Positive incidence angles produce high turbulence and intense transport above the panels, with strong mixing between the canopy and the external flow. Conversely, negative incidence angles yield a skimming flow with large recirculation vortices that dominate the canopy layer. Vertical profiles, ventilation rates and turbulent statistics indicate that even a modest positive incidence significantly enhances ventilation and scalar transport, whereas they are substantially lower in the case of negative incidence. A Lagrangian analysis using synthetic passive particles allowed the estimation of the residence times and turbulent dispersion coefficients. The latter were more than an order of magnitude larger for positive than for negative incidence. The results demonstrate how panel configuration changes mixing and dispersion processes, providing a fluid mechanics basis for refined microclimate models useful for guiding the design of advanced photovoltaic farms.

We present an experimental study of the vertical transport of a passive scalar, released from a localised source at the top of a two-height canopy. The study is based on simultaneous velocity and concentration measurements under neutrally stable conditions in an environmental wind tunnel. By analysing turbulent scalar diffusivity directly from our data, we identify regions of counter-gradient flux in the roughness sublayer above the canopy. Here, turbulent flux opposes the local mean concentration gradient. Quadrant analysis shows that sweep events ($w' \lt 0, u' \gt 0$) entrain low-concentration air from above into the canopy. Ejection events ($w' \gt 0, u' \lt 0$) mostly contribute to down-gradient scalar transport. We developed a semi-analytical model using an orthogonal series expansion of the three-dimensional joint probability density function. This model quantifies the imbalance between sweep and ejection contributions to vertical turbulent scalar flux. The results explain why gradient diffusion based models break down in canopy flows with an elevated source and highlight the importance of coherent motions in turbulent scalar exchange at the canopy–atmosphere interface. In addition, the measured concentration statistics were examined and showed deviations from classical models, particularly at high concentration values.

Characterizing hydrodynamic transport in fractured rocks is essential for carbon storage and geothermal energy production. Multiscale heterogeneities lead to anomalous solute transport, featuring breakthrough curve (BTC) tailing and nonlinear growth of plume spatial moments. We focus on purely advective transport within synthetic geological fractures with prescribed relative closure $\sigma _a/\langle a \rangle$ and correlation length $L_{\textit{c}}$. We adopt a stochastic approach with multiple fracture realizations for each set of geometric parameters. Steady-state depth-averaged Stokes flow is solved under the lubrication approximation. Flow heterogeneity is organized over the correlation length $L_{\textit{c}}$. The ensemble-averaged velocity probability density functions (PDFs) are insensitive to $L_{\textit{c}}$ but strongly influenced by $\sigma _a/\langle a \rangle$, particularly their low-velocity power-law scaling. A time-domain random walk (TDRW) simulation is used to compute plume spatial moments and outlet BTCs. The mean longitudinal plume position scales linearly with time at both early and late stages. The variance shows ballistic scaling at early times and a late-time behaviour controlled by the low-velocity power law of the velocity PDF, with exponent $\alpha$ strongly influenced by $\sigma _a/\langle a \rangle$. The properties of the BTCs are also controlled by $\alpha$, including the broadening of the peak as $\sigma _a/\langle a \rangle$ increases, and the scaling of the power-law tails. Advective transport is also modelled using a one-dimensional continuous-time random walk (CTRW) that relies only on the velocity PDF, flow tortuosity and flow correlation length. The CTRW reproduces the TDRW results and provides analytical predictions for the asymptotic transport scalings.

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.