arXiv subjects

Tanguy Le Borgne

Publications and source records attributed to Tanguy Le Borgne.

At least 19 recordsLinked to original sources

arXiv2026

Scalar mixing by stagnation points on solid surfaces

Recent advances have linked mixing in porous media to the deformation of infinitesimal fluid elements. However, these Lagrangian frameworks generally neglect or simplify interactions with solid surfaces. We consider transport of thin solute blobs (strips or sheets) near single generic obstacles, in particular along separation lines (separatrices) which connect to stagnation points at solid boundaries. In this case, we demonstrate that the coupling between flow and diffusion is analytically tractable. We provide a complete theory for scalar mixing near upstream stagnation points, which we validate by numerical simulations. We identify a mixing time scale $t_B \sim Pe^{1/3}$, where $Pe$ is the Péclet number, which controls whether scalar decay is driven by fluid stretching or by interaction with the solid boundary. Asymptotically, for times $t < t_B$, we find that the maximum concentration $c_{\rm max}$ decays as $t^{-5/2}$ and that the width $σ$ grows like $t^{1/2}$. For $t > t_B$, we find exponential decay of $c_{\rm max}$ and constant width $σ\sim s_B \sim Pe^{-1/3}$, where $s_B$ is a mixing length scale distinct from that which appears in shear flows. The asymptotic scalings of $c_{\rm max}(t)$ and $σ(t)$, and their dependence on $Pe$, are universal to no-slip stagnation points and qualitatively different from stagnation points in open flows.

physics.flu-dyn↗

arXiv2026

How pore-scale disorder controls fluid stretching in two-dimensional porous media

Fluid stretching in porous media governs the mixing of reactants, contaminants, and nutrients, yet how the solid microstructure controls the stretching statistics remains poorly understood. We investigate how porous-medium heterogeneity controls stretching using (i) particle-tracking velocimetry experiments in 3D-printed millifluidic cells, (ii) numerical simulations of solute-plume deformation in the measured flow fields, and (iii) analytical calculations of fluid stretching. The cells contain arrays of cylindrical rods with systematically-varying disorder levels, from ordered to random. Velocity and shear-rate measurements reveal that fluid deformation is strongly localized near solid boundaries for all disorder levels, suggesting that near-wall flow is the main driver of stretching. The mean stretching grows linearly in time for ordered media and quadratically for disordered media, while the stretching distributions are approximately log-normal. We analytically describe the stretching produced by flow around an isolated cylinder and embed this description in a random-walk model that reproduces the observed stretching statistics in random media. These results provide the first quantitative connection between porous-medium structure and fluid-stretching statistics, revealing the extent to which disordered media accelerate mixing relative to ordered media and enabling progress beyond the common mean-field description of stretching in two-dimensional media as a simple shear flow.

physics.flu-dyn↗

arXiv2026

Dynamic multiphase flow triggers chaotic mixing in porous media

Solute mixing plays a pivotal role in a broad spectrum of chemical and biological processes across natural and engineered porous media. However, current understanding of mixing dynamics remains largely constrained to steady flows in fully or partially water-saturated environments. Multiphase flow systems are generally unsteady, with moving fluid interfaces and flow paths that change in time. Despite the widespread occurrence of dynamic multiphase flows, their impacts on solute mixing are largely unknown. Here, we use experiments and numerical simulations to investigate the effect of dynamic two-phase flow on the stretching and folding of fluid elements, a fundamental mechanism driving solute mixing and reactions in porous media. We find that dynamic two-phase flows induce chaotic mixing, characterized by exponential stretching of fluid elements, leading to strongly enhanced mixing compared to steady single phase flows. By extensive numerical multiphase flow simulations, we establish dynamic steady states where we reliably measure the mean fluid stretching rate as a function of flow rate. We show that stretching is maximized at an optimum flow rate which balances fluid shear deformation against the frequency of flow reorientation by the intermittent motion of the fluid interface. The findings are rationalized by a mechanistic model linking basic multiphase flow characteristics to the stretching rate, opening new perspectives to understand and control mixing and reactions in a wide range of multiphase flow systems.

physics.flu-dyn↗

arXiv2026

On the concentration distribution in turbulent thermals

Turbulent thermals emerge in a wide variety of geophysical and industrial flows, such as atmospheric cumulus convection and pollutant dispersal in oceans and lakes. When a buoyant fluid mass rises, or sinks, heat and mass transfers occur by the engulfment of the fresh surrounding fluid inside the thermal - a process that spans over multiple scales from macroscopic entrainment of ambient fluid to microscopic diffusive processes. Turbulent thermals are typically investigated through their integral properties (radius, depth, entrainment rate). However, mixing processes depend on the internal distribution of concentration or temperature inside a thermal, which remains poorly constrained. Here, we use laboratory fluid dynamics experiments and direct numerical simulations to investigate the mixing of a passive scalar in turbulent thermals with large Reynolds numbers. We track the evolution of the concentration field, computing its moments and the probability density function. The concentration distribution exhibits self-similarity over time, except at high concentrations, possibly because of the presence of undiluted cores. These distributions are well approximated by an exponential probability density function. Although diffusion has a strong effect on the spatial structure of the concentration field, we observe no significant effect of diffusivity on the concentration distributions in the investigated range of Peclet numbers.

physics.flu-dyn↗

arXiv2025

Stability of trapped fluid clusters in two-phase porous media flow

A key challenge in multiphase flow through porous media is to understand and predict the conditions under which trapped fluid clusters become mobilized. Here, we investigate the stability of such clusters in two-phase flow and present a simple, quasistatic model that accurately determines the critical Bond number (that is, the critical ratio between the average pressure gradient of the flow and the surface tension) for the onset of cluster mobilization. The model is derived by combining elementary geometrical considerations with mass conservation and a mechanical equilibrium condition, resulting in a system of coupled differential equations. Our derivation sheds new light on the mechanisms that govern cluster stability. In addition, since the number of equations equals the number of cluster openings, our model is significantly faster than direct numerical simulations of the same problem and enables efficient exploration of the system's parameter space. Using this approach, we highlight a discrepancy with the prediction of current mean-field theories, which predict that the largest stable cluster size scales in proportion to $r/\text{Bo}^α$, where $r$ is a typical pore size, $\text{Bo}$ is the Bond number and $α$ is a fixed exponent. We discuss the mechanisms that explain the breakdown of the mean field theories, and we show that a scaling law of this form can only exist if $α$ is allowed to depend on a broad set of flow and geometric parameters.

physics.flu-dyn↗

arXiv2024

A Unified Theory for Chaotic Mixing in Porous Media: from Pore Networks to Granular Systems

Recent studies have revealed the central role of chaotic stretching and folding at the pore scale in controlling mixing within porous media, whether the solid phase is discrete (as in granular and packed media) or continuous (as in vascular networks and open porous structures). Despite its widespread occurrence, a unified theory of chaotic mixing across these diverse systems remains to be developed. Furthermore, previous studies have focused on fluid stretching mechanisms but the folding mechanisms are largely unknown. We address these shortcomings by presenting a unified theory of mixing in porous media. We thus show that fluid stretching and folding (SF) arise through the same fundamental kinematics driven by the topological complexity of the medium. We find that mixing in continuous porous media manifests as discontinuous mixing through a combination of SF and cutting and shuffling (CS) actions, but the rate of mixing is governed by SF only. Conversely, discrete porous media involves SF motions only. We unify these diverse systems and mechanisms by showing that continuous media represents an analog of discrete media with finite-sized grain contacts. This unified theory provides insights into the generation of pore-scale chaotic mixing and points to design of novel porous architectures with tuneable mixing and transport properties.

physics.flu-dyn↗

arXiv2024

Mixing as a correlated aggregation process

Mixing describes the process by which solutes evolve from an initial heterogeneous state to uniformity under the stirring action of a fluid flow. Fluid stretching forms thin scalar lamellae which coalesce due to molecular diffusion. Owing to the linearity of the advection-diffusion equation, coalescence can be envisioned as an aggregation process. Here, we demonstrate that in smooth two-dimensional chaotic flows, mixing obeys a correlated aggregation process, where the spatial distribution of the number of lamellae in aggregates is highly correlated with their elongation and is set by the fractal properties of the advected material lines. We show that the presence of correlations makes mixing less efficient than a completely random aggregation process because lamellae with similar elongations and scalar levels tend to remain isolated from each other. We show that correlated aggregation is uniquely determined by a single exponent which quantifies the effective number of random aggregation events. These findings expand aggregation theories to a larger class of systems, which have relevance to various fundamental and applied mixing problems.

physics.flu-dyn↗

arXiv2023

Dispersion versus diffusion in mixing fronts

Mixing fronts form when fluids with different chemical compositions are brought into contact. They influence a large range of biogeochemical processes in hydrological systems. An important mechanism governing mixing rates in such fronts is stretching by non-uniform flows that accelerates diffusive mass transfer by enhancing concentration gradients. In a range of systems, including porous media at Darcy scale, hydrodynamic dispersion dominates over diffusion to control local mixing rates. As it differs from diffusion through its velocity-dependent dispersion tensor, it is not known how local dispersion interacts with macroscopic mixing front stretching. Here, we investigate the impact of local dispersion versus diffusion on the properties of steady mixing fronts created by both uniform and non-uniform flows. We derive analytical solutions for the concentration profile, mixing scale and mixing rate across the fronts. We validate these predictions by comparison with numerical simulations and experiments performed in quasi two-dimensional tanks over a broad range of Péclet numbers. Without porous media, the mixing scale is governed by local diffusion coupled with flow: it increases diffusively along streamlines in uniform flows while it is constant in converging flows due to the balance between fluid compression and local diffusion. With porous media, the Batchelor scale is no longer sustained and the mixing scale grows with dispersion in non-uniform flows. In addition, the coupling between flow acceleration and dispersion results in a Péclet independent mixing interface, in contrast with the local diffusion scenario. We discuss the consequences of these findings on mixing rates in mixing fronts.

physics.flu-dyn↗

arXiv2023

Impact of pore-scale chaotic mixing on Darcy-scale reaction rates

Prediction of reactive transport in porous media remains challenging when pore scale incomplete mixing is at play. Previous experimental studies investigated chemical reactions in porous media by visualizing reaction product or reactants mostly in uniform flow. However, the local reaction rate, which is necessary to infer mechanisms of reaction in pore space, could not be obtained without considering transport of reaction products and reactants. Thus, the interpretation remained elusive. We visualized the reaction rate field using chemiluminescnece within index-matched 3D porous media under zero acceleration and constant acceleration flow fields to investigate how pore scale chaotic mixing and Darcy scale fluid acceleration rectify reactive transport. We found that the reaction rate kept increasing from upstream to downstream in constant acceleration field, whereas it increased only at the upstream zone in zero acceleration field. The ratio of dispersion rate and size of the mixing interface determined such an effect of acceleration. Moreover, the experimental results showed stronger dependency of reaction rate on velocity compared to the numerical simulations that assume complete mixing in pore space. To explain this, we suggested the mechanistic model that includes the pore scale folding of lamellae due to chaotic mixing and the pore scale concentration gradients against compression. Such a pore scale mechanism was consistent with the experimentally observed change in reaction rate over the space. These results give new insights on underlying mechanisms of reactive transport in porous media.

physics.flu-dyn↗

arXiv2021

Enhanced dispersion in intermittent multiphase flow

Transport in multiphase flow through porous media plays a central role in many biological, geological, and engineered systems. Here, we use numerical simulations of transport in immiscible two-phase flow to investigate dispersion in multiphase porous media flow. While dispersion in the main flow direction is similar to that of a single-phase flow and is governed by the porous media structure, we find that transverse dispersion exhibits fundamentally different dynamics. The repeated activation and deactivation of different flow pathways under the effect of capillary forces lead to intermittent flow patterns, strongly enhancing dispersion. We show that the transverse dispersion is controlled by the length scale of fluid clusters, and thus inversely proportional to the square root of the Bond number, the ratio of the force driving the flow and the surface tension. This result leads to a new scaling law relating multiphase flow dynamics to transport properties, opening a range of fundamental and engineering applications.

physics.flu-dyn↗

arXiv2021

Sharp transition to strongly anomalous transport in unsaturated porous media

The simultaneous presence of liquid and gas in porous media increases flow heterogeneity compared to saturated flows. However, so far the impact of saturation on flow statistics and transport dynamics remained unclear. Here, we develop a theoretical framework that captures the impact of flow reorganization on the statistics of pore-scale fluid velocities, due to the presence of gas in the pore space, which leads to the development of a highly-structured flow field. Preferential flow is distributed spatially through the denoted backbone and flow recirculation occurs in dead-end regions branching from it. This induces a marked change in the scaling of the velocity PDF compared to the saturated case, and a sharp transition to strongly anomalous transport. We develop a transport model based on the continuous time random walk theory that successfully predicts advective transport dynamics for all saturation degrees. Our results provide a new modelling framework linking phase heterogeneity to flow heterogeneity and to transport in unsaturated media.

physics.flu-dyn↗

arXiv2020

Signature of coalescence during scalar mixing in a Rankine vortex

We analyze the dynamics of solute mixing in a vortex flow. The transport of a passive tracer is considered in a Rankine vortex. The action of a shear flow, in general, is to achieve stretching of fluid elements. A vortex flow exhibits stretching and folding of fluid elements in a way which brings adjacent fluid elements closer every turn. A strong stretching along the direction of rotation is accompanied by a concomitant thinning in the radial direction leading to a strong diffusive flux which may cause material from neighbouring regions of the mixing interface to aggregate. Through a Lagrangian concentration evolution technique, the diffusive strip method, we obtain the concentration field and pinpoint the signature of coalescence of two neighbouring concentration regions by analyzing the concentration distribution profiles. We link coalescence with reactivity for mixing-limited reactive flows. The analysis is useful to understand scalar dispersion in vortical flow structures.

physics.flu-dyn↗

arXiv2017

Anomalous transport in disordered fracture networks: spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modelling has remained an open issue. The fundamental reason behind this challenge is that---even if the Eulerian fluid velocity is steady---the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.

physics.flu-dyn↗

arXiv2016

Fluid Stretching as a Levy Process

We study the relation between flow structure and fluid deformation in steady two-dimensional random flows. Beyond the linear (shear flow) and exponential (chaotic flow) elongation paradigms, we find a broad spectrum of stretching behaviors, ranging from sub- to superlinear, which are dominated by intermittent shear events. We analyze these behaviors from first principles, which uncovers stretching as a result of the non-linear coupling between Lagrangian shear deformation and velocity fluctuations along streamlines. We derive explicit expressions for Lagrangian deformation and demonstrate that stretching obeys a coupled continous time random walk, which for broad distributions of flow velocities describes a Lévy walk for elongation. The derived model provides a direct link between the flow and deformation statistics, and a natural way to quantify the impact of intermittent shear events on the stretching behavior, which can have strong anomalous diffusive character.

physics.flu-dyn↗

arXiv2016

Continuous Time Random Walks for the Evolution of Lagrangian Velocities

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of the mean particle velocity, velocity covariance and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions which are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.

physics.flu-dyn↗

arXiv2016

Reactive strip method for mixing and reaction in two dimensions

A numerical method to efficiently solve for mixing and reaction of scalars in a two-dimensional flow field at large Péclet numbers but otherwise arbitrary Damköhler numbers is reported. We consider a strip of one reactant in a pool of another reactant, both of which are advected with the known velocity field. We first establish that the system evolution for such a system under certain conditions is described by a locally one-dimensional reaction-diffusion problem. The approximation of a locally one-dimensional dynamics is true for cases where the strip thickness is smaller than the local radius of curvature and also when the strip thickness is smaller than the distance between adjacent strips. We first demonstrate the method for the transport of a conservative scalar under a linear shear flow, point vortex and a chaotic sine flow. We then proceed to consider the situation with a simple bimolecular reaction between two reactants to yield a single product. The methodology presented herewith essentially generalizes nontrivially the Diffusive Strip Method developed by Meunier and Villermaux (J. Fluid Mech. 662, 134-172 (2010)) to address passive scalar transport, to the generalized situation with multiple reacting species. In essence, the reduction of dimensionality of the problem, which renders the 2D problem 1D, allows one to efficiently model reactive transport under high Péclet numbers which are otherwise prohibitively difficult to resolve from classical finite difference or finite element based methods.

physics.flu-dyn↗

arXiv2016

Enhanced reaction kinetics and reactive mixing scale dynamics in mixing fronts under shear flow for arbitrary Damköhler numbers

Mixing fronts, where fluids of different chemical compositions mix with each other, are typically subjected to velocity gradients, ranging from the pore scale to the catchment scale due to permeability variations and flow line geometries. A common trait of these processes is that the mixing interface is strained by shear. Depending on the Péclet number $Pe$, which represents the ratio of the characteristic diffusion time to the characteristic advection time, and the Damköhler number $Da$, which represents the ratio of the characteristic diffusion time to the characteristic reaction time, the local reaction rates can be strongly impacted by the dynamics of the mixing interface. This impact has been characterized mostly either in kinetics-limited or in mixing-limited conditions, that is, for either very low or very high $Da$. Here the coupling of shear flow and chemical reactivity is investigated for arbitrary Damköhler numbers, for a bimolecular reaction and an initial interface with separated reactants. Approximate analytical expressions for the global production rate and reactive mixing scale are derived based on a reactive lamella approach that allows for a general coupling between stretching enhanced mixing and chemical reactions. While for $Pe Da$, we uncover a "strong stretching" scenario where new scaling laws emerge from the interplay between reaction kinetics, diffusion, and stretching. The analytical results are validated against numerical simulations. These findings shed light on the effect of flow heterogeneity on the enhancement of chemical reaction and the creation of spatially localized hotspots of reactivity for a broad range of systems ranging from kinetic limited to mixing limited situations.

physics.flu-dyn↗

arXiv2016

The Lagrangian Deformation Structure of Three-Dimensional Steady Flow

Fluid deformation and strain history are central to wide range of fluid mechanical phenomena ranging from fluid mixing and particle transport to stress development in complex fluids and the formation of Lagrangian coherent structures (LCSs). To understand and model these processes it is necessary to quantify Lagrangian deformation in terms of Eulerian flow properties, currently an open problem. To elucidate this link we develop a Protean (streamline) coordinate transform for steady three-dimensional (3D) flows which renders both the velocity gradient and deformation gradient upper triangular. This frame not only simplifies computation of fluid deformation metrics such as finite-time Lyapunov exponents (FTLEs) and elucidates the deformation structure of the flow, but moreover explicitly recovers kinematic and topological constraints upon deformation such as those related to helicity density and the Poincaré-Bendixson theorem. We apply this transform to several classes of steady 3D flow, including helical and non-helical, compressible and incompressible flows, and find random flows exhibit remarkably simple (Gaussian) deformation structure, and so are fully characterised in terms of a small number of parameters. As such this technique provides the basis for the development of stochastic models of fluid deformation in random flows which adhere to the kinematic constraints inherent to various flow classes.

physics.flu-dyn↗