Generalized Taylor Dispersion Theory Research Articles

Enhanced dispersion in an oscillating array of harmonic traps.

Experiment, theory, and simulation are employed to understand the dispersion of colloidal particles in a periodic array of oscillating harmonic traps generated by optical tweezers. In the presence of trap oscillation, a nonmonotonic and anisotropic dispersion is observed. Surprisingly, the stiffest traps produce the largest dispersion at a critical frequency, and the particles diffuse significantly faster in the direction of oscillation than those undergoing passive Stokes-Einstein-Sutherland diffusion. Theoretical predictions for the effective diffusivity of the particles as a function of trap stiffness and oscillation frequency are developed using generalized Taylor dispersion theory and Brownian dynamics simulations. Both theory and simulation demonstrate excellent agreement with the experiments, and reveal a "slingshot" mechanism that predicts a significant enhancement of colloidal diffusion in dynamic external fields.

Vertical distribution and longitudinal dispersion of gyrotactic microorganisms in a horizontal plane Poiseuille flow

A more concise and accurate generalized Taylor dispersion theory is applied to dispersion of active gyrotactic microorganisms in a plane Poiseuille flow. The joint effect of boundary conditions, cell shape anisotropy, swimming speed, and flow speed leads to the nonmonotonic variations of the phenomenological dispersion coefficients.

Diffusion and steady state distributions of flexible chemotactic enzymes

Many experiments in recent years have reported that, when exposed to their corresponding substrate, catalytic enzymes undergo enhanced diffusion as well as chemotaxis (biased motion in the direction of a substrate gradient). Among other possible mechanisms, in a number of recent works we have explored several passive mechanisms for enhanced diffusion and chemotaxis, in the sense that they require only binding and unbinding of the enzyme to the substrate rather than the catalytic reaction itself. These mechanisms rely on conformational changes of the enzyme due to binding, as well as on phoresis due to non-contact interactions between enzyme and substrate. Here, after reviewing and generalizing our previous findings, we extend them in two different ways. In the case of enhanced diffusion, we show that an exact result for the long-time diffusion coefficient of the enzyme can be obtained using generalized Taylor dispersion theory, which results in much simpler and transparent analytical expressions for the diffusion enhancement. In the case of chemotaxis, we show that the competition between phoresis and binding-induced changes in diffusion results in non-trivial steady state distributions for the enzyme, which can either accumulate in or be depleted from regions with a specific substrate concentration.

Shear-induced dispersion in peristaltic flow

The effective diffusivity of a Brownian tracer in unidirectional flow is well known to be enhanced due to shear by the classic phenomenon of Taylor dispersion. At long times, the average concentration of the tracer follows a simplified advection–diffusion equation with an effective shear-dependent dispersivity. In this work, we make use of the generalized Taylor dispersion theory for periodic domains to analyze tracer dispersion by peristaltic pumping. In channels with small aspect ratios, asymptotic expansions in the lubrication limit are employed to obtain analytical expressions for the dispersion coefficient at both small and high Péclet numbers. Channels of arbitrary aspect ratios are also considered using a boundary integral formulation for the fluid flow coupled to a conservation equation for the effective dispersivity, which is solved using the finite-volume method. Our theoretical calculations, which compare well with results from Brownian dynamics simulations, elucidate the effects of channel geometry and pumping strength on shear-induced dispersion. We further discuss the connection between the present problem and dispersion due to Taylor’s swimming sheet and interpret our results in the purely diffusive regime in the context of Fick–Jacobs theory. Our results provide the theoretical basis for understanding passive scalar transport in peristaltic flow, for instance, in the ureter or in microfluidic peristaltic pumps.

Dispersion of active particles in confined unidirectional flows

Transport of micro-organisms in confined flows can be characterized by a one-dimensional overall dispersion mechanism, of importance to various biotechnological applications. Based on Brenner’s generalized Taylor dispersion theory, an overall dispersion model is analytically studied in the present work for a dilute suspension of active particles in confined unidirectional flows. With the confined section of the channel and the swimming orientation space taken together as the local space and the longitudinal coordinate standing for the one-dimensional global space, this model is analytically accurate and possessed of wide adaptability in terms of the swimming Péclet number. The Robin boundary condition is introduced to account for wall accumulation of active particles, and compared with a typical reflection boundary condition. Complications associated with the boundary conditions for analytical derivation are removed respectively by a decomposition of the distribution function and an extension of the flow field. Interesting solutions are concretely found and intensively illustrated. Detailed case studies on the transport of spherical and rod-like particles to illustrate the dispersion mechanism are presented with respect to a Couette flow and a plane Poiseuille flow. Associated with the local distribution of particles, extensive descriptions are given for the dynamical system behaviours such as accumulation near both stable points/lines and boundaries, symmetric polarization structure, closed orbits, trapping effect, nematic alignments and bimodalization of swimming direction. For spherical particles, the accumulation is shown leading to a reduction of the overall dispersivity in both of the flows, while for rod-like active particles in the Couette flow, the accumulation can result in an enhancement of dispersion, due to the nematic alignments of particles towards streamlines.

Fluctuation-dissipation in active matter.

In a colloidal suspension at equilibrium, the diffusive motion of a tracer particle due to random thermal fluctuations from the solvent is related to the particle's response to an applied external force, provided this force is weak compared to the thermal restoring forces in the solvent. This is known as the fluctuation-dissipation theorem (FDT) and is expressed via the Stokes-Einstein-Sutherland (SES) relation D = kBT/ζ, where D is the particle's self-diffusivity (fluctuation), ζ is the drag on the particle (dissipation), and kBT is the thermal Boltzmann energy. Active suspensions are widely studied precisely because they are far from equilibrium-they can generate significant nonthermal internal stresses, which can break the detailed balance and time-reversal symmetry-and thus cannot be assumed to obey the FDT a priori. We derive a general relationship between diffusivity and mobility in generic colloidal suspensions (not restricted to near equilibrium) using generalized Taylor dispersion theory and derive specific conditions on particle motion required for the FDT to hold. Even in the simplest system of active Brownian particles (ABPs), these conditions may not be satisfied. Nevertheless, it is still possible to quantify deviations from the FDT and express them in terms of an effective SES relation that accounts for the ABPs conversion of chemical into kinetic energy.

Transport and dispersion of active particles in periodic porous media

The transport of self-propelled particles such as bacteria and phoretic swimmers through crowded heterogeneous environments is relevant to many natural and engineering processes, from biofilm formation and contamination processes to transport in soils and biomedical devices. While there has been experimental progress, a theoretical understanding of mean transport properties in these systems has been lacking. In this work, we apply generalized Taylor dispersion theory to analyze the long-time statistics of an active self-propelled Brownian particle transported under an applied flow through the interstices of a periodic lattice that serves as an idealization of a porous medium. Our theoretical model, which we validate against Brownian dynamics simulations, is applied to unravel the roles of motility, fluid flow, and lattice geometry on asymptotic mean velocity and dispersivity. In weak flows, transport is dominated by active dispersion, which results from self-propulsion in the presence of noise and is hindered by the obstacles that act as entropic barriers. In strong flows, shear-induced Taylor dispersion becomes the dominant mechanism for spreading, with pillars now acting as regions of shear production that enhance dispersion. The interplay of these two effects leads to complex and unexpected trends, such as a non-monotonic dependence of axial dispersivity on flow strength and a reduction in dispersion due to swimming activity in strong flows. Brownian dynamics are used to cast light on the pre-asymptotic regime, where tailed distributions are observed in agreement with recent experiments on motile micro-organisms. Our results also highlight the subtle effects of pillar shape, which can be used to control the magnitude of dispersion and to drive a net particle migration in quiescent systems.

Gyrotactic trapping: A numerical study

Gyrotactic trapping is a mechanism proposed by Durham et al. [“Disruption of vertical motility by shear triggers formation of thin Phytoplankton layers,” Science 323, 1067–1070 (2009)] to explain the formation of thin phytoplankton layer just below the ocean surface. This mechanism is examined numerically using a rational model based on the generalized Taylor dispersion theory. The crucial role of sedimentation speed in the thin layer formation is demonstrated. The effects of variation in different parameters on the thin layer formation are also investigated.

Biased swimming cells do not disperse in pipes as tracers: A population model based on microscale behaviour

There is much current interest in modelling suspensions of algae and other micro-organisms for biotechnological exploitation, and many bioreactors are of tubular design. Using generalized Taylor dispersion theory, we develop a population-level swimming-advection-diffusion model for suspensions of micro-organisms in a vertical pipe flow. In particular, a combination of gravitational and viscous torques acting on individual cells can affect their swimming behaviour, which is termed gyrotaxis. This typically leads to local cell drift and diffusion in a suspension of cells. In a flow in a pipe, small amounts of radial drift across streamlines can have a major impact on the effective axial drift and diffusion of the cells. We present a Galerkin method to calculate the local mean swimming velocity and diffusion tensor based on local shear for arbitrary flow rates. This method is validated with asymptotic results obtained in the limits of weak and strong shear. We solve the resultant swimming-advection-diffusion equation using numerical methods for the case of imposed Poiseuille flow and investigate how the flow modifies the dispersion of active swimmers from that of passive scalars. We establish that generalized Taylor dispersion theory predicts an enhancement of gyrotactic focussing in pipe flow with increasing shear strength, in contrast to earlier models. We also show that biased swimming cells may behave very differently to passive tracers, drifting axially at up to twice the rate and diffusing much less.

The spatial distribution of gyrotactic swimming micro-organisms in laminar flow fields

We compare the results of two-dimensional, biased random walk models of individual swimming micro-organisms with advection–diffusion models for the whole population. In particular, we consider the influence of the local flow environment (gyrotaxis) on the resulting motion. In unidirectional flows, the results of the individual and population models are generally in good agreement, even in flows in which the cells can experience a range of shear environments, and both models successfully predict the phenomena of gravitactic focusing. Numerical results are also compared with asymptotic expressions for weak and strong shear. Discrepancies between the models arise in two cases: (i) when reflective boundary conditions change the orientation distribution in the random walk model from that predicted by the long-term asymptotics used to derive the advection–diffusion model; (ii) when the spatial and temporal scales are not large enough for the advection–diffusion model to apply. We also use a simple two-dimensional flow containing a variety of flow regimes to explore what happens when there are localized regions in which the generalized Taylor dispersion theory used in the derivation of the population model does not apply. For spherical cells, we find good agreement between the models outside the ‘break-down’ regions, but comparison of the results within these regions is complicated by the presence of nearby boundaries and their influence on the random walk model. In contrast, for rod-shaped cells which are reorientated by both vorticity and strain, we see qualitatively different spatial patterns between individual and advection–diffusion models even in the absence of gyrotaxis, because cells are advected between regions of differing rates of strain.

The effect of surface charge distribution on hindered diffusion in pores

An analytic solution is derived for the hindered diffusion of charged, small solutes in charged, cylindrical pores in which the pore wall potential consists of the sum of an average and an oscillatory component. When the oscillatory contribution is absent, the effect of electrostatic interactions on diffusion is negligible. However, when the wall potential or surface charge density varies axially, electrostatic interactions hinder the rate of diffusion significantly, and can stop it completely if “choke points” develop where the solute concentration becomes zero. The degree of hindrance is generally weaker when the electrostatic charge on the pore wall and the charge on the solute have the same signs, leading to a repulsion, than it is in the presence of an attraction. The electrostatic hindrance is also affected by the length scale of the axial variation along the pore wall, becoming stronger as that length grows, until an asymptotic value is reached. The theory for the effect of variations of the electrostatic potential on rates of diffusion is shown to be in good agreement with experimental data taken from the literature. The results here are obtained by using generalized Taylor dispersion theory, and are therefore rigorous predictions of what occurs over times long enough that the solute diffuses through a tube many times longer than a single periodic cell. The electrostatic interactions are calculated using the linear Poisson–Boltzmann equation.

A NUMERICAL SOLUTION FOR THE ANOMALOUS SEDIMENTATION OF A SMALL BROWNIAN SPHERE IN A VERTICAL CYLINDER OF PERIODICALLY VARYING RADIUS

We consider the long-time asymptotic transport of a small Brownian sphere that is sedimenting through a viscous fluid within an infinite vertical cylinder whose radius changes periodically and abruptly between two values. Use of generalized Taylor dispersion theory gives rise to a set of steady-state differential equations that govern the long-time mean velocity. We prove that these equations possess a unique solution. Finite difference discretization of all governing equations and boundary conditions, except for the normalization condition, leads to a homogeneous system of equations that possesses an infinite number of solutions. Application of the normalization condition selects the particular solution of the set that satisfies all conditions. Despite the absence of classical wall effects, the long-time mean velocity of the particle through the pore is generally less than the Stokes sedimentation velocity that applies at every point within the fluid. This anomalous behavior could be used to enhance force-driven separation of Brownian particles that possess identical Stokes velocities but different diffusivities. Furthermore, this numerical approach may be applied to similar problems.

An extension of generalized Taylor dispersion in unbounded homogeneous shear flows to run-and-tumble chemotactic bacteria

In the absence of flow, the biased random walk of bacteria such as Escherichia coli is modeled by straight runs punctuated by random changes in direction, tumbles. We model chemotaxis by allowing the tumble rate of a run to depend on the component of swimming velocity in the direction of the chemoattractant gradient. In the well-studied situation of weak bias in tumble rate, bacteria disperse over a diffusive time scale, and the evolution of density satisfies the classic Keller–Segel advection-diffusion equation. In this paper, we model swimming bacteria being advected and rotated by an unbounded homogeneous shear flow. The flow field alters the trajectories of individuals and thus affects the macroscopic dispersion of a population. We adapt the formal framework of generalized Taylor dispersion theory to make it applicable for run-and-tumble bacteria with an arbitrary bias in tumble rate. This enables us to obtain a macroscopic description of the dispersion of bacteria. For the particular case of simple shear flow, we calculate explicitly the effect of flow on the diffusion tensor and mean swimming velocity.

Anomalous sedimentation of a small Brownian sphere in a vertical circular cylinder of periodically varying radius

The mean velocity of a Brownian sphere sedimenting under the influence of gravity through a viscous fluid confined within a vertical circular cylinder whose radius changes periodically along its effectively infinite length is investigated for circumstances where, owing to the small radius of the sphere relative to the minimum cylinder radius, conventional hydrodynamic wall effects are absent. Using generalized Taylor dispersion theory, explicit results are presented for the case where the periodic changes in cylinder radius occur abruptly, representing the computationally and conceptually simplest configuration. Despite the apparent absence of wall effects, the (mean) settling velocity of the sphere is always less than its Stokes law value, and depends nonlinearly upon the gravity force. Moreover, the sphere’s dispersivity fails to obey a Stokes–Einstein relation. This anomalous behavior may be exploited to enhance the force-driven separation of physicochemically disparate Brownian particles.

Mass-transfer enhancement via chaotic laminar flow within a droplet

The Stokes flow occurring within a non-neutrally buoyant spherical droplet translating by buoyancy through an immiscible liquid which is undergoing simple shear is shown to be chaotic under many circumstances for which the droplet translates by buoyancy through the entraining fluid. This flow is easily produced, for example, when the droplet rises (or falls) through the annular space of a vertical concentric-cylinder Couette viscometer or through a vertical Poiseuille flow. The parameters studied include: (i) droplet/bulk fluid viscosity ratio; (ii) shear strength/bubble rise velocity ratio; and (iii) the angle between the translational bubble velocity vector and the vorticity vector characterizing the undisturbed shear. Streamlines existing within a droplet that translates perpendicular to this vorticity vector are shown to be non-chaotic for all choices of physical parameters. Other relative orientations frequently contain chaotic trajectories. When solute initially dissolved within the droplet is extracted into the bulk fluid, the resulting overall mass-transfer coefficient (calculated via generalized Taylor dispersion theory) quantifying the extraction rate at asymptotically long times is shown to be significantly higher in the chaotic flow case.

Effect of laminar chaos on reaction and dispersion in eccentric annular flow

Generalized Taylor dispersion theory is used to study the chaotic laminar transport of a reactive solute between eccentric rotating cylinders in the presence of an inhomogeneous chemical reaction. The circumstance considered is that of laminar axial ‘Poiseuille’ flow in the annular region between the two non-concentric cylinders, accompanied by a secondary, generally chaotic, flow induced via alternate rotation of the cylinders. A Brownian tracer introduced into the flow is assumed to undergo an instantaneous, irreversible reaction on the surface of the outer cylinder. The resulting effective transversely and time-averaged reaction rate, axial solute velocity, and axial convective dispersivity are computed. When chaos is present, the effective reaction rate is increased to a value several times larger than occurs in the absence of chaotic transport. It is found that an optimum alternation frequency exists, and that this frequency decreases with increasing transverse Péclet number (Peq). It is also observed that the maximum achievable reaction rate increases with (Peq). The effect of laminar chaotic mixing on the mean axial solute/solvent velocity ratio is to drive its value towards the perfectly mixed value of 1.0, despite the removal of solute from the slower-moving axial streamlines near the outer (reactive) cylinder wall. Lastly, in the presence of transverse chaotic transport, the convective Taylor contribution to the axial solute dispersivity acquires a value up to several orders of magnitude smaller than that achievable by means of non-chaotic convection.

GENERALIZED TAYLOR DISPERSION PHENOMENA IN TIME-PERIODIC HOMOGENEOUS SHEAR FLOWS

Generalized Taylor dispersion theory (GTDT) is extended with the goal of obtaining a macroscale description of the transport processes in cases of time-periodic zero-mean convection in a homogeneous shear flow under the action of a spatially-uniform force field. The resulting scheme is uniformly valid throughout the frequency spectrum provided that particle hydrodynamic properties may be obtained from the solution of appropriate steady Stokes problems. The extended theory is illustrated in the context of the sedimentation of Brownian axisymmetric particles in time-periodic homogeneous shear under the action of time-periodic external force. Explicit results are presented for the limiting case when particle rotary oscillations induced by the time-periodic homogeneous shear are small. (The rotary Peclet numbers need not, however, be small.)

TAYLOR DISPERSION OF MULTICOMPONENT SOLUTES

Coupled transport of multicomponent solutes in globally continuous systems is considered in the framework of the Generalized Taylor dispersion theory. Coupling between transports of n different species at the local (or micro-) scale, is considered to result from first-order irreversible surface reactions occurring on the local space boundaries, or from the off-diagonal terms of the solute diffusivity matrices. General expressions are obtained for the global effective (long-time) solute dispersion matrix cofficients: mean global scalar reactivity, velocity vector and dispersivity dyadic. The effect of surface chemical reactions is to partition the matter between different solute constituents. This is manifested in a coupling of the global transport coefficients, which may be mathematically removed by a linear (canonic) transformation applied to the effective global transport equation. This type of coupling does not exist for inert solutes. The second type of the global coupling is represented by the off-diagonal terms of the global velocity and dispersivity matrices. It exists for both reactive and inert solutes. This coupling stems from the convective dispersion process (dependence or the global velocity vector on the local space coordinate). Is shown to be irremovable from the global transport equation by any linear transformation via the solute partition matrix. In the canonic form of the global equation the irremovable coupling is manifested by the traceless parts of the global solute velocity matrix and the global solute dispersivity. The solution scheme is illustrated by calculating the mean global diffusivity of a solute consisting of two components, transport of which is coupled at the microscale via the molecular diffusivity matrix. At the macroscale the coupling is shown to be represented by negative off-diagonal terms of the global diffusivity matrix,

Taylor dispersion of orientable Brownian particles in unbounded homogeneous shear flows

The physical- and orientation-space transport of non-spherical , generally non-neutrally buoyant, Brownian particles in unbounded homogeneous shear flows is analysed with the goal of studying the respective effects of the orientational degrees of freedom of such particles upon their sedimentation and dispersion rates. In particular, the present contribution concentrates on the interaction between the Taylor dispersion mechanism (arising from coupling between the orientational dependence of the particle's translational velocity and the stochastic sampling of the orientation space via rotary Brownian diffusion) and the shear velocity field. Making use of a recent extension of generalized Taylor dispersion theory to homogeneous (unbounded) shear flows, the mean transport process in physical space is modelled by a convection–diffusion problem characterized by a pair of constant phenomenological coefficients, provided that the eigenvalues of the (constant) undisturbed velocity gradient are purely imaginary. The latter phenomenological coefficients – namely, U * , the average ‘slip velocity’ vector (of the particles relative to the ambient fluid), and D * , the dispersivity dyadic or, equivalently, the pair of dyadics $\bar{\bm M}$ and D C (or $\bar{\bm D}^c$ ), the average mobility and the Taylor (or modified Taylor) dispersivity, respectively – are evaluated both asymptotically (in the respective limits of small and large rotary Peclet numbers) as well as numerically (for arbitrary Peclet numbers). It is established that (up to a scalar multiplication factor, independent of Peclet number) the anisotropic portion of the average mobility is formally equivalent to the direct diffusive contribution to the particle stress in the context of suspension rheology. The analysis focuses mainly on the case of simple shear flow. The approximate calculation in the limit of large Peclet numbers, Pe [Gt ] 1, which makes extensive use of the ‘natural coordinates’ along Jeffery orbits previously introduced by Leal & Hinch, verifies that, if the external force is non-orthogonal to the direction along which (undisturbed) fluid velocity variations occur, two of the eigenvalues of D c are proportional to Pe ; moreover, one of these O ( Pe ) eigenvalues is negative . When the external force is parallel to the latter direction, the negative eigenvalue corresponds to the principal direction of contraction in the shear velocity field; this thus relates the non-positive nature of D C to the interaction between the Taylor dispersion mechanism and the (deterministic) convection within the shear field. Explicit results for the variation of the dyadics $\bar{\bm M}$ , D c and $\bar{\bm D}^{c}$ jointly with the respective magnitude of the shear rate and the deviation of the particle geometry from a spherical shape are presented for spheroidal particles. Among other things, it is demonstrated that the proposed definition of the modified Taylor dispersivity coefficient, $\bar{\bm D}^{c}$ , does indeed yield a non-negative dyadic.

Dispersion and reaction in two-dimensional model porous media

Darcy-scale convective–diffusive–reactive phenomenological coefficients characterizing the transport of a reactive solute through the interstices of a two-dimensional, spatially periodic, model porous medium (on whose surfaces the solute undergoes a first-order, irreversible chemical reaction) are herein determined numerically as functions of the microscale Péclet (Pe), Damköhler (Da), and Reynolds (Re) numbers. The role of bed porosity ε and (circular) cylindrical array configuration are also studied, the latter encompassing square, staggered, and hexagonal arrays. Calculations are effected via generalized Taylor dispersion theory. The Darcy-scale reactivity coefficient K̄* characterizing the effective (first-order, irreversible) volumetric reaction rate is found, inter alia, to be (approximately) inversely proportional to Pe, a conclusion confirmed by analytical results for the limiting case of small Da. Configurational properties of the porous medium are observed to significantly influence K̄*, especially for small porosities and large Da. Moreover, it is found that the mean interstitial velocity vector Ū* of the reactive solute generally differs (often dramatically) from the comparable velocity vector (1/ε)Ū of the (inert) solvent as a consequence of the chemical reaction occurring at the surfaces of the cylindrical bed particles. These data reveal that the mean solute speed ‖Ū*‖ through the interstices may be larger or smaller than the comparable solvent speed (1/ε)‖Ū‖, depending upon the existence and nature of a diffusive boundary layer adhering to the cylindrical bed-particle surfaces. Finally, the longitudinal and lateral components of the solute’s (transversely isotropic) dispersivity dyadic D̄*, parallel and perpendicular, respectively, to the direction of mean flow, are generally observed to decrease with increasing Da. This behavior stems from the fact that, in the diffusion boundary-layer limit, an increasing proportion of the total depletion of solute (via microscale reaction at the cylinder surfaces) arises from those interstitial zones characterized by the existence of large velocity gradients.