Transport of a passive scalar in wide channels with surface topography J. V. Roggeveen1 H. A. Stone1 C. Kurzthaler12

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Transport of a passive scalar in wide channels with

surface topography

J. V. Roggeveen1, H. A. Stone1, C. Kurzthaler12

1Department of Mechanical and Aerospace Engineering, Princeton University, New

Jersey 08544, USA

2Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

E-mail: jamesvr@princeton.edu, hastone@princeton.edu,

ckurzthaler@pks.mpg.de

Abstract. We generalize classical dispersion theory for a passive scalar to derive

an asymptotic long-time convection-diﬀusion equation for a solute suspended in

a wide, structured channel and subject to a steady low-Reynolds-number shear

ﬂow. Our theory, valid for small roughness amplitudes of the channel, holds for

general surface shapes expandable as a Fourier series. We determine an anisotropic

dispersion tensor, which depends on the characteristic wavelengths and amplitude

of the surface structure. For surfaces whose corrugations are tilted with respect to

the applied ﬂow direction, we ﬁnd that dispersion along the principal direction (i.e.,

the principal eigenvector of the dispersion tensor) is at an angle to the main ﬂow

direction and becomes enhanced relative to classical Taylor dispersion. In contrast,

dispersion perpendicular to it can decrease compared to the short-time diﬀusivity of

the particles. Furthermore, for an arbitrary surface shape represented in terms of a

Fourier decomposition, we ﬁnd that each Fourier mode contributes at leading order a

linearly-independent correction to the classical Taylor dispersion tensor.

arXiv:2210.02354v1 [physics.flu-dyn] 5 Oct 2022

Transport of a passive scalar in wide channels with surface topography 2

1. Introduction

Transport processes at small scales play a central role in a number of ﬁelds, ranging from

biology [1, 2], where solutes and blood cells are transported through vascular channels

or bacteria spread through soil and tissues, to geophysics [3], where water ﬂows through

complex porous materials and eroded rock particulates sediment in rivers, to microﬂuidic

applications [4, 5, 6, 7, 8], where various biological and synthetic samples are mixed,

sorted, or focused while moving through microchannels. These systems often operate at

low Reynolds number, where viscous forces dominate over inertia and strong stochastic

ﬂuctuations due to Brownian eﬀects dictate the overall transport behavior [9]. In

contrast to dilute, unconﬁned media, natural environments and microﬂuidic applications

exhibit a variety of conﬁning surfaces with diﬀerent properties, including deformable

and elastic materials [10], ﬂuctuating boundaries [11], and structured topographies [12],

which result in complex hydrodynamic ﬂows. Understanding the interaction of Brownian

suspensions with nearby boundaries and ﬂows provides insight into microbiological

processes and enables optimization and design of microﬂuidic devices [5].

The study of solute transport in microchannels dates back to Taylor [13] and

Aris [14] in the 1950’s. In their seminal work, they predicted an enhancement compared

to diﬀusion alone of the streamwise spreading of particles, as diﬀusion across the channel

allows particles to sample diﬀerent velocities of the hydrodynamic ﬂows. Since then,

numerous theoretical and experimental studies have extended their ideas, treating the

transient proﬁle of the solute concentration [15, 16, 17], the eﬀect of a ﬁnite particle

size [18], particle-particle interactions [19], channel corrugations [20], channel wall

slip [21], and absorbing and pulsating channel walls [22]. The latter can lead to a

reduction of particle dispersion due to an entropic slow-down resulting from channel

wall constrictions. Such a slow-down has also been reported for particles diﬀusing

through periodic, corrugated channel geometries [23]. On the other hand, channel wall

topography may enhance dispersion due to the generation of additional hydrodynamic

ﬂows and velocity gradients [11].

Beyond dispersion through channel geometries, solute transport in porous materials

has received substantial scientiﬁc attention dating back to Saﬀman in the 1960’s [24].

Continuing the work of several decades, in the 1970’s de Josselin de Jong noted

that dispersion in porous materials is a tensorial quantity [25], which encodes the

(anisotropic) properties of the medium. In 1980 Brenner developed a systematic theory

to predict the second-order anisotropic dispersion tensor and drift of particles spreading

through spatially periodic, porous media [26], which relies on the details of the ﬂow ﬁeld

within one periodic cell. Brenner’s theory has been extended to account for ﬂuctuations

in the environment that are of the order of the transport process itself [27]. At the same

time, other researchers characterized the tensorial nature of diﬀusion in random porous

media [28, 29]. Building on Brenner’s pioneering work, the study of self-propelled agents

in spatially periodic media has demonstrated that obstacles act as entropic barriers while

producing additional velocity gradients, which can simultaneously reduce or enhance

Transport of a passive scalar in wide channels with surface topography 3

dispersion, depending on the local shear rate [30]. Reminiscent of the transport through

ﬂuctuating channels mentioned above, experiments revealed that ﬂuctuations of the

obstacles enhance transport of particles diﬀusing through the porous matrix [31]. Going

beyond periodic structures, other experiments have demonstrated anomalous dispersal

of solutes in random porous media, where complex ﬂow patterns of Newtonian and

non-Newtonian ﬂuids emerge within the dead end pores [32, 33].

While these studies provide immediate insights into biological and geophysical

transport phenomena, they can also guide the design of new microﬂuidic devices, which

rely on the precise control of transport of particulate suspensions [4, 5, 6, 7, 8]. For

example, the method ‘deterministic lateral displacement’ relies on the use of arrays

of posts within a microﬂuidic channel to eﬃciently separate biological and synthetic

constituents of diﬀerent sizes [34, 35, 36]. The eﬀect of diﬀusion within this context has

been analyzed both experimentally [37] and theoretically [38] in the realm of Brenner’s

theory. In particular, it has been shown that the interactions of ﬁnite-sized particles

with the anisotropic obstacle ﬁeld can generate long-time anisotropic dispersion [37, 38].

Another common approach is the use of tailored surfaces with particular surface

topographies to achieve mixing [39], sorting [40, 41, 42, 43], or focusing of particles in

suspension [44, 45]. In addition, recently, it has been found that particles moving past

herringbone structures have complex, three-dimensional trajectories due to the particle’s

interaction with the surface [46, 47], yet the eﬀect of diﬀusion, which in many contexts

is not negligible, on their long-time transport properties has not yet been addressed.

While in several studies dispersion in narrow channels has been studied, in many of these

microﬂuidic applications channels are signiﬁcantly wider than they are tall, requiring a

new, fully three-dimensional theoretical approach for the characterization of dispersion,

which requires consideration of transport both along and perpendicular to the ﬂow.

Here, we revisit the classical Taylor dispersion theory and extend it for scalar

transport in wide, structured channels. By “structured” we refer to the shape of

the channel walls. We present an asymptotic long-time, two-dimensional convection-

diﬀusion equation, in contrast to Taylor’s one-dimensional equation for dispersion

through narrow channels. In particular, the three-dimensional nature of the surface

structures no longer allows for a reduction to either a two-dimensional or axisymmetric

description of the ﬂow. Our theory, valid for small surface amplitudes, provides an

analytic prediction for the dispersion matrix and the overall drift as a function of

the surface shape. We provide results for diﬀerent surface structures, ranging from

corrugated channel walls, as often used in microﬂuidic applications, to randomly

structured topographies. Finally, we use stochastic simulations to corroborate our

theory.

2. Theoretical Background

Consider a Brownian particle at a position rat time t. The probability density c(r, t) of

the particle (or, equivalently, the solute concentration) is governed by the Fokker-Planck

Transport of a passive scalar in wide channels with surface topography 4

(or convection-diﬀusion) equation,

∂tc=−∇·(uc) + D0∇2c, (1)

where D0is the short-time diﬀusivity of the particle and u= [u, v, w]Tis a quasi-steady

low-Reynolds-number ﬂow ﬁeld. The particle is suspended in a channel with a structured

lower wall, Sw, and a planar upper wall, Sh, at z=h(measured from a reference

surface S0), see ﬁgure 1. The structured wall Swis described by the proﬁle z=aH(rk),

where rk= [x, y, 0]Tare the in-plane coordinates, adenotes the characteristic surface

amplitude, and H(rk) is the surface shape function. In addition, the upper wall is

moving at a speed u0along the x−direction and the lower wall is stationary.

We assume that the channel is not conﬁned along the horizontal directions. At

the top and bottom of the channel the probability density obeys a no-ﬂux boundary

condition,

n·∇c(r, t) = 0 on Shand Sw,(2)

where ndenotes the vector normal to the upper planar surface Shand the lower

corrugated surface Sw, respectively. We are interested in deriving the behavior of the

height-averaged probability density

C(rk, t) = 1

h−aH(rk)Zh

aH(rk)c(r, t) dz≡ hc(r, t)i,(3)

where h·i represents the height-averaging. Unlike in traditional Taylor-dispersion

theory, where averaging over the cross-section of the channel leads to an eﬀective

one dimensional transport equation, we retain two horizontal spatial dimensions. We

decompose the full probability density as

c(r, t) = C(rk, t) + ˜c(r, t),(4)

where ˜cis a perturbation term that accounts for the variations in the probability density

along the z-direction.

Inserting (4) into (1) and deﬁning the in-plane gradient ∇k= [∂x, ∂y,0]Twe obtain

∂t(C+ ˜c) = −∇·(u(C+ ˜c)) + D0(∇2

kC+∇2˜c).(5)

Height averaging (5) and recalling the no-ﬂux boundary condition (2), we ﬁnd

∂tC=−h∇·(u(C+ ˜c))i+D0∇2

kC,

=−∇k·(huiC)− h∂zwiC− h∇·(u˜c)i+D0∇2

kC, (6)

where by deﬁnition h˜ci= 0, h∇2˜civanishes by the Leibniz rule, and we have:

h∂zwi=w(z=h)−w(z=aH)

h−aH(rk).(7)

To derive an equation for ˜cwe subtract (6) from (5):

∂t˜c=−h∇·(uC)−∇k·(huiC) + h∂zwiCi−[∇·(u˜c)− h∇·(u˜c)i] + D0∇2˜c. (8)

As we are interested in the long-time behavior of the particle, we consider time

scales greater than the time scale of diﬀusion in the vertical direction, t>

∼h2/D0[48].

Transport of a passive scalar in wide channels with surface topography 5

ﬂuid U=u0ex

t0t1

z=aH(r∥)

Figure 1. Model sketch. Tracer particles are suspended between an upper planar wall,

Sh, located at z=hand a lower, structured surface, Sw. The latter is described by

z=aH(x, y), where H(x, y) denotes the shape function and ais the surface amplitude.

Further, S0corresponds to the reference surface at z= 0. Sketches of the particle

distributions at time t=t0and a later time t=t1are shown. The upper wall, Sh,

moves at a velocity u0along the x−direction.

In this limit, diﬀusion across the channel averages out and we can consider the steady-

state solution for ˜c,∂t˜c= 0. Further, we expect that the perturbations in the probability

density are smaller than the average, ˜c<

∼C, which leads us to neglect the second term

of (8); this step is standard in this type of theoretical development. Finally, since ˜c

encodes the entire ﬂuctuations of the probability-density in the z-direction, we assume

that the diﬀusion in zis dominant. These assumptions lead to a simpliﬁed form of (8):

D0∂2

z˜c=∇·(uC)−∇k·(huiC) + h∂zwiC.(9)

Subsequently, we rescale times by h2/D0, lengths by h, and velocities by u0.

Introducing the P´eclet number Pe ≡u0h/D0, a dimensionless roughness =a/h, and

exploiting incompressibility, (6) and (9) become

∂tC=−Pe h∇k·(huiC)− h∂zwiC− hu·∇˜cii+∇2

kC, (10a)

∂2

z˜c= Pe hu·∇kC−∇k·(huiC)− h∂zwiCi.(10b)

We note here that the in-plane divergence ∇k·huidoes not vanish in general. Specifying

the surface shape H(rk) and the associated hydrodynamic ﬂows u, allows us to recast

(10a) into a classical, two-dimensional convection-diﬀusion equation with eﬀective

transport parameters.

2.1. Perturbation ﬂow due to surface roughness

The base shear ﬂow u0=u0z/hexthat arises due to the motion of the upper boundary

becomes perturbed by the presence of the surface topography. At low Reynolds numbers,

the ﬂow is governed by the continuity and Stokes equations. Non-dimensionalizing

velocities uby u0and the pressure ﬁeld pby µu0/h, where µdenotes the viscosity of

the ﬂuid, the governing equations are

∇ · u= 0,(11a)

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摘要：

TransportofapassivescalarinwidechannelswithsurfacetopographyJ.V.Roggeveen1,H.A.Stone1,C.Kurzthaler121DepartmentofMechanicalandAerospaceEngineering,PrincetonUniversity,NewJersey08544,USA2MaxPlanckInstituteforthePhysicsofComplexSystems,01187Dresden,GermanyE-mail:jamesvr@princeton.edu,hastone@princeton...

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Transport of a passive scalar in wide channels with surface topography J. V. Roggeveen1 H. A. Stone1 C. Kurzthaler12

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