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Keywords: Granular materials; Fluid dynamics; Filtration; Porous media1. Introduction

Particles in fluids (liquids or gases) appear in many appli-cations in chemical engineering, fluid mechanics, geology andbiology [13]. Also fluid flow through a porous medium is ofimportance in many practical situations ranging from oil recov-ery to chemical reactors and has been studied experimentallyand theoretically for a long time [46]. Due to disorder, porousmedia display many interesting properties that are however dif-ficult to handle even numerically. One important feature is thepresence of heterogeneities in the flux intensities due the vary-ing channel widths. They are crucial to understand stagnation,filtering, dispersion and tracer diffusion.

In the porous space the fluid mechanics is based on theassumption that a Newtonian and incompressible fluid flowsunder steady-state conditions. We consider the NavierStokesand continuity equations for the local velocity u and pressurefields p, being is the density of the fluid. No-slip bound-ary conditions are applied along the entire solidfluid inter-face, whereas a uniform velocity profile, ux(0, y) = V anduy(0, y) = 0, is imposed at the inlet of the channel. For simplic-ity, we restrict our study to the case where the Reynolds number,

* Corresponding author.E-mail address: hjherrmann@ethz.ch (H.J. Herrmann).

defined here as Re VLy/, is sufficiently low (Re < 1) toensure a laminar viscous regime for fluid flow. We use FLUENT[7], a computational fluid dynamic solver, to obtain the numer-ical solution on a triangulated grid of up to hundred thousandpoints adapted to the geometry of the porous medium.

The investigation of single-phase fluid flow at low Reynoldsnumber in disordered porous media is typically performed us-ing Darcys law [4,6], which assumes that a macroscopic index,the permeability K , relates the average fluid velocity V throughthe pores with the pressure drop P measured across the sys-tem,

(1)V = K

P

L,

where L is the length of the sample in the flow direction and is the viscosity of the fluid. In previous studies [814], compu-tational simulations based on detailed models of pore geometryand fluid flow have been used to predict permeability coeffi-cients.

Here we present numerical calculations for a fluid flowingthrough a two-dimensional channel of width Ly and length Lxfilled with randomly positioned circular obstacles [15]. For in-stance, this type of model has been frequently used to studyflow through fibrous filters [23]. Here the fluid flows in thex-direction at low but non-zero Reynolds number and in they-direction we impose periodic boundary conditions. We con-Computer Physics Communicat

Particles

Hans J. Herrmann a,, Jos S. Aa IfB, ETH Zrich, Hngger

b Departamento de Fsica, Universidade Feder

Abstract

For finite Reynolds numbers the interaction of moving fluids with padifferent numerical studies all using the solver Fluent which elucidatecase of fixed particles, i.e. a porous medium and present the distributionthe density of particles and for the fluxes follow an unexpected stretchemassive tracer particles within this fluid. Interestingly a critical Stokesby a critical exponent of 1/2. 2007 Elsevier B.V. All rights reserved.Please cite this article in press as: H.J. Herrmann et al., Particles in fluids, Compute

0010-4655/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.cpc.2007.02.062 () www.elsevier.com/locate/cpc

n fluids

rade Jr. b, Ascnio D. Arajo b, 8093 Zrich, Switzerlando Cear, 60451-970 Fortaleza, Cear, Brazil

les is still only understood phenomenologically. We will present threes issue from different points of view. On one hand we will consider thechannel openings and fluxes. These distributions show a scaling law inxponential behavior. The next issue will be filtering, i.e. the release ofber below which no particles are captured and which is characterizedr Physics Communications (2007), doi:10.1016/j.cpc.2007.02.062

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Fig. 1. Velocity magnitude for a typical realization of a pore space with porosity = 0.7 subjected to a low Reynolds number and periodic boundary conditionsapplied in the y-direction. The fluid is pushed from left to right. The colorsranging from blue (dark) to red (light) correspond to low and high velocitymagnitudes, respectively. The close-up shows a typical pore opening of lengthl across which the fluid flows with a line average velocity v. The local flux atthe pore opening is given by q = vl cos , where is the angle between v andthe vector normal to the line connecting the two disks. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web versionof this article.)

sider a particular type of random sequential adsorption (RSA)model [16] in two dimensions to describe the geometry of theporous medium. As shown in Fig. 1, disks of diameter D areplaced randomly by first choosing from a homogeneous distri-bution between D/2 and Lx D/2 (Ly D/2) the randomx- (y-)coordinates of their center. If the disk allocated at thisposition is separated by a distance smaller than D/10 or over-laps with an already existing disk, this attempt of placing a diskis rejected and a new attempt is made. Each successful plac-ing constitutes a decrease in the porosity (void fraction) byD2/4LxLy . One can associate this filling procedure to a tem-poral evolution and identify a successful placing of a disk asone time step. By stopping this procedure when a certain valueof is achieved, we can produce in this way systems of wellcontrolled porosity. We study in particular configurations with = 0.6, 0.7, 0.8 and 0.9.

2. Results on filtration

Filtration is typically used to get clean air or water and alsoplays a crucial role in the chemical industry. For this reasonit has been studied extensively in the past [17]. In particular,we will focus here on deep bed filtration where the particles insuspension are much smaller than the pores of the filter whichthey penetrate until being captured at various depths. For non-Brownian particles, at least four capture mechanisms can bedistinguished, namely, the geometrical, the chemical, the grav-itational and the hydrodynamical one [17].

Very carefully controlled laboratory experiments were con-ducted in the past by Ghidaglia et al. [18] evidencing a sharptransition in particle capture as function of the dimensionlessPlease cite this article in press as: H.J. Herrmann et al., Particles in fluids, Compute

ratio of particle to pore diameter characterized by the diver-gence of the penetration depth. Subsequently, Lee and Kop-lik [19] found a transition from an open to a clogged state of[m5+; v 1.70; Prn:21/03/2007; 13:22] P.2 (1-4)

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Fig. 2. The trajectories of particles released from different positions at the inletof the periodic porous medium cell. St = 0.25 and the flow field u is calcu-lated from the analytical solution of Marshall et al. [23]. The thick solid linescorrespond to the limiting trajectories that determine .

the porous medium that is function of the mean particle size.Much less effort, however, has been dedicated to quantify theeffect of inertial impact on the efficiency of a deep bed filter.

Here we will concentrate on the inertial effects in capturewhich constitute an important mechanism in most practicalcases and, despite much effort, are quantitatively not yet un-derstood, as reviewed in Ref. [20]. The effect of inertia on thesuspended particles is usually quantified by the dimensionlessStokes number, St V d2pp/18, where dp and p are thediameter and density of the particle, respectively, is a char-acteristic length of the pores, is the viscosity and V is thevelocity of the fluid. Inertial capture by fixed bodies has alreadybeen described since 1940 by Taylor and proven to happen forinviscid fluids above a critical Stokes number [21]. It is our aimto present a detailed hydrodynamic calculation of the inertialcapture of particles in a porous medium. We will disclose novelscaling relations.

Let us first consider the case of an infinite ordered porousmedium composed of a periodic arrangement of fixed circularobstacles (e.g., cylinders) [22]. This system can then be com-pletely represented in terms of a single square cell of unitarysize and porosity given by (1D2/4), where D is the di-ameter of the obstacle, as shown in Fig. 1. Assuming Stokesianflow through the void space an analytical solution has been pro-vided by Marshall et al. [23]. Here we use this solution to obtainthe velocity flow field u and study the transport of particlesnumerically. For simplicity, we assume that the influx of sus-pended particles is so small that (i) the fluid phase is not affectedby changes in the particle volume fraction, and (ii) particleparticle interactions are negligible. Moreover, we also considerthat the movement of the particles does not impart momen-tum to the flow field. Finally, if we assume that the drag forceand gravity are the only relevant forces acting on the particles,their trajectories can be calculated by integration of the fol-lowing equation of motion dup/dt = (u up)/St +Fgg/|g|,where Fg (p )|g|/(V 2p) is a dimensionless parame-ter, g is gravity, t is a dimensionless time, and up and u arethe dimensionless velocities of the particle and the fluid, respec-tively.r Physics Communications (2007), doi:10.1016/j.cpc.2007.02.062

We show in Fig. 2 some trajectories calculated for particlesreleased in the flow for St = 0.25. Once a particle touches theboundary of the obstacle, it gets trapped. Our objective here

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Fig. 3. The dependence of the capture efficiency on the rescaled Stokes num-ber St/( min) for periodic porous media in the presence of gravity. Here weuse Fg = 16, a value that is compatible with the experimental setup describedin Ref. [18]. The inset shows that the behavior of the system without gravity canbe characterized as a second order transition, (St Stc) , with 0.5 andStc = 0.2679 0.0001, 0.2096 0.0001 and 0.1641 0.0001, for = 0.85,0.9 and 0.95, respectively.

is to search for the position y0 of release at the inlet of theunit cell (x0 = 0) and above the horizontal axis (the dashedline in Fig. 2), below which the particle is always capturedand above which the particle can always escape from the sys-tem. As depicted in Fig. 2, the particle capture efficiency can bestraightforwardly defined as 2y0. In the limiting case whereSt , since the particles move ballistically towards the ob-stacle, the particle efficiency reaches its maximum, = D. ForSt 0, on the other hand, the efficiency is smallest, = 0. Inthis last situation, the particles can be considered as tracers thatfollow exactly the streamlines of the flow and are therefore nottrapped.

We show in Fig. 3 the loglog plot of the variation of /Dwith the rescaled Stokes number in the presence of gravity forthree different porosities. In all cases, the variable increaseslinearly with St to subsequently reach a crossover at St, andfinally approach its upper limit ( = D). The results of oursimulations also show that St ( min), where min cor-responds to the minimum porosity below which the distancebetween inlet and obstacle is too small for a massive particleto deviate from the obstacle. The collapse of all data shown inFig. 3 confirms the validity of the scaling law.

We see in the inset of Fig. 3 that the behavior of the system interms of particle capture becomes significantly different in theabsence of gravity. The efficiency remains equal to zero upto a certain critical Stokes number, Stc, that corresponds to themaximum value of St below which particles cannot be captured,regardless of the position y0 at which they have been released.Right above Stc, the variation of can be described in terms ofa power-law, (StStc) , with an exponent 0.5. Our re-sults show that, while the exponent is practically independentof the porosity for > 0.8, the critical Stokes number decreasesPlease cite this article in press as: H.J. Herrmann et al., Particles in fluids, Compute

with , and therefore with the distance from the obstacle wherethe particle is released (see Fig. 3). To our knowledge, this be-havior, that is typical of a second order transition, has never[m5+; v 1.70; Prn:21/03/2007; 13:22] P.3 (1-4)

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been reported before for inertial capture of particles. In order tohave a more realistic model for the porous structure we did alsoinclude disorder [22]. Here we adopted a random pore spacegeometry [16] shown in Fig. 1 and obtained the same results asfor the regular case.

3. Conclusion and outlook

We have found [15] that although the distribution of channelwidths in a porous medium made by a two-dimensional RSAprocess is rather complex and exhibits a crossover at l D,the distribution of fluxes through these channels shows an as-tonishingly simple behavior, namely a square-root stretchedexponential distribution that scales in a simple way with theporosity. Future tasks consist in generalizing these studies tohigher Reynolds numbers, other types of disorder and three di-mensional models of porous media.

We also presented results for the inertial capture of par-ticles in two-dimensional periodic as well as random porousmedia [22]. For the periodic model in the absence of gravity,there exists a finite Stokes number below which particles neverget trapped. Furthermore, our results indicate that the transitionfrom non-trapping to trapping with the Stokes number is of sec-ond order with a scaling exponent 0.5. In the presence ofgravity, we show that (i) this non-trapping regime is suppressed(i.e. Stc = 0) and (ii) the scaling exponent changes to 1. Weintend to investigate in the future the possibility of non-trappingat first contact and the effect on the capture efficiency of simul-taneous multiple particle release.

Acknowledgements

We thank the Max Planck Prize for financial support.

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Particles in fluidsIntroductionResults on filtrationConclusion and outlookAcknowledgementsReferences