Reacting particles in open chaotic flows

2.6 MB·English

by  Alessandro P. S. de Moura

Reacting particles in open chaotic flows
Alessandro P. S. de Moura
Institute of Complex Systems and Mathematical Biology,
University of Aberdeen, Aberdeen, AB24 3UE, UK
Westudythecollision probabilitypofparticlesadvectedbyopenflowsdisplayingchaoticadvec-
tion. We show that p scales with the particle size δ as a power law whose coefficient is determined
by the fractal dimensions of the invariant sets defined by the advection dynamics. We also argue
that this same scaling also holds for the reaction rate of active particles in the low-density regime.
These analytical results are compared to numerical simulations, and we find very good agreement
with thetheoretical predictions.
PACSnumbers: 47.52.+j,47.53.+n,05.45.Ac,47.51.+a
Many fluid flows of interest to science and to engi- tration fields, or modelling the propagation of the reac-
neering are open flows, which are characterised by the tion through reaction fronts. This approach ignores the
2
presence of inflow and outflow regions [1] (see Fig. 1a). factthatanyactivitytakingplaceinthe fluidultimately
1
Thedynamicsofadvectedparticlesinopenflowsistran- arisesfromcollisionsofparticles. If the number ofreact-
0
2 sient, with typical particles leaving to the outflow region ingparticlesisverylarge,sothatthe meanfree pathbe-
in finite time. Advection in many open flows is chaotic tween collisions is much smaller than the typical length
n
[2, 3]; examples of chaotic open flows are found in areas scale of the system, then this continuous description is
a
J such as microfluidics [4], climatology [5], physiology [6], expected to be valid. However, in the low-density limit,

6 population biology [7] and industry [8, 9]. The dynam- where typically particles traverse large distances before
2 ics of advected particles is governed by the chaotic sad- colliding and reacting, the concepts of reactant concen-
dle [10], which is a set of unstable orbits contained in a trationandreactionfronthavenomeaning. Inthis case,
]
D boundedregionofspaceknownasthemixing region (see one must adopt a kinetic theory approach, where activ-
Fig. 1a). Fluid elements are repeatedly stretched and ity is described in terms of the probability of collision
C
folded in the mixing region, and this causes any portion between particles. Although this idea has been used in
.
n of the fluid to be deformed by the flow into a complex closed flows — in the context of chemical catalysis [16],
li filamentary shape, which shadows the unstable manifold particle coalescence [17, 18] and crystallisation [19], for
n
of the chaotic saddle, defined as the fractal set of orbits example — so far this idea has not been pursued in the
[
which converge asymptotically to the chaotic saddle for case of open flows, despite the importance of the low-

1 t [10]. density regime for applications, which is appropriate for
v →Th−e∞fractal structures generated by chaotic advection describing systems as diverse as platelet activation in
3
have dramatic consequences for the dynamics of active blood flows [6], plankton population dynamics [12], and
5
5 processestakingplaceinopenflows,suchaschemicalre- raindrop formation [20, 21].
5 actions and biological processes [3]. The stretching and In this work we investigate the effect of chaotic ad-
1. folding of fluid elements by the flow tends to increase vection on the activity of particles in the low-density
0 the area (or perimeter, in 2D flows) of contact between limit. The rate of reaction in this limit is determined
2 two reacting species, which results in an enhancementof by the collision probability p(δ) of two particles com-
1 the reactiondue purely to the advectivedynamics of the ing within a distance δ of each other before they escape
:
v flow—this has been called dynamic catalysis [3]. It has to the outflow; we assume a reaction event takes place
i been shown [3, 11] that this effect appears as a singular when that happens. δ can be thought of as the size of
X
production term in an effective reaction rate equation, the particles, or alternatively as determined by the re-
r
a which has a power-law dependence on the amount of re- action cross-section σ (with δ √σ). We show that p
∼
actant; the coefficient of the power law is determined by scaleswithδ asapowerlaw,p(δ) δβ,andwederivean
∼
the information fractal dimension of the unstable mani- analyticalexpressionfor the coefficient β in terms of the
fold of the chaotic saddle. Later works have established fractal dimensions of the stable and unstable manifolds
that this enhancementofactivity by chaosis a verygen- of the chaotic saddle. The classical kinetic theory ap-
eralandrobustphenomenon[3],andisafeatureofmany proach, based on the Smoluchowski equation [22], which
kinds of activity and flows, including non-periodic [12], assumes there is a homogeneous mixing of particles in
non-hyperbolic [13, 14] and three-dimensional[15] flows. the mixing region, would predict β = N, where N is
All the works so far on the effects of chaos on chem- the spatial dimension, but we show that because most
ical or biological activity in open flows have adopted collisions happen in the neighbourhood of a fractal set,
continuum descriptions of the reacting material — ei- Smoluchowski’sresultdoesnothold,andβ =N foropen
6
ther describing the reactants using continuous concen- chaotic flows. In the low-density limit, the total number
2
inflow outflow we will refer to this δN scaling as the Smoluchowskipre-
diction.
TheSmoluchowskiscalingderivedabovedoesnottake
into account the fractal nature of the particle distribu-
mixing
region tions generated by chaotic advection in open flows. We
will now derive the correct scaling of p(δ). Let two par-
ticles be chosen randomly within two bounded and dis-
joint regions R and R (see Fig. 1b). As a result of the
(a) 1 2
chaotic dynamics of the flow in the mixing region, those
trajectorieswhichtakelongertoescapeconvergetowards
the unstable manifold W . Consequently, the particles
u
will only have a chance to meet if both come within a
distance δ (the reaction distance) of the unstable man-
u
ifold. This will take place if their initial conditions are
within a distance of order δ of the stable manifold W ,
s
which is the set of initial conditions converging to the
chaotic saddle for t (see Fig. 1b) [10]. The proba-
→∞
bility Ps(1)(δ) that one particle with initial conditions in
R lieswithinadistanceoforderδ ofthestablemanifold
region 1
is proportional to the volume V (or area in two dimen-
(b) 1
sions) of the intersection of R and a δ-covering of the
1
componentofW where the conditionallyinvariantmea-
FIG. 1: (a) Illustration of an open flow. (b) Schematic s
sureisconcentrated[10,23](seeFig. 1b). Ifoneimagines
depiction of the stable (W ) and unstable (W )
s u
manifolds in an open flow. The grey bands are R1 to be covered by a grid of size δ, the number of cells
in the grid intersecting the component of W where the
δ-neighbourhoods of W and W (see text). R and R s
s u 1 2
conditionally invariantmeasure is concentratedscales as
indicate regions where initial conditions are chosen.
N1(δ) ∼ δ−D1s [10], where D1s is the information dimen-
sionofthe stablemanifold. Since thevolumeofeachcell
Q(δ) of reactionevents between reacting particle species in the grid is δN, the volume of the portion of the grid
which takes place in a given flow also scales as δβ. We which intersects Ws scales as V1 N(δ)δN δN−D1s.
arguethatQisagoodmeasureoftheefficiencyofmixing We therefore have Ps(1)(δ) δN−D∼1s, and the s∼ame scal-
∼
in open flows, and thus β characterises the mixing of an ing holds for the probability P(2) that the other particle
s
open flow. These analytical predictions are compared to is close to W . Therefore, the probability P =P(1)P(2)
s s s s
numericalsimulationsin2Dflows,andwefindtheresults that both particles reach a δ-neighbourhood of the un-
agree very well with our theory. stable manifold scales as
We start by deriving the scaling of the collision prob-
ability p(δ) of two particles with randomly choseninitial Ps(δ) δ2N−2D1s. (2)
∼
conditions, by first assuming that after the particles en-
Thetwoparticlescollideiftheyreachtheunstablemani-
terthemixingregion(Fig. 1a),theflowactsasaperfect,
foldwithin adistance δ ofeachother. Consideringagain
uniform mixer, until the particles escape to the outflow.
a grid of size δ covering the unstable manifold W , the
u
Thisperfectmixingassumptionmeansthatthepositions
probability P (δ) that two particles collide can be ap-
u
of the two particles are effectively randomised in a time
proximated by the probability that they end up in the
interval τ which is characteristicof the flow. The proba- same cell in the grid. This is given by P (δ) = P µ2,
bilitythattwoparticlesrandomlyplacedinaboundedN- u i i
where µ is the (conditionally invariant) measure of cell
i
dimensional region are within a distance δ of each other
i, and the sum is over all cells intersecting the unstable
scales as δN. Assuming that δ is small enough so that
manifold. From the definition of the correlation dimen-
the probability p(δ) that the two particles collide before sion Du [10, 23], we find that in the limit δ 0 we get
escaping satisfies p(δ) 1, we conclude that p(δ) is pro- 2 →
portional to δN and t≪o the average time T the particle Pu(δ) δD2u. (3)
∼
staysinthemixingregionbeforeescaping. p(δ)therefore
The overall probability p(δ) = P (δ)P (δ) that the two
scales with δ as s u
particles will collide therefore scales as
p(δ) δN, (1) p(δ) δβ, with β =2N 2Ds+Du. (4)
∼ ∼ − 1 2
The perfect mixing assumption is the assumption used This is our main result. This scaling is clearly different
in Smoluchowski’s classical coagulation theory [22], and from the Smoluchowski prediction: for chaotic flows, in
3
1.0 ns
o 1
cti
a
0.5 e
of r
0.0 no.
y /
−0.5 bilit
ba1e-06
o
−1.0 Pr (b)
−1.5 δ 0.001
−2−.02.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 FIG. 3: Collision probability (squares) and the number
of reaction events (circles) as a function of the reaction
distance δ for the blinking vortex-sink flow, with the
1.0
same parameters as in Fig. 2. The initial conditions
were chosen in the regions R and R defined in Fig. 2.
0.5 1 2
The number of pairs used varied with δ, and it was such
0.0 that it guaranteed that at least 103 collisions took place
for each δ; typically this meant the number of pairs was
−0.5 of order 1010–1011. The straight lines are the fitting of
the data to a power law of the form δβ∗; in both cases
−1.0
∗
fitting yields β =2.23, compared to the prediction
−1.5 β =2.25 of Eq. (4). The collision probability for the
area-preservingbaker map with escape is also shown
−2−.02.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 (diamonds), for contraction factors 0.3 and 0.45 [27].
Fitting yields β∗ =2.30. For this system Ds =1.701
1
FIG. 2: (a) Numerical approximation of the unstable and Du =1.698, and from Eq. (4) the prediction is
2
manifold for the blinking vortex-sink flow, with η =0.5 β =2.298. The triangles show the collision probability
and ξ =10; for these parameters,the dynamics is for the conservative Hénon map for the nonlinearity
chaotic and hyperbolic. (b) Location of collisions for parameter λ=7 [14], for which Ds =1.58 and
1
δ =10−3, for the flow with the same parameters as Du =1.32, leading to a prediction of β =2.16; fitting
2
∗
above. It is clear that collisions take place in the yields β =2.14. The baker map data (diamonds) in
vicinity of the unstable manifold. The initial conditions this plot has been shifted in the graph above to improve
were chosen in the regions R ( 0.5<x<0.5, visibility.
1
−
0<y <0.15) and R ( 0.5<x<0.5, 0.2<y <0.35).
2
−
The time step used in the simulations was ∆t=0.02.
Aref’s blinking vortex flow [26]. It is a 2D incompress-
ible flow onan infinite plane, with two sinks at positions
(+a,0) and ( a,0), which open and close periodically
generalweexpectβ =N. However,β doesapproachthe −
6 in alternation: in the first half of each cycle one sink is
Smoluchowski value of N in the limit where the escape
open and the other one is closed, and in the second half
rate vanishes, when D2,Du N, and from Eq. (4) this
1 2 → the situation is reversed. Each vortex-sink is modelled
implies that β N. This behaviour is expected, since
→ asapointsourceofvorticitysuperimposedto alocalised
in this limit the dynamics within the mixing region can
sink,andflowwhichfallsoneitherofthesinksdisappears
be consideredas that of a closedcontainer being stirred,
fromthe systemanddoesnotcomeback. This systemis
with a small leak, when particles spend a long lime in
clearlyanopenflow,wheretheinflowregioncorresponds
the mixing region and the perfect mixing assumption is
to the whole space beyond the sinks.
approximately valid.
Assuming that the particles advected by the flow be-
ThescalingpredictedbyEq. (4)isvalidforanychoice
haveaspassivetracers,theirequationsofmotionare[24]
of regions R and R , as long as both regions intersect
1 2
the stable manifold Ws. If there is no intersection, the r˙ = C/r, ϕ˙ =K/r2, (5)
two particles never meet because they do not get to the −
mixing region, and collisions cannot take place. where r and ϕ are polar coordinates whose origin alter-
TotestthepredictionofEq. (4),wewillusetheblink- nates between (a,0), during the first half of eachperiod,
ing vortex-sink flow, [24, 25] which is a generalisation of and ( a,0), during the second half of each period; C
−
4
and K are parameters representing the strengths of the the prediction of Eq. (4) (see Fig. 3).
sink andthe vortex,respectively. The advectiondynam- We now examine a system of reacting particles ad-
ics is determined by the two dimensionless parameters vected by a chaotic open flow. Taking as an illus-
η = CT/a2 and ξ = K/C, where T is the flow period. trative example the case of an acid-base-type reaction
We choose units so that a = 1 and T = 1. The dynam- A+B C, consider particles of two species, which we
→
ics is chaotic for a wide range of η and ξ [24]. Fig. 2a will label simply species A and B, such that when an A
shows the unstable manifold for one particular choice of particle comes within a distance δ of a particle of type
parameters, η =0.5 and ξ =10. B, a reaction occurs, resulting in a particle of type C,
We computed the collision probability p(δ) by follow- which is the product of the reaction. If we throw N
1
ing the trajectories of many pairs of particles until one type-Aparticles ina regionR of the flow, and N type-
1 2
particle in each pair escapes, and at every time step ∆t B particles in region R , then we expect that the total
2
we check if the distance between the two particles is less numberN oftype-C particlesproducedinthe systemis
p
thanδ;ifso,theparticlesareconsideredtohavecollided. proportional to N N p(δ) — as long as N and N are
1 2 1 2
Theinitialconditionofoneoftheparticlesineachpairis nottoo large,andthe low-densityassumptionholds. We
chosenrandomlywithinregionR1,definedinthecaption therefore predict that the amount Np of product gener-
of Fig. 2; and the other particle in the pair is started in ated by the reaction scales with the reaction distance δ
region R2 (see Fig. 2). The fraction of the pairs which as δβ, with β given by Eq. (4).
collide before escaping gives us a numerical approxima- We put this prediction to the test in the blinking
tion of p(δ). Figure 3 (the squares) shows the result of vortex-sink system, using an efficient algorithm for find-
applying this procedure for a range of values of δ. The ingtheneighbourswithinadistanceδ[28]. Theresulting
resulting scaling is clearly a power law p(δ) δβ∗, and scalingofN withδ isshowninFig. 3. Wefindapower-
∗ ∼ p
fittingyieldsβ 2.25. Tocomparethiswiththepredic- law scaling as expected, and fitting yields the coefficient
≈
tion of Eq. (4), we use the value D1s ≈1.74 from [24]; to β∗ = 2.25, which is again within 1% of the predicted
compute D2u we first find an approximation Wuspr of the value of β = 2.23. We found the same agreement for
unstable manifold using the sprinkler method [10], and other choices of parameters for the blinking vortex-sink
then we compute the scaling with ǫ of the total number system, and also for the other dynamical systems we in-
of pairs of points on Wuspr separated by a distance less vestigated.
than ǫ; the coefficient of the resulting power law is Du
2 We note that, even though we used the reaction A+
[23]. We found Du 1.71. This is a two-dimensional
2 ≈ B C as an example above, all kinds of reactions and
system, and so N = 2 in Eq. (4). Equation (4) then →
activeprocessesinvolvingparticleswillbeaffectedbythe
∗
predicts β =2.23,whichagreeswith the value β =2.25
scaling (4), since all reactions require particles to come
from the simulation, to within numerical errors.
close together, and the probability of that happening is
Fig. 2a is a snapshot of W at the start of a period;
u governedbythecoefficientβ ofEq. (4). Weproposethat
theshapeofW changesperiodicallywithtime,withthe
u the collisionprobabilityp,andthe wayis scaleswiththe
sameperiodastheflow. Figure2bshowsthepositionsof
collisiondistanceδ,isthenaturalquantitytocharacterise
thecollisionsthattakeplaceatthebeginningofeachpe-
the efficiency of mixing of an open flow (an alternative
riod: eachdotinFig. 2bisonesuchcollision. Comparing
definition is given in [9]). Equation (4) shows that the
Figs. 2a and 2b, it is clear that the collisions take place
mixing efficiency is determined by the fractal properties
in a small neighbourhood of the unstable manifold W ,
u oftheinvariantsetsassociatedwiththechaoticsaddle—
confirming that the assumption made in the derivation
namelyitsstableandunstablemanifolds. Thediscussion
of Eq. (4) is correct.
aboveshowsthatthemixingefficiencyinturndetermines
We have analysed other choices of the parameters η
the efficiency of chemicalreactions(or,in general,active
and ξ for which the flow is chaotic, and we always find
processes)inopenflows,whichmeansthatthe“chemical
that the prediction from Eq. (4) and the results from
efficiency” of a flow is also measured by the coefficient β
the simulations differ by less than 1% in all cases. We
defined in Eq. (4).
found that the choice of the time step ∆t does notaffect
the scaling of p(δ) with δ — although the value of p for
anygivenδ ofcoursedecreaseswith∆t,sincewithlarger
timestepsitismorelikelycollisionswillbemissedbythe
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