Thermodynamics I MECH 213
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VOLUME 93 NUMBER 21 PHYSICAL REVIEW LETTERS week ending 19 NOVEMBER 2004 Experimental Studies of Pattern Formation in a ReactionAdvecti0nDiffusi0n System C R NugentW M Quarles and T H Solomon Department ofPhysics Backnell University Lewisbarg Pennsylvania 17837 USA Received 20 May 2004 published 18 November 2004 Experiments are presented on pattern formation in the BelousovZhabotinsky BZ reaction in a blinking vortex ow Mixing in this ow is chaotic with nearby tracers separating exponentially with time The patterns that form in this ow with the BZ rmction mimic chaotic mixing structures seen in passive transport The behavior is analyzed in terms of a mixing time Tm and a characteristic decorrelation time TBZ for the BZ system Flows with Tm comparable to or smaller than TBZ generate largescale patterns whose features are captured by simulations of mixing elds for the ow DOI 101103PhysRevLett93218301 There is a great deal of interest in understanding how patterns form in nonlinear dynamical systems In par ticular there have been numerous studies of reaction diffusion problems 12 such as chemically reacting uids 34 interacting populations and ecosystems 5 and systems with propagating fronts 6 7 By de nition there are no ows in a reactiondiffusion system any mixing between the different species is achieved solely via diffusion In most real uids however ows in the system signi cantly enhance mixing beyond that due to diffusion alone This advective mixing has a signi cant in uence on the pattern formation process This is par ticularly important in light of recent studies that show that mixing can be chaotic even for simple wellordered laminar uid ows 89 Some recent theoretical and numerical studies have considered the in uence of chaotic advection on chemical and biological patternforming systems 10713 In particular the studies have shown that chaotic mixing in reacting systems typically pro duces lamentary patterns with the laments mimicking the structures used to describe chaotic mixing in the ows 1415 In this Letter we present results of experimental stud ies of pattern formation in a reactionadvectiondiffusion system the BelousovZhabotinsky BZ reaction in a blinking vortex owia simple laminar ow in which the mixing is chaotic The BZ system initially attracted attention since the chemical reaction oscillates when well mixed When con ned to a thin twodimensional layer without any ows the BZ chemicals spontaneously de velop stripe and spiral patterns 1617 We investigate what happens to these patterns when the blinking vortex ow is present in the system In particular the reaction advectiondiffusion patterns that form are compared to the structures associated with chaotic mixing of impuri ties in this ow The blinking vortex ow 8 Fig 1a can be de scribed by the following set of equations M Ms x y 1 2183011 0031900704932121830142250 PACS numbers 8240Ck 05457a 4752j 4754r where xih TnlttltT D 7xb OlttltT2 xs 7 This is a twodimensional laminar uid ow in which uid alternately circles around a left and a right point vortex The blinking period T can be characterized non dimensionally by L ATaz where a is the radius of the entire system The parameter L can be thought of as the ratio between the blinking period and the circulation time for uid elements near the edge of the system Experimentally the ow is generated using a magneto hydrodynamic technique 18720 as shown in Fig lb The system is predominately two dimensional the ow is con ned to a thin 020 cm layer of water containing a small concentration of H2504 0006 M that is needed to conduct an electrical current Two electrodes near the middle of the ow are separated by 2b 19 cm and a b Fl 39dl 2b ui ayer gr Magnet Electrodes 4 23 FIG 1 a Top view of ow and b side view of apparatus for blinking vortex system The ow alternates periodically be tween circulating around the left point vortex labeled as shown and circulating around the right vortex not shown The ow is forced magnetohydrodynamically an electrical current enters the uid through an outer ring electrode then ows radially inward converging on one of the two electrodes nmr the center The interaction between this radial current and a strong magnetic eld produced by a 25 in diameter NdEeB magnet generates the vortical ow The center electrodes are a distance of 2b 19 cm apart and the region of interest within the Plexiglas ring has a diameter of 2a 57 cm and a thickness of 02 cm 2004 The American Physical Society 2183011 VOLUME 93 NUMBER 21 PHYSICAL REVIEW week ending L E T T E R S 19 NOVEMBER 2004 are surrounded by a bounding ring inner diameter 2a 57 cm and an outer electrode The entire apparatus sits on a 25 in diameter NdFeBo magnet which produces a vertical magnetic eld throughout the region of interest within the bounding ring During half of the blinking period current ows radially between the left point elec trode and the outer ring electrode The radial current interacts with the magnetic eld to produce a ow that circles around this electrode as shown in Fig 1a During the other half of the period the current ows between the right point and outer electrodes producing a ow that circles around that electrode Mixing in this ow is characterized by injecting a large blob of a neutrally buoyant impurity 0103 um uorescent polystyrene microspheres with a diffusion coef cient of 45 X 10 8 cmZs into the ow The system is illuminated with black light and images of the mixing are captured by a 12bit chargecoupled device video camera The resulting mixing is shown in Fig 2 As is typical for chaotic systems the mixing is characterized by repeated stretching and folding A region of chaotic mixing occupies the middle of the ow 8 the size of this chaotic region depends on the dimensionless blinking period u Throughout the remainder of this Letter u is held at 052 although the blinking frequency f and the vortex magnitude A are varied 1 w cm 0 O O O L l 0 Jgt FIG 2 Sequence showing the mixing of a passive tracer in the blinking vortex ow period of blinking T 505 u 052 Times after start are as follows a 53 s 1 period b 103 s 2 periods c 203 s 4 periods d Characteristic ensemble average lament width w for evolving dye distri bution determined experimentally by averaging the lament widths at numerous locations in each image The smooth curve is a t of the experimental data to a decaying exponential 2183012 Given a particular value of u the mixing patterns due to advection are determined mixing times however depend on more than just u Complete mixing occurs on a time scale determined by the interplay between advection and diffusion The stretching and folding of chaotic advection reduces the typical thickness w of impurity laments roughly exponentially in time as shown in Fig 2d The smooth curve is a t wt woe A with 079 periods l When the lament width w is suf ciently reduced molecular diffusion which mixes structures on a length scale Ld m nishes the mixing A mixing time scale 7m therefore is then de ned as the time that solves the implicit equa tion wt L dt For wo we use the centercenter vortex spacing 19 cm For the diffusion coef cient D we use a typical value for the chemicals in the BZ reaction D 2 1 X 10 5 cmzs since we are using the mixing time to analyze reactionadvectiondiffusion patterns in the BZ system An increase in the oscillation period T and a proportional decrease in the ow magnitude A maintain the same value of u but increases the mixing time 7m since it takes longer for each period of the blinking oscillation To study reactionadvectiondiffusion patterns in this system we replace the dilute H2804 by the chemicals used in the BZ reaction The solution is mixed from the following recipe 40 ml of 1 M sulfuric acid 1144 g of malonic acid 0418 g of potassium bromate 0044 g of cerium ammonium nitrate and 05 ml of 0025 M fer roin The reaction oscillates between a red and a blue color The system is illuminated with white light and a blue lter is placed on the camera lens such that the red and blue phases show up as dark and light regions re spectively The chemicals are initially well mixed with the whole system oscillating redblue uniformly In the absence of any uid ow the uniform oscillations break up within a couple hundred seconds into a complex pattern of stripes as seen in Fig 3a typical of patterns usually seen in reactiondiffusion systems Figures 3c 3e show pattern formation in the same system with the blinking vortex ow turned on All three cases have u 052 but frequencies f 0010 0030 and 0050 Hz corresponding to mixing times 7m 400 160 and 100 s respectively For each case shown the system is initially well mixed and oscillating redblue uniformly The blinking vortex ow is then initiated and the system is allowed to evolve for over 40 min about 150 periods of the BZ oscillation with the blinking vortex ow Experiments not shown are also done in which the blinking vortex ow is not turned on until after a pattern similar to Fig 3a develops Similar patterns are ob served after a transient period indicating that the longterm behavior is insensitive to the initial conditions The images in Figs 3c 3e reveal patterns that re ect the structures seen in the chaotic mixing of passive 2183012 VOLUME 93 NUMBER 21 PHYSICAL REVIEW week ending L E T T E R S 19 NOVEMBER 2004 1 1 250 200 50 00 50 FIG 3 a Patterns for the Belousov Zhabotinsky reaction in the absence of a driven ow image taken 29 min after start of run b Time series of intensities in arbitrary units taken at the eight different separated but initially dark locations in the absence of a ow c e Patterns for the Belousov Zhabotinsky reaction in the blinking vortex system u 052 The blinking frequency f is c 0010 d 0030 and 6 0050 Hz Time after start of run for c d and e is 41 45 and 46 min respectively impurities with the same value of u compare Fig 3 with Fig 2 similar to numerical results seen in earlier studies with a blinking vortexsink ow 21 23 But there are clear differences as the blinking frequency f and the mixing time 7m are varied Given a large Tm an intricate pattern forms with variations on small length scales Fig 3c On the other hand a ow with a small 7m but same u is characterized by a single largescale contiguous pattern that occupies much of the chaotic region in the center Fig 3e The increase in the size of the patterns with decreasing Tm makes sense after all in the limit of perfect mixing 7m gt 0 the entire system would oscillate redblue in unison The question then is to what time scale Tm should be compared when considering pattern formation in this system The typical period of oscillation for the BZ reaction 16 s without a ow and 16 20 s with the ow is a local time scale which can depend signi cantly on the mixing properties of the ow as shown in previous studies 24 Furthermore even in the absence of any advective mixing distant parts of the system can remain mostly synchronized for several oscillation periods since all parts of the system oscillate with close to the same frequency Consequently a time scale longer than the oscillation period for the BZ reaction is needed 2183013 The time evolution of the BZ pattern found with no ow Fig 3a can be used to approximate a character istic decorrelation time TBZ for pattern formation in the BZ system Figure 3b shows time series of intensities determined at eight different points in Fig 3a The points are chosen such that the oscillations are initially in phase however they are widely separated throughout the system minimum separation of 18 cm or about ten wavelengths of the pattern The large separation assures that there is no diffusive communication between the points during the time scales of the experiment conse a b C FIG 4 Simulated maps of the mixing eld after a 20 b 30 and c 50 oscillation periods u 052 The scaling of the intensity is the same for the three images and has been chosen to highlight the main features The mixing eld for one period of oscillation not shown corresponding to f 0010 Hz is relatively featureless when plotted with the same scaling factors as those shown here 218301 3 VOLUME93NUMBER21 PHYSICAL REVIEW LETTERS week ending 19 NOVEMBER 2004 quently the ensuing calculations do not depend on the separation of the points chosen Since the chosen points do not communicate with each other the phases of the oscillations drift and the eight points become decorre lated with time The time series are summed and the envelope of the combined data is tted to a decaying exponential to determine a decorrelation time TBZ N 100 s more than a factor of 5 longer than the typical BZ oscillation period The patterns shown in Figs 3c7 3e are consistent with an assertion that largescale patterns form if the mixing time Tm is comparable to or smaller than TBZ Insight into the pattern formation process can be found by plotting elds of mixing ef ciencies 25 The trajec tories of triplets of tracers separated initially by Axo 0000 001 cm are simulated numerically from Eqs 1 and 2 on a 600 X 600 grid The mixing ef ciency 5 at time t is de ned by the ratio AxAxo where Ax is the largest separation among the triplet at time t As de ned 5 is related to the nitetime Lyapunov exponent t via t lnf Regions where f is large are those where laments are stretched most in one direction and thinned most in an orthogonal direction consequently these are the regions where advection and diffusion best mix the contents of the uid Fields of the mixing ef ciency are shown in Figs 4a7 4c for two three and ve periods of oscillation re spectively These correspond to the number of periods of oscillation equal to the decorrelation time TBZ N 100 s for blinking frequency f 0020 0030 and 0050 Hz re spectively It can be seen that chemical patterns Figs 3d and 3e form in regions with large mixing ef ciencies dark regions in Figs 4b and 4c although these patterns depend strongly on the mixing time Tm The agreement is quite good even the thin darkened laments connecting the two swirls in the reaction pat terns for the intermediate case Fig 3d are seen in the simulated mixing eld Fig 4b These experiments reinforce the recent theoretical and numerical studies 10 713 that indicate that the tools used to describe chaotic mixing can be applied to analyze and predict patterns in a reacting ow This is evidenced in particular by the good correspondence between the si mulated mixing elds and the experimental reaction advectiondiffusion patterns in the central chaotic region Careful consideration however needs to be made about the time scales both for the mixingiindicated in this work by Tmias well as for the reaction dynamics TBZ Behavior similar to that found in these experiments should be expected for a wide variety of reactionadvec tiondiffusion systems These include but are not limited to phytoplankton blooms in oceanic ows 121326 epidemiological systems plasmas in fusion reactors mi cro uidic processing systems and cellular processes 2183014 This work was supported by the US National Science Foundation Grants No DMR007l77l No DMR 0404961 and No REU0097424 Electronic address tsolomonbucknelledu 1 R Grindrod The Theory and Applications of Reaction Di asion Equations Patterns and Waves Clarendon Press Oxford 1996 2 D BenAvraham and S Havlin Di asion and Reactions in Fractals and Disordered Systems Cambridge University Press Cambridge 2000 AT Winfree Science 175 634 1972 K Showalter J Chem Phys 73 3735 1980 RS Cantrell SIAM Rev 38 256 1996 D Vives J Armero A Marti L RamirezPiscina J Casademunt J M Sancho and F Sagues J Math Chem 23 239 1998 S Theodorakis and E Leontidis Phys Rev E 65 026122 2002 H Aref J Fluid Mech 143 1 1984 TM Ottino The Kinematics of Mixing Stretching Chaos and Transport Cambridge University Press Cambridge 1989 10 Z Neufeld Phys Rev Lett 87 108301 2001 11 Z Neufeld IZ Kiss CS Zhou and J Kurths Phys Rev Lett 91 084101 2003 12 I Scheuring G Karolyi TTA Pentek Toroczkai Freshw Biol 45 123 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77 2003 EEEE 3 EE and Z 2183014