Rotation and alignment of rods in two-dimensional chaotic flow

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1 PHYSICS OF FLUIDS, 4 Rottion nd lignment of rods in two-dimensionl chotic flow Shim Prs, Jeffrey S. Gusto, Monic Kishore, Nichols T. Ouellette, J. P. Gollu, nd Greg A. Voth Deprtment of Physics, Wesleyn University, Middletown, Connecticut 6459, USA Deprtment of Physics, Hverford College, Hverford, Pennsylvni 94, USA Deprtment of Mechnicl Engineering nd Mterils Science, Yle University, New Hven, Connecticut 65, USA Received My ; ccepted 4 Mrch ; pulished online April We study the dynmics of rod shped prticles in two-dimensionl electromgneticlly driven fluid flows. Two seprte types of flows tht exhiit chotic mixing re compred: one with time-periodic flow nd the other with constnt forcing ut nonperiodic flow. Video prticle trcking is used to mke ccurte simultneous mesurements of the motion nd orienttion of rods long with the crrier fluid velocity field. These mesurements llow detiled comprison of the motion nd orienttion of rods with properties of the crrier flow. Mesured rod rottion rtes re in greement with predictions for ellipsoidl prticles sed on the mesured velocity grdients t the center of the rods. There is little dependence on length for the rods we studied up to 5% of the length scle of the forcing. Rods re found to lign wekly with the extensionl direction of the strin-rte tensor. However, the lignment is much stronger with the direction of Lgrngin stretching defined y the eigenvectors of the Cuchy Green deformtion tensor. A simple model of the stretching process predicts the degree of lignment of rods with the stretching direction. Americn Institute of Physics. doi:.6/.5756 I. INTRODUCTION The trnsport of prticulte mteril y fluids is prolem with fr-reching consequences, nd thus long history of study. When the prticles re very smll nd neutrlly uoynt, they tend to ct s Lgrngin trcers nd move with the locl fluid velocity. Prticles with density greter thn or less thn the crrier fluid, however, tend to show different dynmics, such s preferentil concentrtion nd clustering in turulent flow fields. Even prticles tht re neutrlly uoynt cn show dynmics different from the underlying flow when they re lrge compred to the smllest flow scles, since they filter the flow field in complex wys. 4 A lrge numer of ppers on prticle trnsport hve focused on the cse of sphericl prticles. In mny situtions, however, including fier processing in the pper industry 5 nd dynmics of ice in clouds, 6 9 the prticles re not round. The cse of ellipsoidl prticles ws first studied y Jeffery nd Tylor; susequently, Brenner ddressed the cse of generl prticle shpe in series of seminl ppers. 5 Here, we study the motion of rod-like prticles experimentlly in qusi-two-dimensionl flow. We focus on the rottionl dynmics, since we expect tht spheres nd rods will rotte in qulittively different fshions. As long s the Reynolds numer t the prticle scle is smll so tht the locl flow is well pproximted y Stokes flow, spheres will rotte with n ngulr velocity given y hlf the flow vorticity. An nisotropic prticle, however, will lso couple with the strin-rte. In two-dimensionl D Stokes flow with uniform velocity grdients, the rottion rte of n ellipsoid is given y 5, 5 = + + sin u x x u y y cos u y x + u x y, where is the inclintion of the rod with respect to fixed xis, is the spect rtio of the ellipsoid, nd u is the fluid velocity. If the velocity grdient chnges in spce, Eq. is still the first term in series expnsion in higher sptil derivtives of the velocity. The coefficient of the strin-rte portion of the eqution is the eccentricity of the ellipsoid, nd is constrined to lie etween zero for spheres nd one for lines. Even though the right circulr cylinders we study hve shrp corners when compred with idel ellipsoids, we expect ny correction terms to Eq. to e negligile t this order of pproximtion, nd our mesurements confirm this. These insights out how rods couple to strin-rte nd vorticity hve een extended in mny different directions y recent work. Anlytic studies of rod motion in flows with uniform velocity grdients hve explored Jeffery orits nd devitions from them due to wlls nd fluid inerti. 6,7 Szeri et l. 8 developed nlyticl techniques to identify pttern formtion in the orienttion distriution of suspended microstructures in simple flows. In the more complex chotic flows of interest here, the orienttion dynmics ecome nonintegrle nd the velocity grdients often chnge pprecily over the length of prticle. Models of the dynmics of thin rods in turulent flows hve een developed; 5 however, instntneous flow fields re required to determine rod trjectories. In the limit of smll rods with high spect rtio, the rods pproximte mteril lines, nd one cn use theoreticl techniques developed for studying the evolution of mteril 7-66// 4 /4//$., 4- Americn Institute of Physics Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

2 4- Prs et l. Phys. Fluids, 4 R lines in turulence, to study the motion of rods. Numericl simultions provide ccess to the motion of the prticles long with the velocity of the crrier flow nd llow detiled study of the motion of prticles in complex flows; 4 7 however, they must either use drg model for the prticle fluid interctions or fully resolve the prticle oundry lyer. Experimentl studies hve not een le to ccess oth rod motion nd the full fluid velocity field in flows more complex thn uniform velocity grdients. Severl groups hve studied orienttion dynmics in flows with uniform velocity grdients, where the effects of inerti, 8 spect rtio, nd distnce from solid oundries 9, hve een investigted. In more complex flows, the rottionl diffusivity nd orienttion distriution in lortory coordintes, hve een mesured. In our experiments, we hve ccess to high-resolution time-dependent velocity fields, llowing us to chrcterize oth the Lgrngin nd Eulerin flow dynmics. We cn therefore directly compre the orienttion of the rods with properties of the crrier flow. Below, we first study the lignment of rods with the strin-rte tensor s ij =/ u i / x j + u j / x i mesured t the position of the rod, nd susequently consider lignment with the Lgrngin history of the velocity grdients, defined y the Cuchy Green deformtion tensors. To quntify the Cuchy Green deformtion tensors, we use the flow mp x,t, t, which specifies the position t time t + t of fluid element tht ws locted t position x t time t ; see Fig.. The deformtion grdient tensor F ij = i / x j chrcterizes the deformtion of fluid element y the flow mp. Since F ij is not necessrily symmetric, its eigenvlues my not e purely rel. We therefore use the left nd right Cuchy Green deformtion tensors, 4 which re the two possile symmetric inner products of F ij with itself, C L ij = FF T = i j, x k x k C R ij = F T F = k k. x i x j The eigenvlues of C L ij re the sme s the eigenvlues of R, which re rel nd positive. The squre root of the C ij R x r r ( t ) Φ( x r, t, Δt) FIG.. Color online A fluid element t initil position x t time t is mpped to finl position fter time t y the flow. The circulr fluid element is lso deformed y the flow to n ellipse. The eigenvectors of the left ê L nd ê L nd right ê R nd ê R Cuchy Green tensors re shown. L L mximum eigenvlue gives the stretching tht the fluid element hs experienced over the time t. To visulize this process, consider fluid element tht is initilly circulr nd is stretched into n ellipse y the flow, s in Fig.. The stretching is the rtio of the semimjor xis of the ellipse to the rdius of the circle. The sptil distriution of stretching in fluid flows is closely relted to the finite time Lypunov exponents, nd is used to define Lgrngin coherent structures. 5 8 Even though the eigenvlues of the two Cuchy Green tensors re identicl, the eigenvectors re in generl not prllel. As shown in Fig., the eigenvectors of the left Cuchy Green tensor, C L ij, indicte the direction of stretching in coordinte system ligned with the fluid element t time t + t, so tht the eigenvectors give the directions of the principl xes of the ellipse fter stretching. The eigenvectors of the right Cuchy Green tensor, C R ij, on the other hnd, indicte the direction of stretching in coordinte system ligned with the fluid element t time t, so tht mteril lines initilly ligned with the right eigenvectors will end up ligned with one of the principl xes of the ellipse fter stretching. We will denote the eigenvector corresponding to the mximum extensionl eigenvlue of the right Cuchy Green deformtion tensor s ê R, nd the left Cuchy Green tensor s ê L. This method is somewht different from methods used y Szeri et l., nd Wilkinson et l. 7 in their extensive nlysis of the ptterns formed y the orienttion of rods dvected in fluid flows. They compred the orienttion of rods with the direction defined y the eigenvector of the deformtion grdient tensor. As mentioned erlier, the eigenvlues of this tensor my e complex numers in contrst to the eigenvlues of the Cuchy Green deformtion tensor. In simultions of rndom flow, 7 it hs een shown tht rods will e symptoticlly oriented in the direction of the eigenvector corresponding with the lrgest eigenvlue of the deformtion grdient tensor. They lso show tht regions of complex eigenvlues exist t short times, ut dispper t longer integrtion times. We find the method sed on the Cuchy Green deformtion tensors to e more useful for nlysis of the experimentl trjectories tht re limited to reltively short times typiclly few inverse Lypunov exponents, where the deformtion grdient often hs complex eigenvlues. In the reminder of this pper, we present the results of two sets of experiments tht mesured oth the rod motion nd the fluid velocity field tht dvected the rods. We first show tht the rottion rtes of the rods re well descried y Eq.. Next, we consider the lignment of rods with the strin-rte nd the Lgrngin stretching, nd find tht rods lign more strongly with the stretching. Finlly, we develop simple model of the degree of lignment of rods with the stretching experienced y the fluid. II. EXPERIMENTAL METHODS We study the motion of nisotropic prticles right circulr cylinders in chotic qusi-two-dimensionl fluid flows, where time-periodic nd nonperiodic flows re considered. 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3 4- Rottion nd lignment of rods Phys. Fluids, 4 FIG.. Rw imge of mm rods in the periodic flow; rw imge of mm rods long with trcer prticles in the nonperiodic flow. cm cm The flows re produced in shllow electrolytic fluid lyer tht is driven electromgneticlly using Lorentz forcing. 6 A current sinusoidl or constnt flows through the fluid lyer, which intercts with the mgnetic field provided y n rrngement of permnent mgnets locted eneth the plne of the fluid. This results in ody force on the fluid perpendiculr to oth the current nd mgnetic field. The Reynolds numer is defined s Re=UL/, where U is the root men squre fluid velocity, L is the forcing length scle given y the typicl mgnet spcing, nd is the kinemtic viscosity of the fluid. In oth types of flows, Re ws moderte 95 Re 87, ut somewht smller in the periodic cse Re=95. The periodic nd nonperiodic flows were mesured in seprte ut similr pprtuses. 9 In oth cses, we use prticle trcking methods to determine the trjectories nd orienttions of the rods. The time-resolved fluid velocity fields re lso mesured y trcking the motion of smll trcer prticles dvected y the flow. Unlike the time-periodic flow, chrcteriztion of the rod nd fluid motion must e performed simultneously in the nonperiodic cse. This presents some interesting technicl chllenges descried riefly elow nd in detil elsewhere. The electrolyte solutions re chosen to provide electricl conductivity nd to render the rods nd trcer prticles neutrlly uoynt, which ensures tht they coincide in single plne of the fluid. A. Experimentl setup: Periodic flow For the time-periodic flow, the fluid density =. g/cm, % CCl in wter is.6% higher thn the rods =. g/cm, so tht the rods flot t the upper surfce. The rods re mde from fluorescent plstic fier optic cle.5 mm dimeter tht is cut to the desired length.5 mm. The sme mteril is used to mke trcer prticles.5 mm long cylinders, which ensures tht the trcers nd rods flot t the sme height in the fluid. Ultrviolet lmps re used for fluorescence excittion. A rndom rrngement of permnent mgnets with n verge spcing of L=.9 cm is locted eneth the shllow fluid lyer.7 mm deep, cm wide. A sinusoidl electric current, with frequency. Hz period T= s, trvels horizontlly through the fluid, which leds to time-periodic chotic flow Re=95, U=.9 cm/s. B. Imge nlysis: Periodic flow We imge 6 cm 8 4 pixels re in the center of the test section to void edge effects. Figure shows typicl rw imge of mm rods tken t frme rte of 4 Hz. Since the flow is time-periodic, the fluid velocity cn e mesured seprtely without the rods present. For mesurement of the fluid velocity field, the flow is seeded with trcer prticles to n verge concentrtion of prticles per imge, nd their motion is trcked over 5 periods resulting in out 7 trcer prticles per phse. The center of ech trcer prticle is mesured with n uncertinty less thn m.5 pixel. The prticle velocities re mesured y fitting polynomil to the prticle trjectories. The fluid velocity field is then extrcted from ll the trcer velocities occurring t the sme phse y interpoltion onto squre grid of. cm spcing. In seprte experiments, the rod motion is mesured with significntly lower prticle concentrtion 4 depending on rod length to void prticle-prticle interctions, nd repeted 7 9 periods for ech rod length. To determine the center nd orienttion of ech rod, we find ll right pixels corresponding to single rod. The rod position is determined using the intensity-weighted center-of-mss of the pixels. A Hough trnsform gives first guess for the orienttion of the rod. Finlly, we use nonliner fitting lgorithm to optimize the orienttion mesurements y minimizing the difference etween n idel model rod imge nd the rw imge. Using this method, the orienttion of rod is found to within.7 rd ccurcy. C. Experimentl setup: Nonperiodic flow For chotic nonperiodic flows, it is necessry to mesure the fluid motion simultneously with the rod dynmics. To chieve this, trcer prticles re seeded long with the rods t the interfce of density strtified fluid ilyer. The lower lyer is dense electrolytic solution 7% KCl in wter, =. g/cm, the upper fluid lyer is deionized wter, nd the trcers nd rods hve n intermedite density p =.5 g/cm. Surfce tension interctions etween the prticles re eliminted, since the upper nd lower fluids re miscile. 4 This llows for much higher trcer seeding s compred to prticle seeding t free surfce. The trcer prticles 8 m dimeter re significntly smller thn the Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

4 4-4 Prs et l. Phys. Fluids, 4 4 c Rottion rte, Rottion rte, 5 5 d 5 5 e f 4 4 Rottion rte, FIG.. Color online Mesured solid line nd predicted + rottion rte vs time for different rod lengths, d.5 mm,, e 5 mm, c, f mm in periodic flow top row nd nonperiodic flow ottom row. rods.59 mm dimeter,.5. mm long to chieve sufficient resolution of the fluid velocity field in the vicinity of rod. The fluid lyers re ech.5 mm deep with n re cm, nd constnt current is mintined through the lyer to drive the flow. The mgnets re rrnged in squre lttice with lternting poles nd spcing L =.54 cm. When driven t sufficiently high current, the resulting flow is sptiotemporlly chotic with Reynolds numer in the rnge 9 Re 87. D. Imge nlysis: Nonperiodic flow To void edge effects, we imge n cm 6 6 pixels region of interest in the center of the flow cell t frme rte of Hz. The trcer prticles re imged using fluorescence, nd scttered light is used to imge the much lrger rods. Ech imge contins 5 trcers to dequtely resolve the flow field, while the rods re much fewer in numer per imge to void prticleprticle interctions. Figure shows rw imge of mm rods long with the trcer prticles, where rods re esily distinguished from trcers y size. The center nd orienttion of ech rod re identified y n intensity-weighted moment yielding the centroid nd principl xes of the rod. Trcer prticle centers re detected with supixel ccurcy m y Gussin fitting to their diffrction limited imges. Both sets of prticles re trcked using predictive lgorithm, 4 where instntneous prticle velocities nd ngulr velocities re mesured y polynomil fitting to the trjectories. The trcer prticle velocities re ilinerly interpolted to further resolve the fluid velocity in the vicinity of ech rod see Ref. for dditionl detils. This method provides the distinct dvntge of direct mesurement of the fluid velocity field simultneously with the nisotropic prticle dynmics, which is essentil for nonperiodic flows. III. RESULTS A. Rottion rte Eqution shows tht the rottion rte of rod in two dimensions cn e estimted from the crrier fluid velocity grdient t the center of the rod. Figure shows the mesured rottion rte of severl typicl rods of different lengths. Also shown is the predicted rottion rte from Eq. using experimentlly mesured velocity grdients t the position of the rod. The top row dt c re selected from the periodic flow nd the ottom row d f re selected from the nonperiodic flow experiment. Rod length increses from left to right. We mesure the rottion rte of the rods from polynomil fits to the experimentlly mesured orienttions. As shown in Fig., the predicted rottion rte is close to the experimentlly mesured rottion rte for ll rod lengths we hve studied. This my e surprising since our rods hve lengths up to 5% of the length scle of the forcing nd prticle Reynolds numers up to 74 sed on the rod length nd the rms fluid velocity. Eqution gives good predictions despite the fct tht the prticles do not rigorously stisfy the conditions for which it ws derived. The significntly lrger noise in the dt for.5 mm rod in the nonperiodic flow Fig. d is mostly generted y inccurcy in determining the exct orienttion of short rods. Also, difficulties Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

5 4-5 Rottion nd lignment of rods Phys. Fluids, Angle(rd) FIG. 4. Color online Proility density function PDF of the differences etween mesured nd predicted rottion rtes for different rod lengths in oth sets of experiments: Periodic flow, rod lengths re:.5 mm, 5 mm, mm, mm; nonperiodic flow, rod lengths re:.5 mm, 5 mm, + mm. FIG. 5. Color online Proility density function PDF of the ngles etween rod orienttion nd extensionl direction of strin-rte. In oth experiments, the PDF shows wek lignment of rods with the strin-rte. Periodic flow, rod lengths re:.5 mm, 5 mm, mm, mm; nonperiodic, rod lengths re:.5 mm, 5 mm, + mm. in mesuring the velocity grdients in rel time contriute to inccurcies in the predicted rottion rte. The proility distriution of the devition etween the predicted rottion rte nd the experimentlly mesured rottion rte is shown in Fig. 4. For the periodic flow in Fig. 4, the devitions of the prediction from mesurement re out % of the root men squre rottion rte, nd re mostly independent of rod length. The mjor contriution to these devitions is inccurcy in determining the exct velocity grdient of the flow t the center of the rod. The proility distriution in Fig. 4 shows tht for mm rods in the nonperiodic flow, the prediction is much closer to the mesurement thn it is for shorter rods. The smller devition for the long rods is the result of the smller uncertinty in determining the orienttion of the longer rods. B. Alignment of rods with strin-rte The orienttion distriution of rods cn e considered reltive to different directions defined y the flow. First, we will consider lignment with the locl strin-rte. Figure 5 shows the proility distriution of ngles etween the orienttion of rods nd the extensionl direction of the strinrte clculted t the center of the rods. This distriution shows tht the rods tend to lign with the extensionl direction of the strin-rte, lthough the lignment is firly wek. For the rods studied, the lignment with the strin-rte does not show significnt dependence on rod length. C. Alignment of rods with stretching Rods re wekly ligned with the strin-rte in these flows, ut it is relly the history of the velocity grdients long the trjectory of rod tht is responsile for its orienttion distriution. As discussed in the introduction, this Lgrngin history of the velocity grdients cn e quntified using the Cuchy Green deformtion tensors. In Fig. 6, we show snpshots of the pst stretching field, defined y the eigenvlue of the Cuchy Green deformtion tensor t ech sptil point. In oth the periodic nd nonperiodic flows, the stretching field hs mny shrp mxim tht re orgnized into lines. Superimposed on the stretching fields in Fig. 6, we show imges of mm rods tken t the sme time. Animtions of rod motion in time-dependent stretching fields re ville in the enhnced online version of Fig. 6. The rods re preferentilly ligned with the stretching lines, which indictes the importnt role of stretching in orienting rods. As in Ref. 6, we clculte stretching y integrting trjectories of virtul prticles in mesured velocity fields. The grdients in the definition of the Cuchy Green tensors Eq. re evluted using finite differences of prticle trjectories tht re initilly very close to ech other. A rescling method is used to keep the prticles close to ech other even s they experience exponentil stretching. In order to quntify the effect of stretching on the orienttion of rods, we mesure the distriution of ngles etween the orienttion of ech rod nd the direction of pst stretch- Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

6 4-6 Prs et l. Phys. Fluids, FIG. 6. Color online Stretching fields with mm rods superimposed in drk color. Periodic flow with t=t= s. Nonperiodic flow with t=l/u=4 s. enhnced online URL: URL: ing, ê L, t the center of the rod. Figure 7 shows tht this distriution hs lrge proility round zero, indicting tht the rods preferentilly lign with the pst stretching direction. The lignment is significntly stronger thn the lignment with the strin-rte direction in Fig. 5. Surprisingly, the lignment is nerly independent of rod length even though the longest rods mm re 5% of the mgnet spcing in the periodic flow experiment nd 4% of the mgnet spcing in the nonperiodic flow experiment. Even for these reltively lrge rods, there is no mesurle effect either from the rods verging over the sptilly vrying velocity field or from the rottionl inerti of the rods. The lignment in the periodic flow is stronger thn in the nonperiodic flow. This my e prtly result of periodicity, ut it is lso ffected y the lrger stretching in the periodic flow for the integrtion times chosen. Figure 8 shows the proility distriution of lignment of rods with the direction of pst stretching for different integrtion times. The proility of lignment of rods with the direction of stretching increses with incresing integrtion time. This increse in lignment seems nturl, s the it of fluid tht is ccompnying the rod fter longer integrtion hs experienced more stretching. However, t some point the mximum proility sturtes so tht further increses in the integrtion time do not led to dditionl lignment. This sturtion my e sign of limittions on the ccurcy of the experimentl mesurements of the stretching direction of the fluid t the center of the rod. See ppendix We hve lso compred rod orienttion in our flows to the direction defined y the eigenvectors of the deformtion grdient tensor s used in previous studies.,7 Even fter two periods s of the time-periodic flow, there re mny Angle(rd) FIG. 7. Color online PDF of the lignment of rods with the direction of stretching defined y ê L. In oth sets of experiments, ll rod lengths lign with the stretching direction. Periodic flow, t=t, rod lengths re:.5 mm, 5 mm, mm, mm; nonperiodic flow, t=l/u, rod lengths re:.5 mm, 5 mm, + mm Angle(rd) FIG. 8. Color online Dependence of stretching lignment on integrtion time. The PDF of the lignment of rods with the direction of stretching is shown for mm rods with different stretching integrtion times. Alignment increses for longer integrtion times. Periodic flow, T= s nd Lypunov exponent.97t : t= T/6, + T/8, T/4, T/; nonperiodic flow, L/U=4 s nd Lypunov exponent. U/L : t = L/U, + L/U, L/U. Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

7 4-7 Rottion nd lignment of rods Phys. Fluids, 4 R R R θ L x, y R L In Fig. 9, we show the initil circle in the coordinte system ligned with the principl xes of the right Cuchy Green tensor. Here, point on the circle of rdius r is given y the simple prmetric equtions, x =r cos nd y =r sin. After deformtion y the flow over some time intervl, the circle ecomes n ellipse with the sme re. In generl, the flow will hve reoriented the ellipse, so in Fig. 9 c we show the ellipse in the coordinte system defined y the principl xes of the left Cuchy Green tensor. By choosing different coordinte systems in Figs. 9 nd 9 c, we hve used the Cuchy Green tensors to ccount for rottion, leving only the effect of stretching. We cn mp ny point x,y on the circle to the corresponding point x,y on the ellipse y x = sx, c L y = y s, FIG. 9. Color online Deformtion of circulr fluid element into n ellipse y the flow. Initil circle in the coordinte system ligned with the principl xes of the right Cuchy Green deformtion tensor. c Finl ellipse in the coordinte system ligned with the principl xes of the left Cuchy Green deformtion tensor. regions with complex eigenvlues. In these regions, the direction of the eigenvector of the deformtion grdient is undefined, so the method sed on the Cuchy Green deformtion tensors is more useful for compring the lignment of rods with the deformtion mesured in our flows. It would e interesting for future work to mke more detiled comprison of these two methods. D. Theoreticl prediction of lignment of mteril lines with stretching direction Here we use simple model of deformtion to predict the lignment of mteril lines due to stretching in our system. Some theoreticl tools for solving the Fokker Plnck eqution for the orienttion distriution of the microstructure in fluid flows hve een developed, 4 ut we choose to solve simple model tht clerly revels the connection of stretching to the orienttion distriution. The effect of flow is to deform n infinitesiml circle of fluid into n ellipse. The rtio of the semimjor xis of the ellipse to the rdius of the circle is equl to the stretching tht the circle hs experienced, which cn e mesured using the squre root of the mximum eigenvlue of the Cuchy Green tensor s descried in Sec. III C. If we consider stright mteril line segments through the center of the circle with known initil distriution of ngles, we cn clculte the proility distriution of their ngles fter eing stretched y the flow. Figure 9 shows n initilly circulr fluid element tht is then trnsported nd stretched y the flow into n ellipse. θ x, y L where s is the stretching. The ngle etween the point x,y nd the semimjor xis of the ellipse is = rctn y x = rctn tn s so the finl nd initil ngles of ny mteril line through the center of the mteril element re relted y tn = s tn. The numer of lines in rnge d is given y the numer of lines tht re mpped to this rnge from the initil distriution, so the proility distriution of ngles, P, is relted to the initil distriution, P,y P d = P d. From Eq., the differentils re relted y d d = d d rctn s tn = s sin + cos s so the finl distriution of ngles in rnge d for given vlue of stretching, s, is P d = P rctn s tn d. 4 s sin + cos s Eqution 4 implies tht the finl distriution of ngles depends on the initil distriution P nd the mount of stretching, s, tht the mteril lines hve experienced. If rods rotte s mteril lines, we cn use this theory to predict the finl distriution of orienttions of rods. Rods in different regions of the flow experience different vlues of stretching. The proility distriution of orienttions of rods is the sum over ll stretching vlues weighted y the proility density of ny prticulr vlue of stretching P s,, Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

8 4-8 Prs et l. Phys. Fluids, 4.4. ) ( P θ Angle(rd).5.5 FIG.. Color online Experimentl PDF of initil orienttions of rods with respect to the extensionl eigenvector of the right Cuchy Green deformtion tensor. Results re shown for mm rods in the periodic flow for four different integrtion times, t= T/6, + T/8, T/4, T/. Becuse the right Cuchy Green deformtion tensor chnges with integrtion time, this distriution hs wek integrtion time dependence even though the rods lwys hve the sme initil orienttion. P d = P s ds P rctn s tn d. 5 s sin + cos s The stretching distriution P s is mesured from ll rod positions to ccount for their smpling of the flow, nd we further condition P on vlues of stretching s. To compre our mesured orienttion distriutions with the prediction in Eq. 5, we need to know the PDF of initil orienttions of rods, P. Figure shows this initil proility distriution from our experimentl mesurements, where is the ngle etween rod orienttion nd the extensionl eigenvector of the right Cuchy Green deformtion tensor, ê R. Rods show wek lignment with the direction of future stretching, ê R. One might expect tht there would e no lignment with the stretching tht the rod will experience in the future, ut we find wek lignment, which cn e understood s result of the time correltion of the velocity grdients of the flow. From the mesured initil distriution of rod orienttion in Fig., we cn clculte the finl proility distriution of orienttion of rods using Eq. 5. Figure compres this theoreticl prediction of the finl distriution of rod orienttion with our mesurements from the periodic flow Fig. 8 for four different integrtion times. Both the predicted nd mesured distriutions give ngles mesured from the extensionl eigenvector of the left Cuchy Green tensor, ê L. The predicted distriution shows lignment with the direction of stretching in firly good greement with our mesurements; however, the theory predicts somewht stronger lignment thn is oserved. The devition is lrgest for long integrtion times where the theory predicts tht mteril lines re strongly ligned y the stretching to produce shrp pek ner zero in Fig. d. Inccurcies in our mesured velocity fields my led to slightly inccurte mesurements of the stretching direction. These inccurcies would hve the lrgest effect in regions with nerly perfect lignment leding to c d Angle(rd) FIG.. Color online Comprison etween the predicted solid line nd mesured distriution of rod lignment with the stretching direction. Results re shown for mm rods in the periodic flow for four different integrtion times, t= T/6, T/8, c T/4, d T/. smller proility in the experimentl distriution ner =. Another fctor could e tht rods re not mteril lines. Either their length or spect rtio could cuse the mesured lignment to differ from the prediction for mteril lines. However, the lck of rod length dependence in the lignment distriution see Fig. 7 suggests tht this is not lrge effect. Downloded 5 Jun to Redistriution suject to AIP license or copyright; see

9 4-9 Rottion nd lignment of rods Phys. Fluids, 4 IV. CONCLUSIONS Simultneous mesurements of rod motion nd the velocity field dvecting the rods provide powerful new tool for understnding prticles in complex flows. We hve developed methods for mking these mesurements in two types of flows: periodic nd nonperiodic. Prticulrly importnt is the development of techniques for simultneously extrcting the rod nd fluid motion in the nonperiodic flow. The mesured rottion rtes of rods show good greement with the rottion rte predicted for ellipsoidl prticles without inerti in flow with uniform velocity grdient, even for the longest rods studied. We find tht rods lign wekly with the extensionl direction of strin-rte; however, the lignment with the eigenvectors of the Cuchy Green deformtion tensor is much stronger. In oth periodic nd nonperiodic flows, the lignment of rods with the direction of the Cuchy Green deformtion tensor is lmost independent of rod length, even though the rods extend to 5% of periodic flow length scle nd 4% of nonperiodic flow length scle. We developed simple model to predict the lignment of rods with the direction of stretching sed on the ssumption tht rods rotte s mteril lines. The model cptures the min fetures of the lignment distriutions, ut predicts slightly stronger lignment thn the experimentl mesurements show. ACKNOWLEDGMENTS We thnk Bruce Boyes for technicl ssistnce. This work ws supported y NSF Grnt Nos. DMR-85 to Hverford College nd DMR-5477 to Wesleyn University. APPENDIX: EFFECTS OF DEVIATIONS BETWEEN INTEGRATED TRAJECTORIES AND FLUID TRAJECTORIES At longer integrtion times in these chotic flows, smll inccurcies in the velocity field cn led to lrge devitions etween virtul prticle trjectories nd the trjectories of the rods. We cn correct for the devitions of virtul prticle trjectories from rods y forcing the center of group of virtul prticles to follow the center of rod. This wy, we re smpling the sme stretching tht the rod hs experienced. Figure shows the effect of forcing the virtul prticles to follow the rods on the proility distriution of lignment of rods with the direction of pst stretching. The distriution in Fig. shows tht the enefit of forcing the virtul prticles to follow the rods is quite smll. However, if we only look t rods tht experience lrge stretching Fig., the effect of forcing the virtul prticles to follow the rods is to crete significntly stronger lignment. We conclude tht integrtion errors cn hve some effect in regions of lrge stretching, ut the overll effect on the orienttion distriutions we mesure is not significnt. We lso performed this nlysis on the dt from the nonperiodic experiment. Here, forcing the prticles to follow the rods led to somewht worse lignment in some cses. We interpret this s result of the fct tht in the nonperiodic experiment, there cn never e velocity trcer prticles t the position of Angle(rd) FIG.. Color online PDF of lignment of rods with the direction of stretching, showing the effect of forcing virtul prticles to follow the trjectories of rods. Dt re for the periodic flow with mm rod length nd integrtion time t=t. Proility distriution clculted for ll stretching vlues; proility distriution conditioned for stretching vlues lrger thn one rms. virtul prticles follow the rod trjectory, + virtul prticle trjectory my devite from rod trjectory. the rods. Forcing the virtul prticles to follow the rods in this cse cn force them into regions where the velocity field is not resolved s ccurtely. F. Toschi nd E. Bodenschtz, Lgrngin properties of prticles in turulence, Annu. Rev. Fluid Mech. 4, N. M. Qureshi, M. Bourgoin, C. Budet, A. Crtellier, nd Y. Ggne, Turulent trnsport of mteril prticles: An experimentl study of finite size effects, Phys. 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