Eddy Formation and Turbulence in Flowing Liquidscircres.ahajournals.org/content/circresaha/12/5/455.full.pdfEddy Formation and Turbulence in Flowing Liquids By J. E. Meisner, M.D.,

Eddy Formation and Turbulencein Flowing Liquids

By J. E. Meisner, M.D., and Robert F. Rushmer, M.D.

• The steady flow of liquids through straightrigid tubes has been extensively analyzed, butthese data cannot be applied directly to thepulsatile flow of blood through distensible,tapered and branching vascular channels.Since it is very difficult to follow flow patternsin the living animal, certain types of studiesmust be performed in models. Laminar flow,eddy formation and turbulence can be iden-tified with streamers of dye.1-2 This methodis widely used; however, it reveals the flowpattern only where the concentration gradientsof the dye are steep. Furthermore, observa-tion can continue only until the dye is mixedand the flow pattern is obscured. The com-plex hydrodynamics of the cardiovascularsystem,5 however, make more complete inves-tigations necessary.

Detailed analysis of the formation of eddiesand turbulence requires clear detection ofchanges in flow pattern at any time and atany place in the flow channel and continuousobservation of the flow field over a consider-able period of time. These conditions are ful-filled by several colloidal solutions such asvanadium pentoxide, red dye benzopurpurin4B4 and certain clay suspensions. Most ofthese substances have been used for someyears in engineering and chemistry as meansof visualizing flow. The particles in thesefluids obey the principles of streaming bire-fringence, i.e., double refraction in connectionwith shear forces.

A colloidal solution of white Hector ben-

From the Department of Physiology and Bio-physics, University of Washington School of Medicine,Seattle, Washington.

Supported by Training Grant HTS-5147 from theNational Institutes of Health, TJ. S. Public HealthService, and by grants from the American HeartAssociation, Idaho Heart Association and WashingtonState Heart Association.

Received for publication November 19, 1962.

Circulation Research, Volume XII, May 196S

tonite has been used successfully by Hauserand Dewey,5 Wayland,6- 7 and Lindgren8- 9 inflow observations. The technique has beenadapted for this study of changes in flow pat-terns in models designed to simulate flow con-ditions in cardiovascular channels.

MethodsWhite Hector bentonite (National Lead Com-

pany, Los Angeles, California) was prepared foraqueous suspension by fragmenting the clay intotiny particles of less than 0.5 p. equivalent spheri-cal diameter by means of mesh screens (no. 200)and high-speed agitators. A solution containingabout 1% solids was satisfactory for model chan-nels up to 15 mm thick. To stabilize the suspension,0.01 % of tetra sodium pyrophosphate was added.The density of this solution was 1.0847 g/eeat room temperature. Glycerin was added toalter the viscosity at room temperature to 0.036poise.10 No precipitation occurred when the solu-tion was left undisturbed for weeks; such lack ofgravitational effects suggests a colloidal solutionrather than suspension. Thus, the bentonite solu-tion acts as a single unit and the particles faith-fully represent the motion of the fluid flowingalong flat flow channels.

I t is usually assumed for ease of explanationthat crystals of bentonite are elongated slenderparticles of colloidal size, They are probably ellip-tical and have a variable length-to-diameter ratio.At zero flow the particles are randomly oriented,owing to thermal agitation (fig. 1) . With the on-set of viscous shear flow, different velocity gra-dients impinge upon the sides of these elongatedparticles. A particle crosswise to the flow is sub-jected to different velocities at either end and ro-tates until its longitudinal axis is aligned parallelto the shear forces, or perpendicular to the veloc-ity gradient. Viscous fluids may induce relaxationeffects and impede rotation.11 Nevertheless, a verysmall velocity gradient acts on a particle whichis oriented parallel to the laminae of flow. Thus,in steady flow, a particle rotates along an axis per-pendicular to the direction of streaming, but itdoes so with an angular velocity which is minimumwhen the major axis of the particle lies along thestream. At any instant more particles will be inthis position than in any other. Also, there will

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456 MEISNER, RUSHMER

0U4RTEBWAVE PLATE

LAMINAR FLOW

LIGHT SOURCE

TURBULENT FLOW

FIGURE 1

The left side shoivs on sketch of the experimental apparatus. The right side is a schematicrepresentation of the orientation of the bentonite crystals as though the observer movedivith the flow. When the liquid is stationary the particles are oriented, at random due tothermal forces. During laminar floiv the particles tend to be oriented along a certainmean angle to the streamlines. In turbulent flow the particles outline small eddies, whichagain contain tinier eddies.

be no lateral motion across the streamlines. Dur-ing the formation of eddies and turbulence par-ticles move laterally in the fluid and undergomixing across the flow channel.

In addition to these hydrodynamic forces, bom-bardment by the molecules of the surroundingmedium has a rotatory effect on the crystals, alter-ing their orientation.12 Angular displacement ofthe particle by Brownian movement probably hasa mean value which is minimal compared with theeffect of hydrodynamic forces.

When exposed to shear forces, a solution ofbentonite behaves as an anisotropic medium, i.e.,its optical properties are different in different di-rections. Under these conditions the fluid exhibitsdouble refractive properties if viewed by meansof polarized light directed at right angles to thedirection of flow. Briefly, the principle of doublerefraction, or birefringence, is that birefringentmaterial has different refraction indices in differ-ent planes. Normal light falling on such a crystalis broken into two rays, which are polarized intwo planes at right angles to each other. Sincethe indices of refraction differ, the rays travel atdifferent rates. These waves, therefore, differ inphase when they emerge from the birefringent ma-terial. Combination of the two vibrations in somearbitrary phase relation generally yields clliptical-ly polarized light. The elliptiealry polarized light

was converted into plane polarized light by thequarter wave plate, placed between the flow sec-tion and the analyzer (fig. 1). Double refractiondoes not take place when the incident polarizedlight vibrates in a direction parallel to the opticaxes of the material.13 (See Jerrard14 for a de-tailed summary of these mechanisms.)

The optical system used to monitor changes inth« orientation of the bentonite crystals consistedof a white light source, two polarizing filters (com-mercial Polaroid) and a quarter wave plate (fig.1). Light from the source passing through thefirst polarizing filter (the polarizer) emerged vi-brating in one plane. When the second filter (theanalyzer) was placed parallel to the first andturned 90°, the light was maximally attenuated.When a birefringent crystal (e.g., bentonite) wasplaced between such crossed polarizing filters,light emerged from the analyzer. Rotation of thebirefringent substance through 90° resulted inalternate appearance and disappearance of thislight. Thus the observer saw changes in light in-tensity on a dark background as the crystals ro-tated according to the acting shear forces. Al-though a light field would have transmitted morelight, brilliance of color would have been lost.

For visual observation, the dark field and a 100watt light source were satisfactory. For motionpicture photography, four no. 2 photoflood lamps

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TURByLEKTQE IN FLOWING LIQUIDS 457

FIGURE 2

Flow patterns in a flat flow channel downstream from a constriction at different ratesof steady flow. The flow increased from numbers 1 to 6 from 3 cc/sec to 36 cc/sec.

were used, and the display on the dark field wasrecorded on highspeed color film. Monochromaticlight was not necessary.

Flat flow sections, designed to permit essentiallytwo-dimensional flow, were huilt of transparentacrylic plastic 1 mm thick. The flow channel insuch a model was no more than 3.5 mm deep. Itswidth was between 25 and 50 mm and its lengthvaried according1 to the tube inlet conditions. Somemodels were entirely rigid. In others, the sidewalls were made distensible by gluing to the sidesthin plastic sheets under slight tension. Specialconnectors of plastic were made to join the rec-tangular inlet and outlet to tubing. Caxe was takento taper and smooth the inside of these connectorsin order to prevent unnecessary disturbances ofthe flow pattern at these points. Since, understeady flow conditions, the laminar flow pattern inuniform channels of the over-all size found in thecardiovascular system breaks down only at highflow rates, constrictions were introduced into themodels to cause a breakdown of laminar flow atlower rates. In the two-dimensional model theconstriction consisted of a slit which extended tothe full depth of the flow section. Plat models ofthe outflow tract of the left ventricle were con-structed on the same principles.

Round models permitting three-dimensional flow-ere constructed of transparent Tygon tubing withan inside diameter of 22 mm and a wall thicknessof 1.5 nun. The constriction was provided by in-serting a round piece of plastic into the tubing.

Circulation Research, Volume XII, May 1963

This "plastic stenosis" had a tapered smooth inletand a circular hole, with a radius of 0.9 mm inthe center.

All flow sections wci-e mounted horizontallythat gravitational effects were minimized. Thereservoir of the bentonite solution was elevated;the solution ran from it to the flow channel throughrigid pipes. The rate of flow was controlled bymeans of a screw clamp placed about 100 diametersabove the channel inlet.

ResultsAt very low velocities the over-all flow nem

from a constriction in a two-dimensional flowsection remained laminar (fig. 2, strip 1).Shear forces acted at the boundary where theenteringjet made contact with the stable fluid.Changes in light intensity at this boundaryappeared as a bright area. As the rate of flowwas increased to 8 cc/see (fig. 2, strip 2), thecentral core became disturbed and then smalllocalized eddies appeared (strip 3). The fullydeveloped jet pattern is illustrated in strips4 and 5. Eddies formed simultaneously alongthe margins of the issuing jet. The area oflocalized turbulence consisted of eddies of eversmaller size. With further increase in velocity(fig. 2, strip 6), the size of the flow channellimited further development of the jet pat-



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4 5 8 MEISNER, RUSHMEE

FIGURE 3

Flow patterns in a round transparent Tygon tubing doionstream of an orifice at flowrates of 8 ec/sec (3a) and 36 cc/sec (3b).

tern. Vortices appeared near the orifice andthe large vortex produced gave rise to smallervortices swirling in opposite directions closeto the orifice. A numerical analysis of thismodel would be expressed in terras of headloss, which is determined chiefly by differencesbetween the velocities of the entering jet andthe fluid already present in the flow section.Two diameters downstream from this area thedisturbances were damped and the flow wasas undisturbed as it was upstream from theconstriction. This clear display of vorticeswas possible only in the two-dimensional sys-tem; use of a slot rather than a hole as aconstriction practically excluded flow in thethird dimension, i.e., in the direction parallelto the line of sight.

In the three-dimensional system these vor-tices were displayed simultaneously in allradial planes of the tube, so that completeturbulence occupied the area beyond the con-striction. The jet formed downstream fromthe circular constriction is shown in figure 3.At low flow rates changes in light intensityappeared as dark areas some distance down-

stream from the constriction (fig. 3a). Indi-vidual eddies could not be distinguished, al-though the presence of non-laminar flow waseasily detected. Immediately behind the ori-fice the flow was laminar and no changes inlight intensity were present. As the flow ratewas increased, the eddies formed closer to theorifice and the turbulent area became morepronounced (fig. 3b). Turbulence occurredover a wide region starting at the orifice;due to viscous damping this turbulent motiondisappeared completely just a few diametersdownstream from the turbulent area.

Corresponding phenomena were seen in aflat plastic model of the outflow tract of theleft heart with the simulated aortic valvesfixed in an open position. At low rates (60ee/sec) of pulsatile flow, a trail formed, start-ing from the edge of the valve at either sideof the orifice and becoming turbulent fartherdownstream (fig. 4a) ; only the trail originat-ing on the left is clearly visible in figure 4bbecause the inner curve of the aorta obscuresthat on the right. When the flow was steadybut doubled to 120 cc/sec the over-all pattern




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TOBSBUSIBNGE I-HKFLOWINS LIQUIDS 459

FIGURE 4

Flow pattern observed in a flat model of the outflow tract of the left ventricle with rigid"open valves." The floio rate in 4a is pulsatile and 60 ce/$cc; til 4b, 120 co/sec.

was much more disturbed (fig. 4b). The cen-ter stream through the open valve was turbu-lent, as was the flow through the aortic archand the descending aorta. The disturbancesbegan in the ventricle and were induced byboth the high flow velocities and the changesin the caliber of the inlet to the ventricle. Thetrails seen forming from the edges of the valvein figure 4a developed into a train of eddiesat the higher flow rate. The eddies from theright valve edge were visible behind the rightaortic valve. The long eddying trail from theleft valve extended to the crest of the aorticarch; additional eddies in this trail developedat the origin of the branches.

In another model the point of outflow intothe aorta was narrowed to simulate aortic ste-nosis. A jet formed at the constriction andturbulence developed downstream. At a low


flow rate, 20 cc/sec, the pattern in the ven-tricle was only slightly disturbed (fig. 5a).Downstream from the stenosis a clear jet flowpattern developed with eddies aloJig its mar-gins. A big stable vortex originated where thejet first made contact with the outside walls;this vortex swirled clockwise. This area ofturbulence was carried on through the aorticarch and was damped out in the descendingaorta after a few diameters. At the origin, ofthe branches the edd}*- formation was veryclear. Figure 5b shows the patterns at a flowrate of 50 ec/sec. The pattern in. the ventriclebroke down, partly because of this high flowrate and partly because of the changes in thecaliber at the inlet. The jet pattern down-stream from the constriction was greatly ex-tended. The great vortex in the ascendingaorta enlarged and swirled counterclockwise;



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460 MEISNER, BTJSHMER,

FIGURE 5

Development of turbulence downstream of constricted "aortic valves" in a flat channel.The flow wan pulsatile and the picture taken during maximal velocity during the ejectionperiod. The rate of floiv at Sa wus 20 cc/sec; at 5b, 50 cc/sec.

behind the right valve a clear rotating vortexdeveloped. The center core of the issuing jetseemed to hit the opening of the first branch(innominate artery) and split into twostreams: one contributed to the swirling vor-tex and the other continued downstream alongthe vessel wall. The eddying motion at theentrance of the branches was obvious. Anotherstream of vortices seemed to form at the in-side of the aortic arch and was carried down-stream through the descending aorta. No vis-cous damping occurred in the descending aortabecause the flow velocity there was still high.

DiscussionAs this method of streaming birefringence

is based upon the physical characteristics ofa single solution, it precludes some technicaldifficulties which arise when two solutions aremanipulated simultaneously. In the indicator

methods, for instance, it is difficult to intro-duce the dye in such a way that it outlines thesmall areas along the side of the tube wheredisturbed flow appears first. Also, there areno problems of matching velocities and densi-ties, and the effects of varying flow are easilyobserved. Furthermore, changes at all regionsalong the flow section can be observed simul-taneously rather than successively, and theduration of an experiment is not limited bymixing of indicator and substrate. Lindgren9

has mentioned possible distortions of flowpatterns, even with a solution of less thanthe 1% bentonite used in this study ; however,these distortions are more serious in quanti-tative than in qualitative observations.

The major disadvantage of this method inbiological research is the lack of means formaking comparable observations in the living

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TTTBBTJLENCE IN PLOWING LIQUIDS 461

animal. Consequently, extrapolation is neces-sary, but must always be undertaken cautious-ly. The desirability of flat flow sections mayalso be considered a disadvantage since ana-tomical structures usually have other configu-rations. Observation of laminar and turbulentflow in a cylindrical tube is possible, althoughthe flow pattern is not clear and single eddiescannot be distinguished (fig. 3). This problemundoubtedly arises because eddies are simul-taneously shed in all directions along the axisof the space. What appears on the analyzer isan over-all picture of the integral motion ofparticles in space. The flat flow section doesprovide the possibility of observing two-di-mensional flow. Actually, the flow in suchmodels is not ideally two-dimensional becausethe bottom and top of the section establishboundary layers that induce shear forces ina third dimension.

The criteria of the change from laminar toturbulent flow still constitute a moot pointin hydraulics.15'10 In most physiologicalstudies, the distinction between laminar andturbulent flow has been established by thefamiliar curves of sudden change in resistancewhen driving pressure is plotted against rateof flow. This curve is combined with the ratioof inertial to viscous forces and the result isexpressed as Reynolds number. For water inuniform pipes the critical value for transitionfrom steady laminar to turbulent flow is givenas 2000. The critieal Reynolds number forcirculating blood is variously given. Indeed,the validity of any Reynolds number for thecardiovascular system is very doubtful. Theeffect on the flow profile of sudden accelera-tion and deceleration is not yet clearly under-stood, but the results of calculating a Reynoldsnumber for pulsatile flow in channels of non-uniform caliber like the blood vessels areundoubtedly distorted. Figure 3 shows turbu-lence occurring downstream from a constric-tion at very low flow rates. The velocity ofthe fluid through the constriction applied toReynolds formula yielded a value of 800,which would indicate that the jet in the con-striction was not turbulent. It can be seen,

Circulation Research, Volume X/l, May 1SBS

however, that the interaction between the' en-tering jet of relatively high velocity and thealmost stationary fluid in the channel pro-duced instability in the core of the jet andcreated localized disturbances downstreamfrom the constriction. At a slightly highervelocity these areas of eddy formation ex-panded across the whole flow channel, pro-ducing a condition fulfilling the definition ofturbulence : " . . . an irregular condition of flowin which the various quantities show a randomvariation with time and space coordinates." 15

Helps and McDonald17 observed vortices inblood flowing slowly through veins and sug-gested the term "disturbed flow" for thisphenomenon to characterize it as neither lami-nar nor turbulent.

In the model of the aorta depicted in figure4 disturbances of this type were observed withpulsatile flow at rates within the physiologicalrange. At low flow rates (fig. 4a) the pro-duction of turbulence was localized and thedisturbance was carried only a short distancebefore it was eliminated by viscous effects.(Although the disturbances in the ventriclewere artifacts, they can be considered some-what comparable to the turbulent mixing ofblood within a functional ventricle.) As seenin figure 4b the flow pattern became generallydisturbed at higher flow rates and at a critiealvelocity the disturbance reached all points inthe flow channel.

The eddies originating from the edges of thesemilunar valves have two functional ef-fects :18>10 to make the valves close more quick-ly by keeping their cusps from opening com-pletely and touching the side walls, and tokeep the orifices of the coronary vessels open.At a low flow rate, the vortex trail in themodel became turbulent only downstream, butat the high rates, the vortices formed at theedges of the cusps. These events are typicalof eddy formation when an obstacle projectsinto a stream of flowing fluid.

As can be seen in figure 4, the system isextremely complex. The trail originating fromthe right valve cusp collides with the innerwall of the aortic arch. Here the trail is



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462 MEISNER, RUSHMEB,

partly deflected, so that it has some effect onthe center stream, and its progression is gen-erally interrupted. If the channel continuedas a straight pipe, these vortices would notbe deviated and would not affect the centerstream at all. In vivo, the three flaps of theaortic valve would generate simultaneouslyeddying trails. These trails would not con-tinue undisturbed, but would be deflected bythe aortic sinus and by the steep curvature ofthe aortic arch. The aortic sinus would thusact as another impediment and contribute tothe formation of eddies that traveled on down-stream.

A relatively short distance farther along theflow is subjected to centrifugal forces withinthe aortic arch. In photographs of flow in amodel arch, Timm1 observed helical flow gen-erated by the centrifugal forces associatedwith the motion in a curve. Since the axialstream moves fastest, the greatest force actson it; the least force is exerted on and nearthe walls where the liquid moves slowest.Hence, the fluid in the center of the streamforms a secondary flow towards the outerboundary and forces the fluid near that wallto flow towards the inner wall of the curva-ture.1' 3l 1C> 10 As suggested by Timm,1 thepresence of the branches has effects which con-tribute to turbulent flow at low Reynolds num-bers. In figure 4a, eddies appear at the en-trances to the branches, and in figure 4b thistransverse motion is very clearly developedand appeared to affect the flow in the mainchannel.

Clearly, projection of eddies from the edgesof the valves, the induction of secondary flowsin the aortic arch and at the origin of themain branches, and the rapid changes in ve-locity from zero to very high to zero, as wellas the ensuing brief period of backflow, allcontribute to turbulence in the outflow fromthe left ventricle. These observations alsoshow how easily the flow pattern is disruptedat any surface of discontinuity.

SummaryThe hirefringent properties of a colloidal

solution of white Hector bentonite were used

as a means of studying the development offlow patterns in transparent models. Theliquid had approximately the same viscosityas blood; flow rates were within the physio-logical range; and some of the flow channelswere shaped to resemble portions of the car-diovascular system. The light patterns pro-duced by laminar flow were distinct fromthose of eddy formation and turbulence. Tur-bulent flow developed downstream from a con-striction where the velocities of the enteringjet were much below the so-called criticalReynolds number. The same flow pattern wasobserved in a model of aortic stenosis. Alsodisplayed was the flow pattern as it mightdevelop in vivo during midsystole when thesemilunar valves are widely opened. Eddiesoriginating from the edges of the valves, theflow around the aortic arch and the effectsof the openings of the branches all contributedto a complex turbulent pattern. Eddy forma-tion and turbulence can tie produced at lowvelocities when unsteady flow and suddenchanges in diameter occur. This technique isuseful in the study of changes in flow patternunder different physiological or pathological,conditions. Observations from these modelsindicate that a more detailed analysis of flowconditions is both possible and worthwhile.

References1. TIMM, C.: Der Stromungsverlauf in einem Modell

tier mensehliehcn Aorta. Z. Biol. 101: 79, 1942.2. STEHBENS, W. E.: Turbulence of blood flow.

Quart. J. Exp. Physiol. 44: 110, 1959.3. FRY, D. L.: Certain aspects of hydrodynamics

as applied to the living cardiovascular system.IKE Trans. Med. Electron. ME-6: 252, 1959.

4. BINNIE, A. M.: A double refraction method ofdetecting turbulence in liquids. Proe. Phys.Soc. 57: 300, 1945.

5. HAUSER, E. A., AND DEWEY, TJ. R., II.: Visualstudies of flow patterns. J. Phys. Chem. 46:212, 1942.

6. WAYLAND, H.: Streaming birefringence as ahydrodynamie research tool—applied to a rotat-ing cylinder apparatus above the transitionvelocity. J. Appl. Phys. 26: 1197, 1955.

7. WAYLAND, H.: Streaming birefringence of rigidmaeromolecules in general two dimensional laminar flow. J. Chem. Phys. 33: 769, 1960.

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8. LINDGREN, B. R : The transition process andother phenomena in viscous flow. Arkiv forFysik 12: 1, 1957.

9. LINDGREN, E. R : Liquid flow in tubes III .Characteristic data of the transition process.Arkiv for Fysik 16: 101, 1959.

10. GREEN, H. D.: Circulation: Physical principlesin medieal physics. In Medical Physics, editedby O. Glasser. Chicago, Tear Book, 1944,1744 pp.

11. WAYLAND, H.: Streaming birefringence as aqualitative and quantitative flow visualizationtool. Symposium on Mow Visualization, ASMEAnnual Meeting, New York, November 30,1960.

12. BOEDER, P.: uber Strbmungsdoppelbrechung. Z.Physik 75: 258, 1932.

13. SEARS, F. W.: Optics, ed. 3, Cambridge, Addi-sou-Wesley Publishing Co. Inc., 1949, 369 pp.

14. JERRARD, H. G.: Theories of streaming doublerefraction. Chem. Rev. 59: 345, 1959.

15. HLNZE, J. O.: Turbulence; an Introduction toits Mechanism and Theory. New York, McGraw-Hill, 1959, 586 pp.

16. PRANDTL, L.: Essentials of Fluid Dynamics;with Applications to Hydraulics, Aeronautics,Meterology and Other Subjects. New York,Hafner Publ. Co., Inc., 1952, 452 pp.

17. HELPS, E. P, W. AND MCDONALD, D. A.: Obser-

vations on laminar flow in veins. J. Physiol.124: 631, 1954.

18. RuSHMER, E. F.: Cardiovascular Dynamics, ed. 2,Philadelphia, W. B. Saunders Co., 1961, 503 pp.503 pp.

19. MCDONALD, D. A.: Blood Flow in Arteries.(Physiological Society Monograph No. 7.)Baltimore, Williams & Wilkins Co., 1960, 328

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Eddy Formation and Turbulence in Flowing Liquidscircres.ahajournals.org/content/circresaha/12/5/455.full.pdfEddy Formation and Turbulence in Flowing Liquids By J. E. Meisner, M.D.,

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