Magnetorheological effect in the magnetic field oriented along the vorticity

Transcription

1 Magntorhological ffct in th magntic fild orintd along th vorticity Pavl Kuzhir, Cécilia Magnt, Laura Rodriguz Arco, Modsto Lopz-Lopz, Hlmi Fzai, Alain Munir, Andry Zubarv, Gorgs Bossis To cit this vrsion: Pavl Kuzhir, Cécilia Magnt, Laura Rodriguz Arco, Modsto Lopz-Lopz, Hlmi Fzai, t al.. Magntorhological ffct in th magntic fild orintd along th vorticity. Journal of rhology, Amrican Institut of Physics, 014, 58, pp.189. <10.11/ >. <hal > HAL Id: hal Submittd on 10 Oct 014 HAL is a multi-disciplinary opn accss archiv for th dposit and dissmination of scintific rsarch documnts, whthr thy ar publishd or not. Th documnts may com from taching and rsarch institutions in Franc or abroad, or from public or privat rsarch cntrs. L archiv ouvrt pluridisciplinair HAL, st dstiné au dépôt t à la diffusion d documnts scintifiqus d nivau rchrch, publiés ou non, émanant ds établissmnts d nsignmnt t d rchrch français ou étrangrs, ds laboratoirs publics ou privés.

2 1 Magntorhological ffct in th magntic fild orintd along th vorticity P. Kuzhir 1 C. Magnt 1, L. Rodríguz-Arco, M.T. Lópz-Lópz, H. Fzai 1, A.Munir 1, A.Zubarv, and G. Bossis 1 1 Univrsity of Nic-Sophia Antipolis, CNRS UMR76, Laboratory of Condnsd Mattr Physics, 8, avnu Josph Vallot, Nic, Franc Dpartmnt of Applid Physics, Univrsity of Granada, Campus d Funtnuva, Granada, Spain Dpartmnt of Mathmatical Physics, Ural Fdral Univrsity, 51, Prospkt Lnina 6008, Ekatrinburg, Russia Synopsis In this work, w hav studid th magntorhological (MR) fluid rhology in th magntic fild paralll to th fluid vorticity. Exprimntally, th MR fluid flow was ralizd in th Coutt coaxial cylindr gomtry with th magntic fild paralll to th symmtry axis. Th rhological masurmnts wr compard to thos obtaind in th con-plat gomtry with th magntic fild prpndicular to th lowr rhomtr plat. Exprimnts rvald a quasi-bingham bhavior in both gomtris with th strss lvl bing just a fw dozns of prcnt smallr in th Coutt cylindrical gomtry at th sam intrnal magntic fild. Th unxpctdly high MR rspons in th magntic fild paralll to th fluid vorticity is xplaind by stochastic fluctuations of positions and orintations of th particl aggrgats. Ths fluctuations ar inducd by magntic intractions btwn thm. Onc misalignd from th vorticity dirction, th aggrgats gnrat a high strss indpndnt of th shar rat, and thus assimilatd to th suspnsion apparnt (dynamic) yild strss. Quantitativly, th fluctuations of th aggrgat orintation ar modld as a rotary diffusion procss with a diffusion constant proportional to th man squar intraction torqu. Th modl givs a satisfactory agrmnt with th xprimntal fild dpndncy of th apparnt yild strss and confirms th narly quadratic concntration dpndncy Y., rvald in xprimnts. Th practical intrst of this study lis in th dvlopmnt of MR smart dvics with th magntic fild non-prpndicular to th channl walls. I. Introduction Magntorhological (MR) fluids ar suspnsions of micron-sizd magntizabl particls disprsd in a liquid carrir. In th absnc of an applid magntic fild, thy almost bhav as classic hard sphr suspnsions. Whn an xtrnal magntic fild is applid, th particls acquir a magntic momnt, attract ach othr and form structurs alignd with th fild dirction. If th applid magntic fild is prpndicular to th walls of th flow channl, th structurs oppos a high hydraulic rsistanc to th flow, which rsults in a svral ordrs of magnitud incras of th MR fluid viscosity. Furthrmor, a significant thrshold shar strss th yild strss is rquird to onst th flow. Such a fild-inducd yilding (Bingham) bhavior is rfrrd to as th MR ffct [Shulman and Kordonski (198), Gindr t al. (1998), Bossis t al. (00b), d Vicnt t al. (011)]. Th MR ffct is applid in activ

3 vibro-protction and lubrication systms as wll as in prcis polishing [Carlson t al. (1996), Kordonski and Jacobs (1996), Urrta t al. (010)]. Du to th fild-inducd anisotropy, th MR ffct dpnds not only on th magntic fild strngth but also on its orintation. Th MR fluid rhological bhavior has bn xtnsivly studid for th cas of magntic filds prpndicular to both th fluid vlocity and th vorticity. Exprimntally, this configuration has bn ralizd in diffrnt ways: (a) using con-plat or plat-plat rotational rhomtry with an axial magntic fild [cf. rviws by Shulman and Kordonski (198), Gindr t al. (1998)]; (b) in Coutt concntric cylindr or cylindrical bob and cup rhomtrs with a radial magntic fild [Shulman and Kordonski (198), Gindr t al. (1996), Laun t al. (1996), Vkas t al. (001), Gnç and Phulé (00), Ulicny t al. (005), Kordonski and Gorodkin (009)]; (c) in prssur drivn flows through plan, cylindrical or annular channls with a magntic fild applid normally to th channl walls [Shulman and Kordonski (198), Gavin (001), Kuzhir t al. (00), Wang and Gordaninjad (006), Ocalan and McKinly (01)]. Thortically, th MR ffct in this configuration is wll undrstood in trms of both th loss of contact btwn th rhomtr walls and th prcolating particl structurs (rsponsibl for th static yild strss), and th viscous dissipation on th particl aggrgats whos siz dcrass with th shar rat (rsponsibl for th dynamic yild strss). S th rviw by Bossis t al. (00b) and th rfrncs thrin for furthr information. Th studis in othr magntic fild orintations ar lss documntd. As far as w know, pionring works on th ffct of th magntic fild orintation on th MR fluid rhology blong to th group of Profs. Shulman and Kordonski. Thy hav shown a somwhat strongr MR rspons in th pip flow undr a transvrs applid fild as compard to th longitudinal fild [Shulman and Kordonski (198)]. Thy hav also ralizd a flow through a slit channl subjctd to a magntic fild ithr normal to th channl walls or paralll to th fluid vorticity, rporting a somwhat wakr MR rspons for th scond configuration [Kordonski t al. (1989)]. Latr on, Bossis t al. (00a) hav carrid out comparativ xprimnts for: (a) th plat-plat gomtry with th magntic fild prpndicular to th walls; and (b) th Coutt concntric cylindr gomtry with an axially applid magntic fild, i.. th fild alignd with th fluid vorticity. Th MR fluid has shown narly th sam MR rspons in both gomtris at th sam intrnal magntic fild, taking into account th quit strong dmagntizing fild in th paralll plat gomtry. Takimoto t al. (1999) hav rportd a.5 tims dcras of th fild-inducd yild strss in a plan channl as th magntic fild orintation is progrssivly varid from th transvrs to th longitudinal on. Such an angular dpndncy of th MR ffct has bn latr rproducd for prssur-drivn flows by Kuzhir t al. (00) and undrstood in trms of th progrssiv dcras of th angl btwn th particl aggrgats and th flow as th angl btwn th applid magntic fild and th flow varis from 90 to 0. Howvr, non of th abov citd works giv a clar xplanation to th yild strss xistnc in magntic filds paralll to ithr th fluid vlocity or to th fluid vorticity.

4 At first glanc, th particl structurs ar xpctd to b alignd with th dirction of th fild whn this lattr is orintd along ithr th fluid vlocity or th fluid vorticity. This would rsult in a rlativly low viscous dissipation and low strss lvls. Lt us xamin in mor dtails th configuration with th magntic fild alignd with th vorticity th cas of th prsnt study. If w considr a dilut rgim and assum that th particl aggrgats do not intract with ach othr, w may dscrib thir angular motion by Jffry quations with an xtrnal torqu includd. Chaffy and Mason (1964) and Almog and Frankl (1995) hav carrid out ths calculations for non-brownian rod-lik particls. Thy hav shown that, whatvr th fild strngth is, initially misalignd particls prform prcssional motion around th vorticity axis and thir trajctoris convrg to th stady-stat orintation alignd with th vorticity (and fild) dirction. Thus, in th stady-stat rgim, th rhology of th suspnsion of vry long and fully alignd aggrgats is xpctd to b similar to that of a suspnsion of infinit cylindrs alignd with th vorticity. Th viscosity of such a suspnsion dpnds on th particl concntration [Christnsn (1991), Christnsn (199)] but should b indpndnt of th magntic fild intnsity as far as th rotation of aggrgats around thir axis is not affctd by th fild and fully dtrmins th suspnsion strss. Thus, no yild strss is xpctd. Of cours, hr w considr magntic filds strong nough to induc particl aggrgation (typically H >1 ka/m). In a typical MR fluid with a particl volum fraction of about 10%, intractions btwn particl aggrgats bcom important and induc priodic misalignmnts of aggrgats with rspct to th vorticity dirction. Misalignd aggrgats could gnrat substantial viscous strsss dpnding on th aggrgat orintation distribution and lngth, which dpnd, in thir turn, on th intrplay btwn th magntic and hydrodynamic intractions. Rcntly, Kuzhir t al. (011) hav thortically studid th MR fluid flow in th longitudinal fild and shown that stochastic aggrgat misalignmnts from th flow dirction (causd by magntic intractions btwn aggrgats) could rsult in a Bingham bhavior with an apparnt (dynamic) yild strss comparabl to that masurd in th transvrs fild. W xpct thrfor that th sam physics may govrn th MR fluid rhology in th magntic fild paralll to th fluid vorticity. Rcall that th considrd mchanism rquirs rlativly strong intractions btwn aggrgats. Othrwis, th ffct can b th opposit, i.., a dcras of th ffctiv viscosity with th incrasing fild, as it has bn obsrvd in suspnsions of wakly magntizabl rd blood clls [Tao and Huang (011)]. In th prsnt papr, w prform both thortical and xprimntal studis on th MR fluid flow in th magntic fild paralll to th fluid vorticity. In th xprimnts, w us a concntric cylindr rotational gomtry with an axially applid magntic fild, and mak th masurmnts in a wid rang of applid magntic filds and MR fluid concntrations. A spcial attntion is paid to th comparison of th MR ffct for th magntic fild paralll to th fluid vorticity and paralll to th vlocity gradint, th lattr configuration bing ralizd with th hlp of a con-plat gomtry. In our thory, w modl th stochastic misalignmnts of th particl aggrgats as a rotary diffusion procss with a diffusion constant dfind by th strngth of th magntic intractions btwn aggrgats. Our main goal is to dmonstrat

5 4 that such a fild-inducd rotary diffusion procss may b at last on of th possibl mchanisms xplaining th MR ffct in th magntic fild alignd with th fluid vorticity. Apart from th fundamntal intrst, this study is motivatd by th dvlopmnt of activ hydrodynamic/hydrostatic barings of machin tools with th magntic fild applid coaxially with th rotating shaft. In particular, this configuration corrsponds to barings quippd with a non-magntic shaft [Urrta t al. (010)]. Th us of non-magntic shafts instad of magntic ons allows avoiding th problm of magntic attraction btwn th shaft and th pol pics of th lctromagnt, which hindrs an ffctiv ral-tim control ovr th baring prformancs. Th prsnt papr is organizd as follows. In Sction II, w prsnt th xprimntal apparatus and xprimntal protocols. In Sction III, w ovrviw th xprimntal rsults obtaind both in th Coutt cylindrical and th con-plat gomtris. Th thortical modl for th MR ffct in th magntic fild alignd with th vorticity is dscribd and compard with xprimnts in Sction IV. Conclusions and prspctivs ar outlind in Sction V. II. Exprimntal procdur Th magntorhological fluids of particl volum fractions ranging from 10 to 6% wr prpard by disprsing carbonyl iron (BASF) microparticls (of a man diamtr of µm and magntization saturation, M S = A/m) in a silicon oil (Rhodorsil 47V500; VWR Intrnational, dynamic viscosity at 5 C is 0 =0.48 Pa s) undr vigorous stirring. Subsquntly, th MR fluids wr stabilizd against aggrgation du to colloidal intractions by adding an appropriat amount of aluminum starat (Sigma Aldrich), as dscribd in dtails in Lópz-Lópz t al. (008). Th rhological masurmnts wr prformd with th hlp of th controlld-strss rotational rhomtr Thrmo Haak RS 150 using a standard titanium-mad cylindrical Coutt gomtry (bob and cup gomtry Z0 DIN/ISO 19) shown schmatically in Fig. 1. Th intrnal cylindr had a diamtr of D=0 mm and hight h=0 mm; th radial distanc btwn both cylindrs (calld hrinaftr th gap) was qual to g=0.8 mm. Th bottom con had an apx angl =118 and th distanc btwn th con dg and th bottom of th cup was adjustd to h 1 =1 mm. Th amount of MR sampl pourd into th cup was chosn in such a way to kp th sam distanc (qual to h = 15 mm) btwn its fr surfac and th uppr bob surfac for all xprimnts. A uniform xtrnal magntic fild was applid co-axially with th masuring gomtry, and thus, in th dirction paralll to th MR fluid vorticity. This fild was gnratd by a coil (80 mm in intrnal diamtr and 10 in hight) placd co-axially with th masuring gomtry. Th magntic fild H 0 was masurd in th cntr of th mpty coil with th hlp of a Caylar GMH-10 gaussmtr as a function of th lctric currnt I applid to th coil, and th following linar rlationship was stablishd: H 0 =6.11 I, whr I is xprssd in A and H 0 in ka/m.

6 5 Fig.1. Cylindrical Coutt (bob and cup) gomtry with th magntic fild applid paralll to th rhomtr axis Th masuring protocol can b summarizd as follows. Bfor th masurmnts th MR fluids wr dgasifid for 15 minuts and placd immdiatly aftr that insid th rhomtr. Firstly, th MR fluid was pr-shard in th absnc of th magntic fild for 5 minuts at a shar rat qual to 100 s -1. Thn, a magntic fild of an intnsity of about 6 ka/m was applid and th pr-shar stag continud for anothr 5 minuts. Aftr th pr-shar stag, th magntic fild intnsity was adjustd to a dsird valu (ranging from 6 to 0.6 ka/m), th suspnsion was lft at rst for minuts, and thn, a strss ramp was applid to th MR sampl with duration of ach stp qual to 0 s. Th shar rat was masurd for ach stp of th strss ramp and th dpndncy of th shar strss on th shar rat th flow curv was obtaind. Onc th strss ramp ndd, th magntic fild was again radjustd to th valu of 6 ka/m, and th pr-shar stag was rpatd bfor a nw strss ramp prformd at a nw valu of th applid fild. Not that th magntic fild (of an intnsity of at last 6 ka/m) was maintaind from th bginning to th nd of th masurmnts in ordr to rduc particl sttling bcaus of gravity. W hav chckd that th MR fluid viscosity at th nd of th masurmnts was, in th worst cas, 10% lowr than that at th bginning. It is important to not that th masurd flow curvs corrspond to th shar flow in th whol MR sampl, covring not only th masuring gap btwn cylindrical parts of th gomtry but also th rgions blow and abov th masuring gap. Thus, th valus of th imposd strss nd to b corrctd by subtracting vntual contributions of th bottom and uppr parts of th sampl. Th contribution of th uppr part is dducd by comparing th masurd friction torqus (xrtd by th MR fluid on th rotating bob) for diffrnt hights h abov th uppr bob surfac [cf. Fig.1]. This contribution appars to b lowr than th instrumntal rror of th rhomtr of 1%. Th contribution of th bottom part is inspctd by masurmnts of th friction torqu for th MR sampls filling th cup up to th lvl, at which th bottom con is fully immrgd into th MR fluid but th masuring gap rmains unfilld. This torqu appars to b about % of th total torqu corrsponding to th whol bob immrsd into th sampl. To corrct this systmatic rror, w subtractd th friction torqu T con masurd for th immrsd con from th torqu T bob masurd for th fully

7 6 immrsd bob. Th shar strss in th gap btwn rotating cylindrs is thrfor givn by:, with R=10 mm and L=0 mm bing, rspctivly, th radius and ( Tbob Tcon) /( R L) th lngth of th cylindrical part of th rotating bob. Whn th matrial undrgos a yilding bhavior, th friction torqu xrtd on a considrd part of th bob is xpctd to b proportional to th product of th yild strss, Y, and th gomtry radius, r, and th latral surfac, S, of th rspctiv bob part. Th product S r appars to b much smallr for th bottom and th uppr parts of th bob as compard to th cntral cylindrical part corrsponding to th masuring gap. Furthrmor, analysis shows that th avrag magntic fild in th uppr and th bottom parts is a fw tims lowr than in th cntral part (cf. Appndix B); this could rsult in lowr valus of th yild strss in ths parts as compard to th cntral part. Thus, combination of ths both ffcts could xplain th ngligibl masurd contributions of th bottom and th uppr parts of th sampl to th ovrall friction torqu. In ordr to compar th rhological rsults obtaind for diffrnt magntic fild orintations, w also conductd th rhological masurmnts in a con-plat gomtry (mad of titanium, diamtr 5 mm and apx angl ) in th prsnc of a uniform magntic fild alignd with th rhomtr axis and gnratd by th sam coil placd co-axially with th masuring gomtry. W usd a masuring protocol similar to th on usd for th cylindrical Coutt gomtry without prforming any rhomtric corrctions. III. Exprimntal rsults A. Analysis of th intrnal fild Th main goal of this Sction is to confirm th ffct of th magntic fild orintation (ithr along th vorticity or along th vlocity gradint) on th MR rspons of th MR suspnsions within a wid rang of xprimntal paramtrs, such as th shar rat, th magntic fild intnsity and th suspnsion concntration. For this purpos, w will compar rhological rsults obtaind both in Coutt and con-plat gomtris, in which th fild is alignd along th vorticity and along th vlocity gradint, rspctivly. Comparison at th sam xtrnal magntic fild dos not allow a corrct analysis of th intrinsic MR proprtis of th suspnsion bcaus ths proprtis dpnd principally on th intrnal magntic fild, H, insid th MR sampl rathr than on th xtrnal on H 0. Thrfor, w nd to dtrmin th distribution of th intrnal magntic fild as a function of th shap of th MR sampl and of th applid homognous fild, H 0. Th intrnal fild H dpnds, among othr things, on th suspnsion microstructur, which, in its turn, is strongly affctd by th shar rat. Dirct masurmnts of th magntic fild distribution insid narrow gaps of th shard sampl ar subjctd to various tchnical problms. Altrnativly, th fild distribution can b obtaind by finit lmnt calculations, which rquir knowldg of th magntic prmability,, of th suspnsion as a function of th fild and th shar rat. Exprimntal dtrmination of in shard suspnsions poss problms similar to thos rlatd to masurmnts of H. Thrfor,

8 7 w shall procd to thortical stimations of th suspnsion prmability using two diffrnt approachs. In our xprimnts, th yild strss was obtaind by a linar fit of th flow curvs at rlativly high shar rats. Thrfor, both stimations wr don for th high-shar rgim, i.. at shar rats 100 s -1. Th first stimation is basd on th assumption of compltly unstructurd suspnsion at high shar rats. This situation corrsponds to an isotropic stat with particl structurs dstroyd by shar, as opposd to th low-shar rgim with strongly alignd structurs. D Vicnt t al. (00) hav rportd valus of th magntic prmability of th unstructurd MR suspnsion at diffrnt particl volum fractions, [s Fig. 5 of that papr]; for xampl,.6 at =0.. Th scond stimation is basd on th assumption of longatd aggrgats. In th cas of th Coutt cylindrical gomtry, th aggrgats ar supposd to b alignd with both th xtrnal fild and th fluid vorticity (in rality, th aggrgat orintation may fluctuat around th vorticity axis s Sc. IV but w nglct ths fluctuations for stimations of ). At such conditions, th suspnsion proprtis possss a strong anisotropy and th axial, zz, and radial, rr, componnts of th prmability tnsor ar found using th Maxwll-Garntt man fild thory [Brthir (199)], as dscribd in dtails in Appndix A. In th cas of th con-plat gomtry, w adopt th modl of cylindrical particl aggrgats tiltd by th shar flow at som angl with rspct to th applid fild. This modl has bn dvlopd by Gómz-Ramírz t al. (011) [Eqs. (5), (6) of that papr] and th basic quations ar prsntd in Appndix A. Th modl provids a valu of th tilt angl as a function of th applid fild [Eq. A.], but indpndnt of th shar rat in th high-shar rgim. Using this stimatd orintation of th aggrgats, w calculat th componnts of th magntic prmability tnsor [Eqs. A.4-A.5]. This scond stimation provids th following man valus (avragd ovr th rang of th xtrnal fild 0<H 0 <0.6 ka/m) of th magntic prmability componnts at =0.: zz 5.8 and rr.7 for th Coutt gomtry, and zz.8 and rr.7 for th con-plat gomtry. Analysis shows that th magntic fild distribution in con-plat gomtry is much mor snsitiv to th magntic prmability valus than th fild in th masuring gap of th Coutt gomtry. Fortunatly, both stimations giv narly th sam valu of th magntic prmability for th con-plat gomtry. Thrfor, for this gomtry, w took an avrag valu of th magntic prmability, zz = rr =.7±0.1 for stimations of th intrnal magntic fild. Onc th magntic prmability is stimatd, w procdd to calculation of th magntic fild distribution insid th MR sampls in both considrd gomtris. Dtails of th simulations ar givn in th Appndix B along with Figurs B1 and B showing th magntic fild distribution. Figur B shows that th intrnal fild rmains narly homognous in th whol sampl filling th con-plat gomtry xcpt nar th sampl mniscus. Th man valu of th intrnal fild is only 5% highr than th valu H 0 / zz stimatd undr th assumption of a plain infinitly thin layr of MR fluid [Fig. Bb]. Figur B1 shows that th magntic fild rmains narly homognous within th masuring gap btwn two cylindrs in th cylindrical Coutt gomtry. Th man valu of th intrnal

9 8 fild in th masuring gap appars to b only 4% highr than th valu H 0 of th xtrnal fild masurd in th cntr of th mpty coil [Fig. B1b]. Dspit this rlativ homognity insid th masuring gap, th magntic fild outsid th gap (in th spac btwn th bottom con and th cup as wll as abov th bob), is a fw tims lowr than th on insid th gap. This diffrnc appars purly bcaus of th spcific shap of th MR fluid sampl filling th bob and cup gomtry (dmagntizing ffct) and it is not rlatd to magntic parts of th gomtry (mad of titanium). Such non-uniformity of th magntic fild outsid th masuring gap could induc migration of th magntic particls toward th rgion of th highst fild, i.. insid th masuring gap, which would rsult in a considrabl incras of th suspnsion viscosity with tim. To chck this point, w prformd a kind of crp tst for which a constant shar strss was applid to th MR sampl and th magntic fild was abruptly switchd on. Th shar viscosity was masurd as a function of tim during a priod of 1h. Th rsults of this tst (not shown hr for brvity) rvald a small but gradual viscosity dcras with tim at long tim scal, totally inconsistnt with th particl crowding ffct. Th insignificanc of particl migration is also consistnt with an stimation of th charactristic tim t of particl migration from th uppr or th lowr parts of th Coutt gomtry to th masuring gap. A simpl balanc of th magntic forc and th hydrodynamic drag acting on th particls givs valus of t of th ordr of t ( X ) /( H a ) s th priod strongly xcding typical xprimntal tims, whr = 0.5 Pa s is th suspnding liquid viscosity, X10 mm is a charactristic scal of th fild gradint, a1 µm is th particl radius, 1 is th particl magntic contrast factor, and µ 0 = H/m is th magntic prmability of vacuum. B. MR rspons in Coutt and con-plat gomtris Lt us first xamin th MR rspons of th MR suspnsion in both gomtris at th sam xtrnal magntic fild H 0. Th flow curvs obtaind in ths gomtris ar shown in Fig.a for th MR fluid of particl volum fraction =0. and at an xtrnal magntic fild H 0 =18. ka/m. Rcall that in th first cas, th magntic fild is orintd along th vorticity, whil in th scond on, along th vlocity gradint. As it is sn in this figur, th strss lvl in th Coutt gomtry is highr than that in th con-plat gomtry at th sam xtrnal magntic fild. This diffrnc can b attributd to two diffrnt ffcts: (a) th dmagntizing ffct of th MR sampl, which lads to a substantial diffrnc in intrnal magntic filds in both gomtris subjctd to th sam xtrnal fild; (b) MR fluid anisotropy, i.. dpndnc of th suspnsion proprtis on th fild dirction causd by th anisotropy of th MR microstructur. As mntiond in Sc. III-A, th comparison of th MR rspons at th sam xtrnal magntic fild dos not allow sparating ths two ffcts. On th contrary, comparison at th sam intrnal fild allows us to liminat (within th rrors rlatd to th thortical dtrmination of this fild) th first ffct and to analyz th ffct of th fild orintation. Th flow curvs masurd for both gomtris at th sam intrnal magntic fild, H 1 ka/m ar shown in Fig. b for th sam particl volum fraction =0.. As a charactristic valu of H in both gomtris, w us th avrag valu of th magntic

10 9 fild in th masuring gap of th considrd gomtry, calculatd in Appndix B and shown in Figs. B1b and Bb. Analysis of Figs. a and b shows that, if th strss lvl is typically highr in Coutt gomtry at th sam xtrnal fild (Fig. a), th tndncy rvrss if th comparison is mad at th sam intrnal fild (Fig. b): th strss lvl in th Coutt gomtry appars to b lowr than that in th con-plat on. This diffrnc could b xplaind in trms of th orintation of th suspnsion microstructur with rspct to th flow. Th dgr of misalignmnt btwn th flow and th aggrgats in th con-plat gomtry is xpctd to b highr than th dviation of th aggrgat orintation from th vorticity axis in th Coutt gomtry. Strongr misalignmnt gnrats highr viscous dissipation and should rsult in highr strss lvl in th con-plat gomtry with th fild prpndicular to th rhomtr walls. Fig.. Comparison btwn th MR fluid flow curvs masurd in th con-plat and th cylindrical Coutt gomtris at th sam xtrnal magntic fild H 0 =18. ka/m (a), and at th sam intrnal magntic fild H 1 ka/m (b). For both figurs, th particl volum fraction is =0. W also rmark that th initial part of th flow curv in th cylindrical Coutt gomtry has a mor roundd shap than that in th con-plat gomtry. Such an initial roundd shap of th flow curvs has alrady bn obsrvd in magntic fibr suspnsions in th plat-plat gomtry and has bn attributd to gap-spanning aggrgats whos lngth is

11 10 boundd by th rhomtr gap [Gómz-Ramírz t al. (011)]. Th flow curvs obsrvd in th prsnt study ar likly govrnd by th sam physics (i.. intractions btwn th aggrgats and th walls); howvr, at this stag, w ar unabl to answr why th flow curv in th Coutt gomtry is smoothr than th on in th con-plat gomtry. Finally not that th flow curvs shown in Figs. a-b xhibit a quasi-linar bhavior at rlativly high shar rats, 100 s -1. Th high-shar parts of th flow curvs wr fittd by a linar function, Y, xtrapolatd to lowr shar rats. Th intrcpt, Y, of th linar fit to zro shar rat is dfind as th apparnt yild strss (oftn usd in magntorhological litratur and somtims calld th dynamic yild strss). Th valu Y should not b confoundd with th ral yild strss (usually calld th static yild strss) corrsponding to th failur of th prcolatd MR structurs and to th onst of th flow. This apparnt yild strss is considrd to b th most important charactristic of th MR suspnsions. Thrfor, in what follows, w shall focus our attntion on this magnitud. Bcaus of th abov mntiond diffrnc in dmagntizing ffcts, w shall compar th apparnt yild strss obtaind in both gomtris at th sam intrnal magntic filds, H. Th xprimntal dpndncis of th apparnt yild strss on th intrnal magntic fild ar shown in Fig.a for th MR suspnsion of particl volum fraction, =0.. As sn in this figur, in both gomtris, th apparnt yild strss incrass with an incrasing magntic fild, which is undrstood in trms of fild-inducd nhancmnt of magntic intractions, which maks th particl structurs mor robust and mor rsistant to sharing forcs. At th sam intrnal magntic filds, th apparnt yild strss in th con-plat gomtry (in th magntic fild paralll to th vlocity gradint) appars to b somwhat highr than th on in th cylindrical Coutt gomtry (in th magntic fild longitudinal with th vorticity). This fact sms to b quit intuitiv. As alrady mntiond, in th con-plat gomtry, th fild crats particl structurs prpndicular to th rhomtr walls. Ths structurs hindr significantly th flow and gnrat substantial viscous dissipation. On th othr hand, in th cylindrical Coutt gomtry, th magntic fild orints th particl structurs prfrably along th vorticity, so a smallr strss lvl is xpctd. Howvr, if th magntic fild is strong nough to induc th particl structurs, and if ths structurs ar prfctly alignd with th fild, th intnsity of th magntic fild in this scond cas should not influnc th strss lvl of th suspnsion, and, instad of th Bingham bhavior, a Nwtonian on is xpctd with th viscosity dfind by th particl volum fraction. Such apparnt inconsistncy will b xplaind in th nxt sction.

12 11 Fig.. Comparison of th xprimntal dpndncis of th apparnt yild strss on th intrnal magntic fild (a), and on th particl volum fraction (b), obtaind for th con-plat and th cylindrical Coutt gomtris. For th rsults prsntd in (a), th particl volum fraction is =0.. For th rsults prsntd in (b), th intrnal magntic fild is adjustd to H 10 ka/m. Th intrnal magntic fild is calculatd in Appndix B as a function of th MR sampl shap, its magntic proprtis and th xtrnal fild. Exprimntal dpndncis of th apparnt yild strss on th particl volum fraction,, ar shown in Fig. b for both gomtris at th sam intrnal magntic fild, H 10 ka/m. Obviously, in both cass th MR ffct is a growing function of th concntration. Within th concntration rang , th strss lvl appars to b largr in con-plat gomtry, as compard to th Coutt on, for th rasons xplaind abov. A dtaild analysis of th magntic fild and concntration ffcts on th apparnt yild strss in Coutt gomtry will b studid in dtails in th nxt sction IV. IV. Thory and comparison with xprimnts In this sction w focus on th apparanc of th apparnt yild strss whn th fild is applid in th dirction of vorticity. Thortical xplanations for th fild-inducd apparnt yild strss whn th fild is applid in th dirction of th vlocity gradint ar xtnsivly

13 1 rportd in th litratur [cf. th rviw by Bossis t al. (00b) and th rfrncs thrin] and will not b tratd hr. Qualitativly, w adopt th sam xplanation for th MR ffct in th magntic fild alignd with th vorticity as th on usd by Kuzhir t al. (011) for th magntic fild alignd with th vlocity. If, for som rason, th particl aggrgats gt misalignd with th vorticity, thy will gnrat substantial hydrodynamic strsss proportional to th squar of thir lngth-to-diamtr ratio, r, a paramtr calld aspct ratio: 0 r. Onc misalignd with th vorticity, tnsil hydrodynamic forcs start acting on th aggrgats, such that thir lngth, or rathr aspct ratio, dcrass with th shar rat as follows: r 1/ [Shulman t al. (1986), Martin and Andrson (1996), Gómz-Ramírz t al. (011)]. Thrfor, th hydrodynamic part of th aggrgat strss appars to b indpndnt of th shar rat and can b considrd as th apparnt yild strss th strss associatd to th hydrodynamic dissipation on th aggrgats rathr than to a thrshold strss rquird for braking th structurs and to th onst of th flow. On of th rasons for th aggrgat misalignmnt from th fild and vorticity dirctions may com from magntic intractions btwn aggrgats. In th prsnc of an xtrnal fild, th aggrgats possss an inducd magntic momnt and, to a first approximation, can b considrd as long dipols with th magntic pols locatd at thir xtrmitis. In shar flow, th aggrgats initially orintd along th vorticity, mov with diffrnt vlocitis bcaus thir cntrs of mass displac along diffrnt stramlins. Th pols of th nighboring aggrgats may attract or rpl ach othr. This may caus th aggrgats to chang thir orintation. Such angular motion should not b confoundd with continuous spinning undr shar flow. As mntiond in th Introduction, th aggrgat flipping is avoidd by th magntic torqu xrtd on th aggrgats by th applid magntic fild. In a rlativly concntratd suspnsion, th aggrgats ar spacd in an irrgular mannr, thir numbr is important and thir lngth may vary bcaus of th complicatd kintics of thir formation/dstruction. This will mak intr-aggrgat intractions stochastic and lad to random fluctuations of thir orintation. In summary, th combination of th stochastic misalignmnt of th aggrgats with thir high hydrodynamic rsistanc and vntual dstruction by sharing forcs may rsult in a yilding bhavior of th MR fluid undr th fild alignd with th vorticity. Having proposd a qualitativ xplanation for th MR ffct in th considrd gomtry, in what follows, w shall try, on this basis, to mak a quantitativ prdiction of th MR fluid apparnt yild strss in this gomtry. For this purpos, lt us considr a simpl shar flow btwn two infinit plans with th xtrnal magntic fild, of an intnsity H 0, orintd paralll to th plans and prpndicular to th flow, cf. Fig.4. Th Cartsian rfrnc fram is introducd in such a way that th axis 1 is orintd along th fluid vlocity, th axis along th vlocity gradint and th axis along th vorticity and along th applid magntic fild. Bcaus of th infinit siz of th channl in th dirction, th dmagntizing ffcts ar absnt and th magntic fild, H, insid th MR fluid is th sam as th xtrnal fild: H=H 0. Not that such

14 1 stimation of th intrnal magntic fild rmains tru for th concntric cylindr gomtry usd in our xprimnts. Actually, finit lmnt simulations show only 4% dviation of th intrnal fild H in th masuring gap from th xtrnal on H 0 (cf. Sc. III-A and Appndix B). Fig.4. Sktch of th problm gomtry Th magntic fild inducs th formation of aggrgats, of man lngth L and man radius A, orintd along th applid fild. Thir stochastic misalignmnt is supposd to b a Gaussian dlta-corrlatd procss and is mimickd by a rotational diffusion procss. This diffusion is not causd by Brownian motion of aggrgats but is inducd by magntic intractions btwn nighboring aggrgats, rsulting in thir stochastic angular displacmnt. According to th random walk principl [van d Vn (1989)], th rotational diffusion procss may b sn as a sris of stochastic angular jumps of th aggrgat orintation charactrizd by a man squar amplitud and a man duration t. In th framwork of this modl, th rotary diffusivity associatd to this procss is dfind as [cf. Eq. (9.7) in Doi and Edwards (1986)]: Dr / t t (1) whr / t is th man squar angular vlocity of th aggrgats. Th man duration of th jumps can b stimatd as th invrs of th shar rat, 1 t, and th man squar angular vlocity through th stochastic torqu inducd by magntic intractions with nighboring aggrgats th so calld intraction torqu Tint / fr, whr T int is th man squar valu of th intraction torqu and f r is th rotational friction cofficint of th aggrgat. This lads to th following xprssion for th rotary diffusivity: D r T int r f () Th man squar intraction torqu can b stimatd using an ffctiv fild approach as follows. Considr an aggrgat moving in th shar flow and subjctd to a magntic fild

15 14 H. Th nighboring aggrgats moving around our givn aggrgat will induc a supplmntary magntic fild at th location of th givn aggrgat. This fild is xpctd to b a sum of a prmannt componnt, H p, and a stochastic componnt H st. Th first on could b attributd to th statistical avrag intraction btwn th givn aggrgat and a surrounding ffctiv mdium having a prmannt man magntization, M. Th trm H p may b thrfor sn as a local Lornz fild, whos componnts ar qual to ( H ) n M, with p i ik k n ik bing th tnsor of dmagntization factors of th aggrgat [Brthir (199)]. Th fild H p contributs to an nhancmnt of th suspnsion magntization du to rgular intractions btwn aggrgats and xists vn in th absnc of fluctuations in thir orintation. Th scond trm, H st, appars du to stochastic magntic intractions btwn th aggrgats bcaus of irrgular spacing btwn thir magntic pols undr shar flow. Th orintation of this stochastic fild is xpctd to xprinc random fluctuations during tim, such that H st 0, whil its quadratic man valu is supposd to b proportional to th squar of th suspnsion magntization: Hst M. Th instantanous intraction torqu inducd by th stochastic fild is thrfor givn by a vctor product of th aggrgat magntic momnt and th stochastic fild: Tint mh st. Th aggrgat magntic momnt and th suspnsion magntization can b stimatd for th cas of a prfct alignmnt of th aggrgats with th magntic fild, as follows: m 0 ahva and M ( / a) ah with 0 = H/m bing th magntic prmability of vacuum; a - th aggrgat magntic suscptibility, V a =A L - th aggrgat volum, and a - th volum fraction of particls in th suspnsion and th intrnal volum fraction of th aggrgats, so that th ratio (/ a ) stands for th concntration of aggrgats in th suspnsion. Prforming th ncssary substitutions and taking into account th xprssion fr for th friction cofficint, w obtain 8 0L /(ln ) th following xprssion for th rotary diffusivity: D r 0 a H a0 1 () whr 0 is th viscosity of th suspnding liquid, is th form-factor 4 r /(ln ) dscribing th hydrodynamic rsistanc of th aggrgats and coming from th slndr body thory [Batchlor (1970)]; is th hydrodynamic scrning lngth of th aggrgats normalizd by thir radius A; th numrical constant is introducd as a corrction factor calld intraction constant, which is takn as a fr paramtr of th prsnt modl. Th intraction constant should not b confoundd with th intraction paramtr, which is oftn rfrrd to th ratio of th magntic-to-thrmal nrgy and it is altrnativly calld th dipolar coupling paramtr. Th aggrgat aspct ratio r or rathr th form-factor, can b found by a balanc btwn tnsil hydrodynamic forcs and cohsiv magntic forcs, as usually don in classical magntorhology [Shulman t al. (1986), Martin and Andrson (1996)]. For this

16 15 purpos, lt us considr an aggrgat whos orintation is dscribd by th angls and in th sphrical coordinat systm, as shown in Fig.4, whr is th angl that th aggrgat forms with th magntic fild (axis ) and is th angl btwn th flow dirction (axis 1 ) and th aggrgat projction onto th shar plan 1. Altrnativly, th aggrgat orintation cay b dscribd by a unit vctor,, orintd along its major axis. Th projctions of this vctor onto th thr basis axs rad: 1 sincos, sinsin, cos. Applid to th gomtry considrd in th prsnt work, th instantanous forc balanc [drivd in dtails in Kuzhir t al. (011)] rads: L 0 sin sin cos A afmcos, (4) ln whr th lft-hand part corrsponds to th tnsil hydrodynamic forc and th right-hand part to th magntic cohsiv forc; f m is th magntic forc btwn nighboring particls constituting th aggrgats pr unit cross-sctional ara of th particls. Th magntic forc f m and th aggrgat magntic suscptibility a, intrvning into Eqs. () and (4), ar calculatd as a function of th magntic fild H using finit lmnt simulations, as xplaind in dtails in Kuzhir t al. (011). Th simulation rsults for f m and a hav bn fittd by th following xprssions, valid in th rang of th magntic fild intnsitis, 0 H 0 ka/m: f ( H) ( H / M ) ( H / M ) 168 H and 4 m S S 0, with M S = A/m bing th a( H) ( H / M S ) 50.9( H / M S ) 9.9 saturation magntization of th carbonyl iron particls. During thir motion, th aggrgats ar continuously dstroyd by th hydrodynamic forcs and rformd by attractiv magntic forcs. Th avrag aggrgat lngth is dfind by th kintics of this procss rathr than by th instantanous forc balanc (4). At this stag, w do not nd to dscrib this kintics in ordr to undrstand th mchanism of th MR ffct in th cas of th magntic fild alignd with th vorticity. A rapid stimat of th aggrgat lngth can b obtaind by assuming th prcssion of an aggrgat around th vorticity axis at som prcssion angl,. This aggrgat is supposd to b dstroyd onc it achivs th polar angl, =/4, at which th tnsil hydrodynamic forc is maximal. Exprssing th aggrgat aspct ratio r =L/A from Eq. (4), as a function of th prcssion angl and avraging ovr all possibl valus of, w gt th following quation for th form-factor : 4r a fm 4a fm cot ln 0 0 1, (5) whr th approximation cot cos / 1 cos is usd, making it possibl to xprss th man cotangnt of th azimuth angl through th man squar componnt cos of th unit vctor.

17 16 Th nxt stp is to find th man orintation of th aggrgats undr shar and magntic filds. Th aggrgat orintation is commonly dscribd by th scond and th fourth rang tnsors, i k and ik l m calld, rspctivly, th scond and th fourth statistical momnts. Th quation dscribing th tmporal volution of th scond statistical momnt coms from th avrag of th torqu balanc and, bing applid to th cas of long aggrgats possssing an inducd magntic momnt, rads [Pokrovskiy (1978), Kuzhir t al. (011)]: d i k dt H il l k i l lk il l k i l lk i k l m lm (1 / ) 0 a a hi hl lk hk hl li ik lm hl hm Dr ik ik 0 a(1 / a) (6) whr t is th tim; (1/ ) ( v / x v / x ) and (1/ ) ( v / x v / x ) ar ik i k k i ik i k k i th rat-of-strain and vorticity tnsors, rspctivly; h i is th i-th componnt of th unit vctor h orintd along th intrnal magntic fild H; ik is th Kronckr dlta. In th prsnt cas, w hav only two non-zro componnts of th rat-of-strain and vorticity tnsors, / and on non-zro componnt of th fild unit vctor, h 1. Th factor (1 / ) / (1 / ) a a a a apparing in th last quation coms from th magntic torqu xrtd on th aggrgats by th applid fild and has bn calculatd taking into account th statistical avrag of th intractions btwn th givn aggrgat and th surrounding ffctiv mdium (using th abov dfind Lornz local fild H p ). In ordr to solv quation (6), a closur rlation btwn th scond and th fourth statistical momnts must b introducd, and w choos th quadratic closur, ik lm ik lm, which appars to corrctly dscrib th orintation stat of highly alignd suspnsions of rod-lik particls or molculs [Doi and Edwards (1986)]. Using this rlation along with th xprssions () and (5) for th rotary diffusivity and th aggrgat form-factor, rspctivly, and considring th stady stat ( d / dt 0), th quation (6) rducs to th following st of algbraic quations: i k

18 C 1 C C C C C C 1 6C (7) whr 0 a (1 / a) /[4 a m( a(1 / a))] C1 0 a H /(4 a fm) and C H f ar dimnsionlss factors indpndnt of th shar rat. As xpctd, th sum of th lft-hand sids of th thr first quations givs zro undr th condition. Th shar rat coms out from quations (6) 1 1 and (7) bcaus both th magntic and th diffusion trms (th two last trms at th righthand-sid of Eq. (6)) appar to b proportional to th shar rat du to th form-factor 1 [cf. Eq. (5)], which maks th rotary diffusivity linar in th shar rat, 1 Dr [cf. Eq. ()]. momnts, Th systm (7) is solvd numrically with rspct to th four unknown statistical 1,, and 1, th componnts 1 and bing zro for symmtry rasons. Analysis shows that th aggrgat orintation is strongly influncd by th intraction constant, rsponsibl for th intnsity of th rotary diffusion procss. Thortical dpndncis of th four scond statistical momnts on th intraction constant ar shown in Fig. 5a in a smi-logarithmic scal for th particl volum fraction =0. and th magntic fild, H=18. ka/m. At =0, th diffusion is absnt, all th aggrgats ar prfctly alignd with both th fild and th vorticity, which implis 1 and. As th intraction constant incrass, th diffusion bcoms mor important, th aggrgat orintation xhibits strongr dviation from its quilibrium position, so, th statistical momnt (man squar cosin of th angl btwn th aggrgats and th fild) dcrass and th othr statistical momnts incras. Intrstingly, th statistical momnt through a minimum and th momnts 1 and passs 1 through a maximum at intrmdiat valus of. At a furthr incras of th intraction constant, th diffusion bcoms vry strong and maks th orintation distribution mor and mor isotropic, such that th thr

19 18 diagonal componnts bcom qual: dcrass to zro., and th componnt 1 1 1/ Fig.5. Thortical dpndncis of th scond statistical momnts on th intraction constant (a) and on th suspnsion volum fraction (b). In both cass, th magntic fild is paralll to th vorticity and its intnsity is qual to H=18. ka/m. In figur (a), th particl volum fraction =0.; in figur (b), th intraction constant is qual to = Anothr paramtr affcting th orintation distribution is th suspnsion volum fraction,. Thortical dpndncis of th scond statistical momnts on th volum fraction ar prsntd in Fig. 5b for th magntic fild H=18. ka/m and a fixd valu of th intraction constant, = In th limit of infinit dilution, 0, th aggrgats do not fl th prsnc of ach othr, and thir orintation dos not dviat from th quilibrium on along th vorticity axis which rsults in 1. As th concntration incrass, th magntic intractions btwn th aggrgats incras, th rotary diffusivity grows as [cf. Eq. ()], which lads to an incras of stochastic fluctuations of thir orintation. This implis an initial grows of th statistical momnts 1, and 1 with th concntration and

20 19 a dcras of th momnt. Crtainly, at high concntrations, whn th orintation stat bcoms mor isotropic, collisions btwn aggrgats may occur. This may chang th diffusion procss in a way that th orintation distribution will achiv th isotropic stat at lowr concntrations. W xpct, howvr, that th diffusion procss with collisions will kp th sam qualitativ bhavior as th on solly inducd by magntic intractions. Th last stag of our modling is th calculation of th shar strss. Applid to th cas of long aggrgats with inducd magntic momnts subjctd to both shar and magntic filds, th xprssion for th strss tnsor of a smi-dilut suspnsion rads [Pokrovskiy (1978), Kuzhir t al. (011)]: 1 ik p ik 0 ik 04 ik ik lm ik lm lm a (1 / ) a a 0H ik lm hl hm hi hl lk 0Dr ik ik a a (1 / a ) a (8) whr p is th prssur in th suspnsion. Rplacing th diffusion constant and th form-factor with appropriat xprssions, and using th quadratic closur approximation, quation (8) rducs to th Bingham rhological law for th shar strss ( 1 componnt of th strss tnsor), as follows: Y 1 Y 0 1 a f H m 4 a (1 / a) (1 / ) a a a 0 a H 1 f 4 a m 1 (9a) (9b) Th apparnt yild strss, Y, is th sum of th contributions coming from th hydrodynamic strss, th xtrnal torqu xrtd on th aggrgats by th applid magntic fild, and th rotary diffusion of th aggrgats, which tnds to forc th orintation distribution to an isotropic stat; ths thr contributions corrspond to th first, th scond and th third trms on th right-hand sid of Eq. (9b). Rplacing th scond statistical momnts in Eq. (9b) by appropriat numrical valus found from Eq. (7), w ar abl to calculat numrical valus of th apparnt yild strss as a function of th magntic fild and th particl volum fraction. Th thortical dpndncy of th thr yild strss componnts on th magntic fild ar shown in Fig. 6a for th MR fluid of particl volum fraction =0. and for th intraction constant, = W s that th strss componnt coming from th rotary diffusion of aggrgats sms to b th most important and th strss componnt coming from th magntic torqu is th last. In figur 6b, w compar th thortical and

21 0 xprimntal fild dpndncis of th total apparnt yild strss for thr diffrnt particl volum fractions. Our modl fits rasonably wll th xprimntal data, and th bst fit is obtaind with th valu =0.045 of th intraction constant. As xpctd, th apparnt yild strss grows monotonically with th magntic fild intnsity du to incrasing magntic intractions both btwn nighboring aggrgats and btwn th particls blonging to th sam aggrgat. Fig.6. Magntic fild dpndncis of th apparnt yild strss in th cas of th magntic fild alignd with th vorticity. Th figur (a) shows thr diffrnt componnts of th apparnt yild strss calculatd with th hlp of Eq. (9b) at th particl volum fraction =0.. Figur (b) illustrats a comparison btwn xprimnts and thory for th total apparnt yild strss at diffrnt particl concntrations. In both figurs th valu =0.045 of th intraction constant is usd, which corrsponds to th bst fit of th thory to th xprimntal data. Th thortical and xprimntal dpndncis of th apparnt yild strss on th particl volum fraction ar prsntd in Fig.7 for th magntic fild, H=18. ka/m. Th thortical curv is calculatd using th valu =0.045 of th adjustabl paramtr, similar to th on usd for th fild dpndncy of th apparnt yild strss [cf. Fig. 6b]. Both th xprimnts and th thory show a narly quadratic growth of th apparnt yild strss with th concntration; th powr-law dpndncy, Y., is bttr distinguishd in th inst of th Fig.7 whr th apparnt yild strss is plottd vrsus.. Such a powr-law

22 1 concntration dpndncy can b xplaind by a combination of two ffcts as follows. Firstly, in th smi-dilut rgim, th strss is roughly proportional to th concntration of aggrgats, and thus, to th particl volum fraction. Scondly, with growing concntration, th fluctuations of th aggrgat orintation bcom mor important, rsulting in an incras of th statistical momnt 1 [cf. Fig. 5b]. Sinc th apparnt yild strss is also a growing function of 1 [cf. Eq.9b], both ffcts lad to a concntration dpndncy strongr than linar. Fig. 7. Exprimntal and thortical dpndncis of th apparnt yild strss on th particl volum fraction in th cas of th magntic fild alignd with th vorticity. Th magntic fild intnsity is qual to H=18. ka/m and th intraction constant is =0.045 Not that th thortical rsults for th concntratd suspnsions hav bn obtaind considring a smi-dilut rgim and nglcting vntual collisions btwn aggrgats. Such approximations apparntly do not induc mislading rsults and allow us to captur th main physics bhind th unxpctd apparanc of th apparnt yild strss in th magntic fild alignd with th vorticity. Howvr, th thory should b improvd in th futur by considring lubrication contact forcs btwn aggrgats and thir collisions. Not that collisions btwn rod-lik particls may also rsult in rotary diffusion, oftn rportd for concntratd suspnsions of non-brownian fibr suspnsions [cf. original papr of Folgar and Tuckr (1984)]. Both our mchanism and th collision-inducd rotary diffusions lad to th sam scaling dpndncis for th diffusion constant, D. Howvr, th physics is quit diffrnt in both cass. In th cas of fibr suspnsions, th random orintation fluctuations ar only dictatd by th collision rat btwn particls proportional to th shar rat, whil in our cas, it is th intrplay btwn long-rang dipolar forcs and shar forcs, which producs th sam scaling. It is worth mntioning that a similar rotational diffusion concpt has bn prviously mployd to xplain th unxpctdly high MR ffct in th magntic fild paralll to th flow through capillaris [Kuzhir t al. (011)]. W xpct that th mchanism of orintation fluctuations is also inhrnt to th MR rspons in th magntic fild paralll to th vlocity r

23 gradint (and prpndicular to th walls). In this latr cas, th dynamic yild strss is convntionally attributd to viscous dissipation on th aggrgats inclind with rspct to th flow. Convntional modls considr a stabl stady-stat aggrgat orintation dfind by th quilibrium of th magntic and hydrodynamic torqus. This approach usually lads to rlativly small angls btwn th aggrgats and th flow and, consquntly, to undrstimatd valus of th apparnt yild strss in th prpndicular fild [Bossis t al. (00b)]. In rality, th aggrgat orintation is xpctd to undrgo fluctuations bcaus of magntic intractions and collisions btwn thm. This ffct should lad to a mor isotropic orintation distribution and to highr strss lvls as compard to thos prdictd by classical thoris. Anothr important ffct, which may influnc th rhology at any fild orintation, is connctd to th rorganization of th intrnal aggrgat structur undr flow. Th hydrodynamic tnsil forcs tnd to xtnd th aggrgat along thir major axis. Whn th aggrgats ar progrssivly misalignd from th dirction of th applid fild, th tnsil forcs incras and caus an incras of th intrnal mchanical strsss. This can induc a progrssiv ruptur of th intrparticl contacts and dcras th magntic cohsiv forc with th aggrgat rotation. A similar ffct occurs in th absnc of flow, during th lastic dformation of th gap-spanning particl structurs as thy ar xtndd by th motion of th rhomtr plats. In th prsnc of flow, this ffct can modify th forc balanc (4) and, consquntly, th aspct ratio of th aggrgats, as wll as thir orintation distribution. Thrfor, furthr improvmnt of th thory should includ th contribution of local magntic forcs btwn pairs of particls rlatd to th xtnsion and collisions btwn aggrgats as wll as gnralization to all orintations of th magntic fild. V. Conclusions In this work, w hav studid th MR fluid rhology in th magntic fild paralll to th fluid vorticity. Exprimntally, th MR fluid flow was inducd in th Coutt coaxial cylindr gomtry with th magntic fild paralll to th symmtry axis. Th rhological masurmnts wr compard to thos obtaind in th con-plat gomtry with th magntic fild prpndicular to th lowr rhomtr plat. Exprimnts rvald a strong shar-thinning bhavior in both gomtris with th strss lvl bing just a fw dozns of prcnt smallr in th Coutt cylindrical gomtry at th sam intrnal magntic fild. This rsult agrs qualitativly with som prliminary xprimnts rportd by Bossis t al. (00a). Th main goal of th prsnt work is th xplanation to th unxpctdly high MR rspons in th magntic fild paralll to th fluid vorticity. Th ky hypothsis is that th particl aggrgats ar not prfctly alignd with th vorticity, but undrgo stochastic fluctuations of thir positions and orintations. Ths fluctuations ar inducd by magntic intractions btwn aggrgats. Irrgular spacing btwn th moving aggrgats along with thir polydisprsity impart a stochastic natur to fluctuations. Onc misalignd from th vorticity dirction, th aggrgats gnrat high hydrodynamic and diffusion strsss, th latr bing th masur of th stochastic intraction

24 torqu. Sinc th aggrgat aspct ratio, r, is a dcrasing function of th shar rat, th strss gnratd by th aggrgats appars to b indpndnt of (in th rang of whr r 1) and it is considrd as th suspnsion apparnt (dynamic) yild strss. Quantitativly, th fluctuations of th aggrgat orintation ar modld as a rotary diffusion procss with a diffusion constant proportional to th man squar intraction torqu. Th orintation distribution is found from th quations rlating th scond and th fourth statistical momnts using th quadratic closur approximation. Th aggrgat man aspct ratio is found from th balanc of th hydrodynamic tnsil and magntic cohsiv forcs acting on particls constituting th aggrgats. Th modl givs a satisfactory agrmnt with th xprimntal fild and concntration dpndncis of th dynamic yild strss, mploying a singl adjustabl paramtr. Th rathr strong concntration dpndncy, rvald by th xprimnts, is xplaind by incrasing intractions btwn aggrgats lading to strongr misalignmnts from th vorticity dirction with a growing particl concntration. Furthr dvlopmnt of th thory will b rlatd to th modling of th ruptur of contacts btwn particls insid th aggrgats du to th xtnsion and collisions btwn th aggrgats subjctd to hydrodynamic forcs, as wll as gnralization of th thory to all orintations of th magntic filds. Th practical intrst of this study lis in th dvlopmnt of MR smart dvics with th magntic fild non-prpndicular to th channl walls. Acknowldgmnts This work has bn supportd by Projcts P09-FQM-4787 (Junta d Andalucía, Spain), Factoris of th Futur (Grant No. 6007, DynExprt FP7) and PICS projct: Magntic nanocomposits for mchanical and biological applications with Ural Fdral Univrsity, Russia. In addition, L. Rodríguz-Arco acknowldgs financial support by Scrtaría d Estado d Educación, Formación Profsional y Univrsidads (MECD, Spain) through its FPU and Estancias Brvs programs. Appndix A. Magntic prmability of th aggrgatd suspnsion Following th work of Gómz-Ramírz t al. (011), w assum that th applid magntic fild crats cylindrical aggrgats of vry high aspct ratio r >>1, alignd with th applid magntic fild in th absnc of shar. Th diagonal componnts of th magntic prmability tnsor ( along th aggrgat s major axis and along th minor axis) ar found using th Maxwll-Garntt man fild thory [Brthir (199)]: 1 a / a, (A.1) a 1 / a (A.) 1 / a a Y.

25 4 Rcall that a /6 is th intrnal volum fraction of th aggrgats and th aggrgat magntic suscptibility a is givn as a function of th intrnal magntic fild by th formula citd blow Eq. (4). For calculation of th intrnal magntic fild in th cylindrical Coutt gomtry, w shall nglct disprsion of th aggrgat orintation with rspct to th dirction of th applid fild. Undr such a condition, th axial (along th symmtry axis) and th radial componnts of th magntic prmability tnsor ar simply givn by: zz and rr. In th con-plat gomtry, th aggrgats ar misalignd from th applid fild by an angl, whos valu appars to b indpndnt of th shar rat (in th high-shar rgim) and givn by [Gómz-Ramírz t al. (011)]: f tan m 0H a a 1 / 1 / a a, (A.) whr th magntic forc btwn particls f m pr unit particl cross sction is dfind as a function of th intrnal fild by th formula givn blow Eq. (4). Not that in th original papr of Gómz-Ramírz t al. (011), Eq. (A.) is writtn in a slightly diffrnt form [Eq. (6) in thir papr], in which th magntic forc f m is alrady rplacd by a spcific xprssion. Th axial and radial componnts of th magntic prmability tnsor in th con-plat gomtry ar rlatd to th main componnts and and to th angl by th following xprssions: zz cos sin rr tan 1 tan, (A.4), (A.5) whr tan can b dirctly substitutd into Eq. (A.4) from Eq. (A.). Appndix B. Distribution of th intrnal magntic fild Th magntic fild distribution insid th MR sampl is found by numrical solution of Maxwll magntostatic quations using a finit lmnt mthod and with th hlp of FEMM softwar [Mkr (009)]. In simulations w us th ral gomtry of th MR sampl and ral paramtrs of th coil applying th xtrnal magntic fild H 0 (magntic fild in th cntr of th mpty coil), as wll as th valus of th axial and radial componnts, zz and rr, of th suspnsion magntic prmability tnsor, which hav bn stimatd in Appndix A. Figur B1a shows th magntic fild distribution in th cylindrical Coutt gomtry at H 0 =0.6 ka/m (corrsponding to an intnsity of th lctric currnt of th coil qual to I=5A) and at th particl volum fraction =0.. W rmark that th magntic fild is

26 5 rlativly homognous insid th masuring gap and appars to b rathr clos to th xtrnal fild H 0. This is clarly sn in Fig. B1b, whr th fild distribution in th masuring gap along th axial dirction is compard to th xtrnal fild. At th sam tim, w confirm that th magntic fild in th rgions blow and abov th masuring gap is a fw tims lowr that th fild insid th gap. Intrstingly, th avrag valu of th magntic fild insid th gap sms to b rathr insnsitiv to th magntic prmability valu, zz =5.4 (structurd suspnsion) or zz =.7 (unstructurd suspnsion), whil th diffrnc btwn th filds insid th gap and in th MR sampl rgions abov or blow th gap is highr for a highr magntic prmability. Fig. B1. Magntic fild distribution in th cylindrical Coutt gomtry at th xtrnal fild orintd coaxially with th gomtry axis of symmtry: D distribution (a) and 1D distribution along th vrtical axis in th middl btwn th cylindrical surfacs of th masuring gap (b). Th lctric currnt supplid to th coil is I=5A; th

27 6 magntic fild in th cntr of th mpty coil is H 0 =0.6 ka/m; th particl volum fraction of th MR suspnsion is =0.. Figur Ba shows th magntic fild distribution in th con-plat gomtry at th sam valus of H 0 and as in Fig. B1. Th magntic fild also sms to b rathr uniform in th masuring gap. As sn in Fig. B1b, its avrag valu is quit clos to th stimatd on, H=H 0 / zz. Fig. B. Magntic fild distribution in th con-plat gomtry with th xtrnal fild orintd coaxially with th gomtry axis of symmtry: D distribution (a) and 1D distribution along th radial dirction in th middl btwn th lowr surfac and th flat part of th uppr surfac of th masuring gap (b). Th lctric currnt supplid to th coil is I=5A; th magntic fild in th cntr of th mpty coil is H 0 =0.6 ka/m; th particl volum fraction of th MR suspnsion is =0..