Spin-Up Flow of Ferrofluids: Asymptotic Theory And

PHYSICS OF FLUIDS 20, 053102 ͑2008͒

Spin-up flow of ferrofluids: Asymptotic theory and experimental measurements Arlex Chaves,1 Markus Zahn,2 and Carlos Rinaldi1 1Department of Chemical Engineering, University of Puerto Rico, Mayagüez Campus, P.O. Box 9046, Mayagüez PR 00681-9046, Puerto Rico 2Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA ͑Received 28 May 2007; accepted 1 February 2008; published online 28 May 2008͒ We treat the flow of ferrofluid in a cylindrical container subjected to a uniform rotating magnetic field, commonly referred to as spin-up flow. A review of theoretical and experimental results published since the phenomenon was first observed in 1967 shows that the experimental data from surface observations of tracer particles are inadequate for the assessment of bulk flow theories. We present direct measurements of the bulk flow by using the ultrasound velocity profile method, and torque measurements for water and kerosene based ferrofluids, showing the fluid corotating with the field in a rigid-body-like fashion throughout most of the bulk region of the container, except near the air-fluid interface, where it was observed to counter-rotate. We obtain an extension of the spin diffusion theory of Zaitsev and Shliomis, using the regular perturbation method. The solution is ␣ Ӷͱ / ␣ rigorously valid for K 3 2, where K is the Langevin parameter evaluated by using the applied field magnitude, and provides a means for obtaining successively higher contributions of the nonlinearity of the equilibrium magnetization response and the spin-magnetization coupling in the magnetization relaxation equation. Because of limitations in the sensitivity of our apparatus, experiments were carried out under conditions for which ␣ϳ1. Still, under such conditions the predictions of the analysis are in good qualitative agreement with the experimental observations. An estimate of the spin viscosity is obtained from comparison of flow measurements and theoretical results of the extrapolated wall velocity from the regular perturbation method. The estimated value lies in the range of 10−8–10−12 kg m s−1 and is several orders of magnitude higher than that obtained from dimensional analysis of a suspension of noninteracting particles in a Newtonian fluid. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2907221͔

I. INTRODUCTION free surface tracers limits the accuracy of measurements A. Spin-up flow of ferrofluids close to the container wall. Furthermore, interfacial curvature coupled with magnetic surface shear stresses drives a surface The phenomenon of spin-up flow of a ferrofluid, a stable flow,7 making free surface measurements unsuitable in deter- colloidal suspension of permanently magnetized nanopar- mining the bulk flow. ticles, in a cylindrical container and subjected to an applied Related works using either unstable or concentrated sus- rotating magnetic field which is uniform in the absence of pensions of particles or using non-uniform magnetic fields the ferrofluid has received considerable attention in the de- include Brown and Horsnell,15 Kagan et al.,16 and Calugaru velopment of the field of ferrohydrodynamics.1–14 Typically, et al.17 These authors report observations where the magnetic experiments are carried out by placing a sample of ferrofluid fluid switches between co-rotation and counter-rotation with in a cylindrical container subjected to a rotating magnetic respect to the applied magnetic field depending on magnetic field. This setup is suitable for studying the coupling between field amplitude and frequency. Explicitly, Brown and internal angular momentum, related to the “spin” of the sub- 15 continuum magnetic particles, and the macroscopic or “ex- Horsnell observed corotation of field and fluid for low applied fields and counter-rotation for high applied magnetic ternal” angular momentum, related to the experimentally ob- 16 17 servable macroscopic flow of the suspension. In a stationary fields, whereas Kagan et al. and Calugaru et al. observed cylindrical container, surface velocity profiles show the fer- counter-rotation for low applied magnetic fields and corota- rofluid rotating, rigid-body-like, in a direction which depends tion for higher applied fields. on the applied magnetic field amplitude, frequency, and ro- Observations of counter-rotation of field and fluid led 15 tation direction. Such essentially rigid-body motion is ob- Brown and Horsnell to investigate the direction in which served in the inner core of the ferrofluid and extends close to the cylindrical container would rotate if it could freely do so. the stationary cylindrical vessel wall. To make such observa- This represents an indirect measurement of the torque ex- tions, tracer particles are often placed on the rotating free erted by the ferrofluid on the container wall. One would ex- surface—ferrofluid is opaque making probing of bulk flow pect the counter-rotating fluid to drag the cylindrical con- profiles through optical methods difficult. The usage of such tainer with it, but experiments show the container corotating

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with the field, whereas the fluid counter-rotates. Such obser- winding distribution is used as the source of the applied vations have since been corroborated by Kagan et al.16 and magnetic field. Hence, according to Glazov, experimental ob- Rosensweig et al.7 servations of spin-up flow are entirely attributed to imperfec- Various authors1–5,7,10,11,14 have attempted theoretical tions in the experimental apparatus and the concomitant analyses aimed at explaining experimental observations of higher-order spatial harmonics in the magnetic field distribu- free surface flow profiles. The spin diffusion theory, due to tion. Still, the predictions of Glazov fail to explain the ex- Zaitsev and Shliomis,2 assumes that the magnetic field perimentally observed counter-rotation of field and ferrofluid throughout the ferrofluid region is uniform, with concomitant at the free surface.7 uniform magnetization of the ferrofluid. The resulting At least three other hypotheses have subsequently been magnetic body couple is uniform, whereas the magnetic advanced in the literature to explain the observed phenomena body force is exactly zero. Within the assumptions of their ͑i͒ that the flow is entirely driven by free surface curvature analysis, the ferrofluid flow field is analytically determined and surface excess forces;7 ͑ii͒ that the observed phenom- by using the phenomenological structured continuum enon results due to a shear stress dependent slip boundary theory18–20 which includes the effects of antisymmetric condition26 on the ferrofluid translational velocity coupled stresses, body couples, and couple stresses representing the with a spin-slip boundary condition;10,27 and ͑iii͒ that an en- short-range surface transport of internal angular momentum. ergy dissipation mechanism gives rise to temperature gradi- The spin-diffusion results of Zaitsev and Shliomis2 were sub- ents, with concomitant gradients in magnetic properties, re- sequently used to estimate the magnitude of the phenomeno- sulting in magnetic body forces which drive flow in the logical coefficient of spin viscosity, appearing in the com- direction opposite field rotation.14,28 monly accepted constitutive form of the couple stress That surface curvature dependent flows indeed occur in pseudodyadic, as well as to analyze/interpret experimental ferrofluids was demonstrated by the experiments of Rosens- observations.6,7 Unfortunately, as pointed out by Rosensweig weig et al.7 In their study, free surface rotation rate and di- et al.,7 this analysis fails to correctly predict the experimen- rection were measured for the meniscus formed by a dilute tally observed free surface flow magnitude and direction. water based ferrofluid in a 9.5 mm internal diameter capil- The spin-diffusion theory2 and other7,10,11 analyses lary tube subjected to uniform rotating magnetic fields of adapted therefrom assume that the fluid magnetization is pro- increasing amplitude. The shape of the meniscus was portional to the magnetic field, limiting the solution range to switched from concave to convex by modifying the capillary low values of the applied magnetic field where the Langevin surface properties. With a concave meniscus the magnetic parameter ␣, appearing in the equilibrium magnetization re- field and fluid were observed to counter-rotate, whereas mag- lation, is negligible ͑␣→0͒. Also, these analyses neglect or netic field and fluid were observed to corotate with a convex are inconsistent with the spin-magnetization coupling in the meniscus. Additionally, Rosensweig et al.7 observed that the phenomenological magnetization relaxation equation ͑7͒ of angular rotation rate of the free surface increased as the in- Shliomis.21 This can be most clearly seen in Ref. 2 where the ternal diameter of the cylindrical container decreased. This, magnetization equation lacks the ␻͑x͒ϫM͑x͒ term on the they remarked, is contrary to usual expectations in viscous right hand side of Eq. ͑7͒. It is neglected in, or rather is flows. These observations led Rosensweig et al.7 to conclude inconsistent with, the analysis the Rosensweig et al.7 and that “surface stress rather than volumetric stress is respon- Kaloni10 when they assume that the field and magnetization sible for the spin-up phenomenon.” are uniform throughout the ferrofluid. This is illustrated Another explanation was suggested by Kaloni,10 who at- when one considers that a position dependent spin field, as tributed the experimental observations of Ref. 7 to a slip obtained in their contributions, is inconsistent with a uniform velocity boundary condition. Kaloni obtained the transla- magnetic field and magnetization, as assumed in their treat- tional and spin velocity fields between two stationary coaxial ments, in the magnetization relaxation equation ͑7͒. The cylinders subjected to an externally applied uniform rotating importance of the spin-magnetization coupling term magnetic field. However, Kaloni failed to make definite pre- ␻͑x͒ϫM͑x͒ in predicting novel ferrofluid behavior has been dictions based on the magnitude and direction of spin-up shown previously by Zahn and co-workers,22–25 where the flows in ferrofluids under typical experimental conditions. situation of plane-Pouseuille/Couette flow of ferrofluid sub- A third theory, due to Shliomis et al.,28 suggested that jected to alternating and rotating magnetic fields was ana- spin-up flow is driven by nonuniformity of magnetic perme- lyzed. ability due to radial temperature gradients produced by vis- An alternative theoretical explanation for the observed cous dissipation in the microeddies that arise around the ro- flow phenomena was developed by Glazov.3–5 The analysis tating particles, especially at high magnetic field frequencies. considers the effect of higher-order spatial harmonics, ex- This view was analytically extended by Pshenichnikov pected to occur in practice due to nonidealities in the stator et al.,14 predicting counter-rotation of fluid and field for fre- winding distribution, e.g., due to slot effects. Essentially, quencies below 105 s−1͑16 kHz͒ and a maximum of the ve- /ͱ3 ͑ Glazov’s analysis considers the same phenomenological locity profile located at r=RO 3 RO is the radius of the theory used by Zaitsev and Shliomis2 and others,7,10,11 but outer container͒, independent of the properties of the fluid. In neglects contributions due to couple-stresses in the ferrof- addition, Pshenichnikov et al. claimed that for low frequen- luid. Glazov’s analysis, in its various forms and stages, es- cies and amplitudes of the magnetic field the flow is driven sentially concludes that the flow would not occur in the ab- only by tangential magnetic stresses on the free boundary of sence of the higher-order harmonics when a two pole the fluid.

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Chaves et al.29 reported experimental bulk velocity profile measurements obtained by using the ultrasound velocity profile ͑UVP͒ method for ferrofluid under a uniform rotating magnetic field. These measurements showed rigid-body-like azimuthal velocity profiles co-rotating with the field throughout the bulk of the container, in qualitative agreement with the predictions of Ref. 2. Likewise, it was observed that the flow direction at the free surface is opposite to the direction of field rotation as in Ref. 7. These experiments suggest that two different mechanisms operate in driving the flow; surface shear stresses for the fluid near the top free surface and volumetric effects for the bulk. Furthermore, these observations demonstrate that surface flow measurements are inappropriate in the assessment of bulk flow theories. Herein, we present a complete characterization of bulk flow during spin up using the UVP method, with corresponding torque measurements on the surface of the cylindrical container for two different volumes of fluid. Water and kerosene based ferrofluids were used with several volumetric particle fractions obtained by dilution of the more concentrated FIG. 1. Schematic illustration of the coupled ferrohydrodynamic problem commercial ferrofluid. Because the experiments were carried for spin-up flow. The azimuthal velocity component and axially directed out at non-negligible values of the magnetic field, whereas spin velocity are obtained for a long column of ferrofluid of external radius 2 the spin diffusion theory strictly applies in the limit of zero RO subjected to a uniform rotating magnetic field perpendicular to the axis of the cylinder. The magnetic field source is modeled as a z-directed surface field, we begin by presenting an asymptotic solution of the ͑␪ ͒ current distribution K ,t iz, which is backed by a material of infinite mag- ferrohydrodynamic problem, extending the results of Ref. 2. netic permeability, ␮→ϱ.

B. Analytical approach zeroth-and first-order terms in the regular perturbation ex- Motivated by the experimental evidence presented by 29 pansion of all relevant field quantities for the spin-up flow of Chaves et al., where bulk velocity profiles for ferrofluid in a ferrofluid in a cylindrical container when the externally a cylindrical container showed qualitative agreement with applied rotating magnetic field is uniform in the absence of the spin-diffusion theory, we extend this theory using the ferrofluid. The solution procedure consists of determining the regular perturbation method. To this end, the magnetization zeroth-order magnetic quantities, from which the resulting ͑ ͒ relaxation equation 7 used in our analysis includes a term magnetic force and couples are calculated. These are used in O͑␣3͒ of in the equilibrium magnetization expression, Eq. turn to obtain the zeroth-order translational and spin velocity ͑ ͒ ␹ 8 , in contrast to the usual assumption of Meq= iH, which fields. The zeroth-order fields thus obtained are subsequently O͑␣͒ only includes the term of order . In addition, the term used to determine the first-order magnetic field and magne- ␻͑ ͒ϫ ͑ ͒ x M x enters the analysis at first order. tization with concomitant first-order magnetic forces and In order to solve the coupled ferrohydrodynamic equa- couples used in determining the first-order flow fields. These tions, we will apply a regular perturbation expansion in pow- steps illustrate the procedure used to obtain the effects of the ers of the small parameter spin-magnetization coupling to any order in ␧, with conver- gence for ␧Ͻ1. ␮ ␹ K2␶ ␧ ϵ 0 i , ͑1͒ ␨ II. ANALYSIS OF FLOW AND TORQUE ␮ ϵ ␲ϫ −7 / where 0 4 10 H m is the permeability of free space, Conceptually, we consider the experimental situation il- ␹ i is the ferrofluid’s initial susceptibility, K is the amplitude lustrated in Fig. 1, consisting of an infinite column of ferrof- ␶ of the magnetic field, and is the effective relaxation time of luid contained in a nonmagnetic cylinder of radius RO and ⍀ ␶ ␮ the particles. A similar approach, using f as the perturba- magnetic permeability 0 whose outer wall is constrained to tion parameter, was used by Rinaldi and Zahn25 to analyze be stationary due to an externally applied mechanical torque. the problem of plane-Poiseuille/Couette flow of ferrofluid The column of ferrofluid is assumed to be long enough in subjected to externally applied alternating and rotating mag- the z-direction such that end effects may be considered netic fields. There, it was shown that, as a consequence of the negligible. We are therefore solving for the fields in the bulk simplified geometry, the magnetic field and magnetization and thereby eliminating all aforementioned free-surface- could be analytically obtained to any order n independently curvature effects, the analysis and effects of which we leave of the translational and spin velocity field. This ceases to be to future contributions. The cylinder is placed in the gap of a the case in the analogous problem of spin-up flow of ferrof- magnetic field source, which, for definiteness, we choose to luid in a cylindrical container, requiring determination of the be the idealization of a two-pole magnetic machine stator. ͑n−1͒th order fields before obtaining the nth order fields. The stator winding is modeled as a surface current distribu- Thus, in the present contribution we consecutively obtain the tion in the z-direction with real amplitude K͑␪,t͒ located at

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radius r=RO, and which sinusoidally varies with time at ra- D␻ 1 ⍀ ␳I = ␮ M ϫ H −2␨ͩ ٌ ϫ v − ␻ͪ + ␩Јٌ2␻, ͑6͒ dian frequency f and is sinusoidally distributed with angle Dt 0 2 ␪, where I is the moment of inertia density of the suspension ͑⍀ ␪͒ ͑␪ ͒ ͕ j ft−m ͖ ͑ ͒ K ,t =Re Ke iz. 2 and ␩Ј is the coefficient of spin viscosity. The first term on the right hand side of Eq. ͑6͒ represents the external body The current distribution is backed with a materialͱ of infinite couple density acting on the structured continuum whenever magnetic permeability, ␮→ϱ. Here, jϵ −1 is the imagi- the local magnetization is not aligned to the applied magnetic nary number, m is an integer which for a uniform magnetic field. The second term is the antisymmetric vector of the field source is 1, and K is the surface current peak amplitude, Cauchy stress and represents the interchange of momentum taken to be real so that, without loss of generality, we have between internal angular and macroscopic linear forms. The chosen a phase angle of zero for the current distribution. We third term represents the diffusion of internal angular mo- have made the radii of the ferrofluid container and stator mentum between contiguous material elements, which has equal for simplicity in the analysis. An air gap would have been generally neglected by fixing the coefficient of spin introduced another region where Maxwell’s equations, in the viscosity to zero. This assumption results in zero flow, as can magnetoquasistatic limit, would have to be applied and be easily verified from the analysis presented below. solved. The suspension magnetization M represents the local alignment of subcontinuum units and as such must also obey A. Governing equations a balance equation. The original equation21 has been the sub- 32–37 We use the set of ferrohydrodynamic equations summa- ject of much recent debate, but is still regarded as correct rized in Refs. 25 and 30. The first is the continuity equation, for low applied field amplitude and frequency, i.e., when the which assumes that the ferrofluid is a homogeneous dilute local magnetization is not far from its equilibrium value. The colloidal dispersion of magnetic particles ͑subcontinuum magnetization relaxation equation, as it is usually called, is units͒ in an incompressible liquid carrier, then M 1ץ v = 0, ͑3͒ + ٌ · ͑vM͒ = ␻ ϫ M − ͑M − M ͒. ͑7͒ · ٌ t ␶ eqץ where v is the mass average velocity of the fluid. The second In Eq. ͑7͒, the first term on the right hand side represents the expresses conservation of linear momentum, effect of particle rotation on M͑x͒, whereas the second term M͑x͒ Dv 1 accounts for orientational diffusion of toward an equi- ␳ ␳ ␮ ٌ ٌ ␨ ٌ ϫ ͩ ٌ ϫ ␻ͪ = g + 0M · H − p +4 v − librium value. The equilibrium magnetization Meq for a su- Dt 2 perparamagnetic ferrofluid is given by the Langevin 11 ,␩ٌ2v. ͑4͒ relation + M H 1 H ␮ M HV This equation takes into account the asymmetry of the vis- eq = L͑␣͒ = ͫcoth ␣ − ͬ , ␣ = 0 d c . ␾ ͉ ͉ ␣ ͉ ͉ cous stress tensor, a consequence of body couples exerted on Md H H kBT the subcontinuum units. In Eq. ͑4͒ ␳ is the density, g is the ͑8͒ gravitational acceleration, p is the pressure, M is the magnetization vector of the suspension, H is the applied magnetic Here, L͑␣͒ is the Langevin function and ␣ is the Langevin field vector, ␨ is the vortex viscosity ͑a dynamical parameter parameter, a measure of the relative magnitudes of magnetic analogous to the shear viscosity and which characterizes the and thermal energy. Note that H is the local amplitude of the asymmetric component of the Cauchy stress͒, ␻ is the spin magnetic field, not to be confused later in our solution with ͑ ͒ velocity vector, and ␩ is the suspension-scale shear viscosity. K, the scale of the field. In Eq. 8 Md is the domain mag- The second term on the right hand side of Eq. ͑4͒ represents netization of the magnetic nanoparticles, kB is Boltzmann’s the magnetic body forces due to field inhomogeneities, constant, T is the absolute temperature, and Vc is the volume whereas the fourth term represents the antisymmetric com- of the magnetic cores. The use of an effective relaxation ponent of the Cauchy stress, occurring when the particles time, rotate at a rate different than half the local vorticity of the 1 1 1 flow. A dilute limit expression for monodisperse nanoparticle = + , ͑9͒ ␶ ␶ ␶ suspensions in Newtonian fluids is available for the vortex B N viscosity, is commonly assumed because relaxation can simultaneously occur by the Brownian and Néel mechanisms. The Brownian ␨ ␾ ␩ ͑ ͒ = 1.5 h 0, 5 relaxation time is given by ␾ ␩ where h is the hydrodynamic volume fraction of suspended 3 0Vh ␩ ␶ = , ͑10͒ particles and 0 is the shear viscosity of the suspending B k T fluid.31 B The spin velocity in the ferrofluid is governed by the with Vh the hydrodynamic particle volume, and the Néel re- internal angular momentum equation, laxation time is given by

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20 1 K V Dahler and is a simple ad hoc alternative, with a single ␶ ͩ a c ͪ ͑ ͒ N = exp , 11 adjustable parameter ␥ to the general tensorial boundary con- f0 kBT dition proposed by Aero.27 where f0 is a characteristic frequency of the magnetic 11 material, Ka is the magnetocrystalline anisotropy constant C. Scaling of the governing equations of the magnetite nanoparticles, and V is the magnetic c Scaling of the governing equations is required in order to volume. formulate a regular perturbation problem which permits an All that remains to complete our description are analytical solution by decoupling the ferrohydrodynamic Maxwell’s equations in the magnetoquasistatic limit,38,39 equations ͑3͒–͑12͒. Field quantities are scaled with respect to ϫ H = 0, ٌ · ͑M + H͒ = 0, ͑12͒ their expected characteristic values. The resulting scaled ٌ variables are denoted by an overtilde and are assumed to be and the corresponding interfacial boundary conditions for the of order unity, magnetic field, namely, continuity of the normal component of magnetic induction and the jump in the tangential mag- M H r ,M˜ = , H˜ = , ٌ˜ = R ٌ , ˜r = , ˜t = ⍀ t netic field due to surface currents, ␹ O f iK K RO ͓͑ ͒ ͑ ͒ ͔ ͑ ͒ n · H + M a − H + M b = 0, 13 p v ␻ ˜p = , ˜v = , ␻˜ = , ͑17͒ ϫ ͓ ͔ ͑ ͒ ⌸ ␼ n Ha − Hb = K, 14 Ul ␼ in which n is a unit vector locally normal to the interface where Ul and are the characteristic translational and spin pointing from phase b to phase a. velocities44 and ⌸ is the characteristic pressure assumed to be ␩␼. As the linear velocity is driven by rotation of the B. Fluid boundary conditions particles, we have chosen for the scale of the linear velocity U =␼R . The characteristic spin velocity is then determined In an attempt to provide an alternate explanation for the l O by substituting the scaled variables of Eq. ͑17͒ in the internal observations of Rosensweig et al.,7 Kaloni10 considered the angular momentum equation ͑6͒, neglecting the moment of effect of different boundary conditions on the translational inertia term. Grouping terms we obtain and spin velocity fields, in the framework of the Zaitsev and Shliomis2 analysis. Boundary conditions in structured con- ␮ ␹ K2 ␩Ј i M˜ ϫ H˜ +2ٌ˜ ϫ ˜v −4␻˜ + ٌ˜ 2␻˜ . ͑18͒ 0 = 0 tinuum theories have been under debate for a long ␨␼ ␨ 2 RO time.20,26,27,40–42 The boundary conditions used by Kaloni10 are a coefficient-of-friction ␤ dependent slip velocity, The M˜ ϫH˜ term in Eq. ͑18͒ is not order unity, as the cross product is order sin ␦ with ␦ the lag angle between magne- 1 v − v = ͓n ϫ ͑n · T ϫ n͔͒ ͑15͒ tization and magnetic field. Under conditions for which s ␤ ⍀ ␶Ӷ ␦ϳ⍀ ␶ f 1, which usually apply in ferrofluids, sin f . and a generalized spin-slip condition, Hence, to make the first term on the right hand side of Eq. ͑18͒ of order unity, we choose as the characterisitc spin ve- ␥ locity ␻ − ␻ = ٌ ϫ v. ͑16͒ s 2 ␮ ␹ K2⍀˜ ␼ = 0 i f , ͑19͒ Here n is an outward-directed normal unit vector at the ␨ ␻ boundary, T the Cauchy stress tensor in the fluid, vs and s are the wall linear and spin velocities, respectively, and ␥෈ and consequently ͓0,1͔ is an adjustable coefficient. ␨p ␨␻ ␨v The translational slip velocity boundary condition ͑15͒ ˜p = , ␻˜ = , ˜v = . 43 ␮ ␩␹ 2⍀˜ ␮ ␹ 2⍀˜ ␮ ␹ 2⍀˜ dates back to Lamb’s arguments of slip being opposed by 0 iK f 0 iK f 0 iK fRO frictional forces at the wall and has been championed by ͑20͒ various authors.26,27 However, as the slip “friction coefficient” ␤ is inversely proportional to particle radius,26 these The resulting scaled governing equations, neglecting inertial effects should be negligible in ferrofluids, where characteris- terms in the linear momentum equation, are tic particle diameters are of the order of 10 nm. ␨ ␨ ␩ The spin-slip boundary condition ͑16͒ is a linear combi- 0 = M˜ · ٌ˜ H˜ − ٌ˜˜p +2 ٌ˜ ϫ ␻˜ + e ٌ˜ 2˜v, ͑21͒ ␩⍀˜ ␩ ␩ nation of the two cases of zero spin and zero asymmetric f stress boundary conditions and is the subject of even further controversy.20,26,27,40–42 The first case ͑␥=0͒ corresponds to 1 4␩ M˜ ϫ H˜ +2ٌ˜ ϫ ˜v −4␻˜ + ٌ˜ 2␻˜ , ͑22͒ = 0 ␩ ␬2 the possibility of no spin slip of the particles near the wall ⍀˜ e due to strong particle-wall interaction. In the second case f ͑␥=1͒, the particles rotate at the local vorticity of the fluid, M˜ Mץ hence antisymmetric stresses are absent in the wall. This ⍀˜ + ⍀˜ ␧˜v · ٌ˜ M˜ = ⍀˜ ␧␻˜ ϫ M˜ − M˜ + eq , ͑23͒ f f f ␹ t iK˜ץ boundary condition was initially proposed by Condiff and

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␧ -ϫ H˜ = 0, ͑24͒ Thus, choosing as the perturbation parameter allows de ˜ٌ coupling of the equations as well as considering the effects of non-negligible magnetic fields. ͒ ͑ ͒ ˜ ˜ ␹͑ ˜ٌ · iM + H = 0. 25 ␩ ϵ␩ ␨ D. Zeroth order problem In the preceding equations, we have defined e + and introduced the dimensionless parameter ␬, In order to apply the regular perturbation method, we expand all field quantities in powers of the parameter ␧ and 4␩R2 ␨ ␬2 ϵ O . ͑26͒ group coefficients of like powers of ␧ to obtain the set of ␩ ␩Ј e equations for each order of the problem. The corresponding ͑ ͒ ͑ ͒ ⍀˜ set of equations for the zeroth order problem are As can be observed in Eq. 21 – 23 , the parameter f ͒ ͑ ˜ could also be used as a perturbation parameter, as done by ٌ˜ ϫ H0 =0, 32 Rinaldi.45 Though that would permit decoupling of the equations, the resulting solution would still be limited to the case ␹ M˜ + H˜ ͒ =0, ͑33͒͑ · ˜ٌ ␣→0 as the equilibrium magnetization would have to be i 0 0 assumed linear with the applied field. For the case of a dilute ˜Mץ ͑␾→0͒ ferrofluid, the perturbation parameter ␧ is related to ⍀˜ 0 ˜ ˜ ͑ ͒ f = H0 − M0, 34 t˜ץ -the Langevin parameter ␣, as can be shown using the follow ing expression for the initial susceptibility of a dilute ferrofluid,11 ␨ 2␨ ␩ M˜ · ٌ˜ H˜ − ٌ˜˜p + ٌ˜ ϫ ␻˜ + e ٌ˜ 2˜v, ͑35͒ = 0 0 0 0 ␩ 0 ␨ 0 ␲␾␮ M2d3 ⍀˜ ␩ ␹ 0 d ͑ ͒ f i = , 27 18kBT 1 4␩ M˜ ϫ H˜ +2ٌ˜ ϫ ˜v −4␻˜ + ٌ˜ 2␻˜ . ͑36͒ = 0 the Brownian relaxation time ͑10͒, and the vortex viscosity 0 0 0 0 ␩ ␬2 ⍀˜ e ͑5͒, obtaining f 2 2 ␣2 As can be seen, the zeroth order magnetic field and magne- ␧ ␣2 ͑ ͒ = K = , 28 tization can be independently solved of the hydrodynamic 3 3 ͉H˜ ͉2 problem. ␣ where K is the Langevin parameter evaluated for the characteristic magnetic field amplitude K, 1. Zeroth order magnetic field and magnetization ␮ M KV ␣ 0 d c ͑ ͒ K = . 29 We assume that the magnetic field and magnetization kBT have a functional form similar to Eq. ͑2͒ with m=1, Note that ␧ is a phenomenologically valid perturbation pa- ˜ ͑ ␪ ˜͒ ͕͓ ˆ ͑ ͒ ˆ ͑ ͒ ͔ j͑˜t−␪͖͒ ͑ ͒ rameter, not limited to dilute ferrofluids, as it has been de- H0 ˜r, ,t =Re Hr,0 ˜r ir + H␪,0 ˜r i␪ e , 37 rived from the phenomenological equations describing ferro- ˜ ͑ ␪ ͒ ͕͓ ˆ ͑ ͒ ˆ ͑ ͒ ͔ j͑˜t−␪͖͒ ͑ ͒ hydrodynamics and is not a function of position. The M0 ˜r, ,˜t =Re Mr,0 ˜r ir + M␪,0 ˜r i␪ e , 38 Langevin parameter ␣ would not be an appropriate perturba- where the hat denotes a complex quantity. By using Eq. ͑38͒ tion parameter as it would depend on the local magnetic in Eq. ͑34͒, we obtain expressions for the zeroth order mag- field, and hence on position. Furthermore, the connection netization components as a function of the zeroth order com- ͑28͒ between ␧ and ␣ is only applicable in the dilute limit K ponents of the magnetic field, for which Eqs. ͑5͒, ͑10͒, and ͑27͒ apply, hence we choose to ␧ continue our solution in terms of the parameter . Hˆ Hˆ ˆ ͑ ͒ ␪,0 ˆ ͑ ͒ r,0 ͑ ͒ Expanding the Langevin function in a power series in ␣ M␪,0 ˜r = , Mr,0 ˜r = . 39 ͑ ͒ 1+j⍀˜ 1+j⍀˜ and using Eq. 28 , the Meq term can be expressed as a f f ␧ function of the perturbation parameter , Now, substituting this in Eq. ͑33͒ and using Eq. ͑32͒,we M ␧ ␧2 obtain the following system of differential equations for the eq = H˜ ͫ1− ͉H˜ ͉2 + ͉H˜ ͉4 + O͑␧3͒ͬ. ͑30͒ ␹ azimuthal and radial components of the magnetic field: iK 10 70 d͑˜rHˆ ␪ ͒ Equation ͑30͒ is substituted in Eq. ͑23͒ to obtain ,0 + jHˆ =0, ͑40͒ dr˜ r,0 ˜Mץ ˜⍀˜ + ⍀˜ ␧˜v · ٌ˜ M ͒ ˆf d͑˜rH ˜ץ f t r,0 ˆ ͑ ͒ = jH␪,0, 41 ⍀˜ ␧␻ ϫ ˜ ˜ ˜ dr˜ = f ˜ M − M + H ␧ ␧2 with solution given by ϫͫ1− ͉H˜ ͉2 + ͉H˜ ͉4 + O͑␧3͒ͬ. ͑31͒ ˆ −2 ͑ ͒ 10 70 Hr,0 = C1,0 + C2,0˜r , 42

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ˆ ͑ −2͒ ͑ ͒ 1 d d␻˜ ␬2 ␩ ␬2 H␪,0 =−j C1,0 − C2,0˜r . 43 ͩ z,0ͪ ␬2␻ e ͑ ͒ ˜r − ˜ z,0 =− C1 − . 50 ˜r dr˜ dr˜ 2 4␩͑1+⍀˜ 2͒ Because the solution range includes ˜r=0, the constant C2,0 f must be zero, so that the magnetic field remains finite at the The corresponding solution is origin. The constant C1,0 is obtained using the tangential ͑ ͒ C ␩ jump condition, Eq. 14 written in dimensionless form, ␻ ͑ ͒ 1 ͑␬ ͒ e ͑ ͒ ˜ z,0 ˜r = + C2I0 ˜r + , 51 2 ␩͑ ⍀˜ 2͒ ˆ ͑ ͒ ͑ ͒ 4 1+ f H␪,0 ˜r =1 =−1. 44 ͑ ͒ The resulting expressions for the complex amplitudes of the where I0 x is the modified Bessel function of first kind, zeroth order magnetic field are order zero. The constant that accompanies the modified ͑ ͒ Bessel function of the second kind of order zero, K0 x , has ˆ ˆ ͑ ͒ Hr,0 =−j, H␪,0 =−1. 45 been fixed equal to zero so that the solution remains finite at ˜r=0. Using Eq. ͑51͒ in Eq. ͑49͒ and integrating, we can This result demonstrates that the zeroth order magnetic field determine the zeroth order translational velocity. The expres- is uniform in the cylinder gap. sions for the constants C1 and C2 were determined using the boundary conditions ͑15͒ and ͑16͒ to obtain 1 I ͑␬˜r͒ I ͑␬͒ 2. Zeroth order magnetic body force and body couple ␻˜ ͑˜r͒ = ͭ␩ ͫ1− 0 ͬ − ␥␨ 2 ͮ, z,0 e ͑␬͒ ͑␬͒ 4␩*͑1+⍀˜ 2͒ I0 I0 With expressions for the zeroth order magnetization ͑39͒ f ͑ ͒ and magnetic field ͑45͒, we obtain expressions for the mag- 52 netic body force and magnetic body couple. Since the zeroth ␨ I ͑␬͒ I ͑␬˜r͒ order magnetic field is uniform, the magnetic body force ͑ ͒ 1 ͩ 0 ͪ ͑ ͒ ˜␪ ˜r = ˜r − . 53 v ,0 ͑␬͒ ͑␬͒ M˜ ٌ˜ H˜ ␬␩*͑ ⍀˜ 2͒ I0 I1 0 · 0 is clearly zero. The zeroth order body couple is 2 1+ f ⍀˜ In Eqs. ͑52͒ and ͑53͒, we have defined the parameter ␩* as ˜ ͑ ˜ ϫ ˜ ͒ f ͑ ͒ lz,0 = M0 H0 z = . 46 1+⍀˜ 2 I ͑␬͒ f ␩* ϵ ␩ − ␨͑␥ −1͒ 2 . ͑54͒ ͑␬͒ I0 Therefore, the linear and spin velocity fields are intimately 3. Zeroth order linear and spin velocity fields related to the value of ␬ and therefore to the value of the spin viscosity ␩Ј.As␩Ј→0 ͑or ␬→ϱ͒, the translational velocity The zeroth order magnetic body force is zero, whereas tends to zero, whereas the spin velocity becomes uniform in the zeroth order body couple is constant. Thus, to obtain the the gap. Comparison between Eq. ͑53͒ and the expression translational and spin velocity fields, we solve the linear and reported by Ref. 2, shows that our zeroth order analysis cor- ͑ ͒ ͑ ͒ internal angular momentum equations 35 and 36 , assum- responds to the spin diffusion result, albeit allowing a more ing steady flow in axisymmetric cylindrical coordinates. The general boundary condition for the spin velocity. ͑ ͒ ␻ ͑ ͒ resulting equations governing ˜v␪,0 ˜r and ˜ z,0 ˜r are the azimuthal component of the zeroth order linear momentum, E. First order problem

␩ d 1 d͑˜r˜v␪ ͒ 2␨ d␻˜ e ͫ ,0 ͬ − z,0 =0, ͑47͒ The set of equations for the ﬁrst order problem are ob- ␩ dr˜ ˜r dr˜ ␩ dr˜ tained using the same procedure as for the zeroth order problem, resulting in and the z-component of the zeroth order internal angular mo- ͒ ͑ ˜ mentum, ٌ˜ ϫ H1 = 0, 55

1 d d␻˜ 4␩ 2 d͑˜r˜v␪ ͒ 1 ͩ˜r z,0ͪ + ,0 −4␻˜ + =0. ␩ ␬2 z,0 ٌ˜ · ͑␹ M˜ + H˜ ͒ = 0, ͑56͒ ˜r dr˜ dr˜ e ˜r dr˜ ⍀˜ 2 i 1 1 1+ f ͑48͒ ˜Mץ ˜ ˜ٌ ˜⍀˜ 1 ⍀ ͑ ͒ ␻ ͑ ͒ f + f˜v0 · M0 t˜ץ In order to solve for ˜v␪,0 ˜r and ˜ z,0 ˜r , we introduce the vorticity ⍀˜ of the zeroth order translational velocity in Eq. z,0 ˜ ͑47͒ and then carry out the integration to yield an expression H0 = ⍀˜ ͑␻˜ ϫ M˜ ͒ − M˜ + H˜ − ͉H˜ ͉2, ͑57͒ for the zeroth order vorticity, f 0 0 1 1 10 0 2␨ ␩ ⍀˜ = ␻˜ + C , ͑49͒ ␨ ͒ ˜ ˜ٌ ˜ ˜ ˜ٌ ˜ ͑ z,0 ␩ z,0 ␩ 1 e e 0 = M0 · H1 + M1 · H0 ␩⍀˜ f which we replace in Eq. ͑48͒ to yield a nonhomogeneous 2␨ ␩ modiﬁed Bessel equation of order zero for the z-component − ٌ˜˜p + ٌ˜ ϫ ␻˜ + e ٌ˜ 2˜v , ͑58͒ of spin velocity, 1 ␩ 1 ␨ 1

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1 2. First order magnetic body force and body couple ͑ ˜ ϫ ˜ ˜ ϫ ˜ ͒ 0 = M0 H1 + M1 H0 ⍀˜ f The first order magnetic body force is 4␩ ͒ ͑ ˜ ˜ٌ ˜ ˜ ˜ٌ ˜ ͒ ͑ ˜ ϫ ˜v −4␻˜ + ٌ˜ 2␻˜ . ͑59͒ ˜+2ٌ 1 1 ␩ ␬2 1 f1 x = M0 · H1 + M1 · H0. 67 e In the same form as for the zeroth order problem, the mag- Owing to the homogeneity of the zeroth order magnetic field, netic field and magnetization can be independently obtained the second term of Eq. ͑67͒ is clearly zero. As can be shown of the hydrodynamic problem. The solution strategy will be by taking the curl of this equation, and using the fact that the similar to the zeroth order problem. zeroth order magnetic field is uniform, the first order body force is potential, 1. First order magnetic field and magnetization ͒ ˜ Focusing on the first order magnetization relaxation Eq. ٌ˜ ϫ˜ ͑ ͒ ٌ˜ ϫ ͑ ˜ ٌ˜ ˜ ͒ ˜ ٌ˜ ٌ͑˜ ϫ f1 x = M0 · H1 = M0 · H1 = 0, ͑57͒, it is clear that the second term of the left hand side is ͑ ͒ zero owing to the uniformity of the zeroth order magnetic 68 field, whereas the norm of the zeroth order magnetic field equals unity, and can be expressed as the gradient of a scalar field f͑x͒=−ٌ␸. This is relevant as a potential magnetic force ͉˜ ͉ ͑ ͒ H0 =1. 60 does not result in macroscopic flow. The first order body couple is given by We obtain the components of the first order magnetization equation from Eq. ͑57͒, ˜ ͑ ͒ ˜ ϫ ˜ ˜ ϫ ˜ ͑ ͒ l1 x = M0 H1 + M1 H0, 69 1 Hˆ ˆ ͫ ˆ ⍀˜ ␻ ˆ r,0ͬ ͑ ͒ Mr,1 = Hr,1 − f ˆ z,0M␪,0 − , 61 ⍀˜ 10 which can be shown to have the form 1+j f

1 Hˆ ˜l ͑˜r͒ = B + GI ͑␬˜r͒, ͑70͒ ˆ ͫ ˆ ⍀˜ ␻ ˆ ␪,0ͬ ͑ ͒ z,1 0 M␪,1 = H␪,1 + f ˆ z,0Mr,0 − . 62 1+j⍀˜ 10 f where the constants B and G are defined as Using Eqs. ͑61͒ and ͑62͒ together with Eqs. ͑55͒ and ͑56͒, we obtain a system of equations similar to Eqs. ͑40͒ ͑7−⍀˜ 2 +2⍀˜ 4͒⍀˜ ͑ ͒ ϵ͑␥␨ ␩ ͒ f f f ⍀˜ ͑␬͒ and 41 , however, in this case the flux continuity equation B − e −2 fI1 ␩*͑ ⍀˜ 2͒3 has a forcing function that involves the radial derivative of 20 1+ f the z-directed component of spin velocity. This system of ϫ͕ ␩ ␹ ͑ ␹ ͒⍀˜ 2 ␨͓͑ ␹ ͒2 ⍀˜ 2͔ equations can be reduced to an equidimensional differential 5 e i 2+ i f + 1+ i + f equation with a Bessel forcing function for the azimuthal ϫ͓7␥ −2−͑4+␥͒⍀˜ 2 +2͑␥ −1͒⍀˜ 4͔͖/ component of the magnetic field, f f ͕ ␩*␬͑ ⍀˜ 2͒3͓͑ ␹ ͒2 ⍀˜ 2͔ ͑␬͖͒ ͑ ͒ 20 1+ f 1+ i + f I0 , 71 d d͑˜rHˆ ␪ ͒ ͫ ,1 ͬ ˆ ˆ ͑␬ ͒ ͑ ͒ ˜r − jH␪ = A˜rI ˜r , 63 dr˜ dr˜ ,1 1 ␩ ⍀˜ ͓͑1+␹ ͒2 − ⍀˜ 4͔ where Aˆ has been defined as G ϵ e f i f . ͑72͒ 4␩*͑1+⍀˜ 2͒3͓͑1+␹ ͒2 + ⍀˜ 2͔I ͑␬͒ ␩ ␬␹ ⍀˜ f i f 0 Aˆ ϵ e i f . ͑64͒ ␩ ͑ ␹ ⍀˜ ͒͑⍀˜ ͒2͑ ⍀˜ ͒ ͑␬͒ 4 * 1+ i + j f f − j j + f I0

The particular solution can be determined by using the 3. First order linear and spin velocity fields method of variation of parameters. The solutions for the azimuthal and radial components of the magnetic field are To solve the internal angular and linear momentum equations, we use the same procedure as for the zeroth order Aˆ I ͑␬˜r͒ ˆ ͑ ͒ ͫ 1 ͑␬͒ͬ ͑ ͒ problem to obtain the following nonhomogeneous modified H␪ ˜r = − I , 65 ,1 ␬2 ˜r 1 Bessel differential equation with Bessel forcing function for the spin velocity, ˆ ͑␬͒ A 2I1 ˆ ͑ ͒ ͫ ͑␬ ͒ ͑␬ ͒ ͬ ͑ ͒ 2 2 Hr,1 ˜r = j I0 ˜r + I2 ˜r − . 66 1 d d␻˜ ␬ ␩ ␬ 2␬ ␬ ͩ z,1ͪ ␬2␻ e ͓ ͑␬ ͒ ͔ ˜r − ˜ z,1 =− C1 − B + I0 ˜r G , ˜r dr˜ dr˜ 2 ␩⍀˜ ͑ ͒ ͑␬ ͒/ 4 f In Eq. 65 , it can be shown that the limit of I1 ˜r ˜r when ˜r tends to zero is ␬/2. By using Eqs. ͑65͒ and ͑66͒ in Eqs. ͑61͒ ͑73͒ and ͑62͒, one can obtain the components of the first order magnetization. with particular solution given by

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C ␩ ␬ 1 ␻ ͑ ͒ 1 ͑␬ ͒ e ͫ ͑␬ ͒ ͬ ͑ ͒ ␻ ͑ ͒ ͕ ␨͑␥␨ ␩ ͓͒ ͑␬͒2 ͑␬͒ ͔ ˜ z,1 ˜r = + C2I0 ˜r + B − ˜rI1 ˜r G . 74 ˜ z,1 ˜r = 2 − e I1 G + I2 B 2 ␩⍀˜ 2 ␩␩ ⍀˜ ͑␬͒ 4 f 8 * fI0 ͑ ͒ ␩ ͑␬ ͓͒ ␩ ͑␩ ␥␨͒␬ ͑␬͒ ␨͑␥ By using Eq. 74 , the first order vorticity is obtained and + eI0 ˜r −2 B + e − I1 G +2 then the first order azimuthal component of velocity. The ͒ ͑␬͒ ͔ ͑␬͓͒␩ ␩ ͑ ͑␬ ͒␬ ͒ −1 I2 G + I0 e * 2B −˜rI1 ˜r G constants C1 and C2 were determined from the first order ␨͑␩ ␥␨͒ ͑␬͒ ͔͖ ͑ ͒ no-slip boundary condition for the velocity and the first order +2 e − I2 G , 75 spin velocity boundary condition. By substituting C1 and C2 in Eq. ͑74͒, one obtains the expressions for the z-directed spin velocity, and the azimuthal component of the linear velocity,

␨ I ͑␬˜r͓͒−2␩B + ͑␩ − ␥␨͒␬I ͑␬͒G +2␨͑␥ −1͒I ͑␬͒G͔ + B G ˜ ͑˜͒ ͭ 1 e 1 2 v␪,1 r = − ␩⍀˜ ␩*␬I ͑␬͒ ␩*␬ 4 f 0 ˜r͑␥␨ − ␩ ͕͒I ͑␬͒2G + I ͑␬͓͒B − I ͑␬͒G͔͖ + e 1 2 0 +˜r͓B − I ͑␬˜r͒G͔ͮ. ͑76͒ ␩ ͑␬͒ 2 *I0

F. Second order problem The second order magnetization equation is

M˜ H˜ H˜ 1ץ ˜ ˜͑ ͒ ⍀˜ 2 ⍀˜ ͑ ٌ˜ ˜ ٌ˜ ˜ ͒ ⍀˜ ͑␻ ϫ ˜ ␻ ϫ ˜ ͒ ˜ ˜ 1 ͉˜ ͉2 0 ͫ ͑˜ 4 ˜ 4 f + f ˜v0 · M1 + ˜v1 · M0 = f ˜ 0 M1 + ˜ 1 M0 − M2 + H2 − H0 + Hr,0 + H␪,0 − Hr,0Hr,1 t 10 5 14˜ץ 1 ˜ ˜ ͒ ˜ 2 ˜ 2 ͬ ͑ ͒ + H␪ H␪ + H H␪ . 77 ,0 ,1 7 r,0 ,0 In Eq. ͑77͒, the ﬁfth, sixth, and seventh terms on the right hand side correspond to the ␧2 term of Eq. ͑30͒. These terms are equivalent to ˜ ͑˜ 4 ˜ 4 ͒ ͓ ͑ ␪͒ ͑ ␪͒ ͔ ͑ ͒ H0 Hr,0 + H␪,0 =2 sin 5t −5 ir − cos 5t −5 i␪ , 78

1 H˜ ͑H˜ H˜ + H˜ ␪ H˜ ␪ ͒ = ͕͓b␬˜rI ͑␬˜r͒ + ͑a − b͒˜rI ͑␬͒ − ͑a + b͒I ͑␬˜r͔͒cos͑3t −3␪͒ + ͓ar˜␬I ͑␬˜r͒ − ͑a + b͒˜rI ͑␬͒ 0 r,0 r,1 ,0 ,1 ˜r␬2 0 1 1 0 1 1 + ͑− a + b͒I ͑␬˜r͔͒sin͑3t −3␪͖͒i + ͕͓br˜␬I ͑␬˜r͒ + ͑a − b͒˜rI ͑␬͒ − ͑a + b͒I ͑␬˜r͔͒cos͑3t −3␪͒ 1 r ˜r␬2 0 1 1 ͓ ␬ ͑␬ ͒ ͑ ͒ ͑␬͒ ͑ ͒ ͑␬ ͔͒ ͑ ␪͖͒ ͑ ͒ + ar˜ I0 ˜r − a + b ˜rI1 + − a + b I1 ˜r sin 3t −3 i␪ 79

˜ ˜ 2 ˜ 2 second order magnetic ﬁelds. This will require assuming the H H H␪ = ͓− sin͑5t −5␪͒i + cos͑5t −5␪͒i␪͔, ͑80͒ 0 r,0 ,0 r following functional forms for the magnetic ﬁeld and mag- where netization:

␹ ␩ ␬2⍀˜ 2͑ ␹ ͒ i e f 2+ i ͑ ͒ ͑⍀ ˜ ␪͒ ͑ ⍀ ˜ ␪͒ a =− , 81 H˜ =Re͕Hˆ 1ej ft− + Hˆ 3ej 3 ft−3 ͖, ͑83͒ ␩*͑ ⍀˜ 2͒2͓⍀˜ 2 ͑ ␹ ͒2͔␬ ͑␬͒ 1 2 4 1+ f f + 1+ i I0

2 ˜ ˜ 2 ␹ ␩ ␬ ⍀ ͑1+␹ − ⍀ ͒ ͑⍀ ˜ ␪͒ ͑ ⍀ ˜ ␪͒ i e f i f ˜ ͕ ˆ 1 j ft− ˆ 3 j 3 ft−3 ͖ ͑ ͒ b =− . ͑82͒ M =Re M1e + M2e . 84 ␩*͑ ⍀˜ 2͒2͓⍀˜ 2 ͑ ␹ ͒2͔␬ ͑␬͒ 4 1+ f f + 1+ i I0 The contributions due to the fifth and seventh terms of Thus two magnetic field problems would have to be solved Eq. ͑77͒ cancel, as seen from Eqs. ͑78͒ and ͑80͒. From Eq. to obtain the second order force and couple. While this is not ͑79͒, one finds that the fundamental and third harmonic of difficult, it is lengthy and tedious and we therefore leave our the applied field will have to be considered in solving for the solution at the first order.

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G. Torque on the cylindrical container variables of Eq. ͑16͒ and carrying out the integration, we obtain the general dimensionless expression for torque on the Having obtained the translational and spin velocity inner surface of the cylinder, fields, we proceed to derive an expression for the torque on the surface of the cylinder. The hydrodynamic torque on the ␩ d ˜v␪ d ˜ ␹ ⍀˜ ͫ 2 ͩ ͪ ͑ ͒ ␻ L =2 ˜r + ˜r˜v␪ −2˜r˜ container is given by O i f ␨ dr˜ ˜r dr˜ z ͑ ϫ ͒ ͑ ͒ 4␩ d␻˜ LO = Ͷ dS · − T r + C , 85 + z ͬ , ͑93͒ s ␩ ␬2 e dr˜ ˜r=1 where T is the stress tensor with a symmetric Newtonian part where ˜L =L /͑␮ K2V ͒ and V is the fluid volume. Expand- and an asymmetric part whose phenomenological form was O O 0 f f ing ˜v␪ and ␻˜ in powers, of the parameter ␧ and grouping presented by Condiff and Dahler,20 z coefficients of like powers, we obtain the dimensionless ,T =−pI + ␩ٌ͓v + ٌvT͔ + ␭ٌ͑ · v͒I + ␨⑀ · ٌ͑ ϫ v −2␻͒. z-directed torque ͑ ͒ 86 L ⍀˜ GI ͑␬͒ ˜L = O,z =−␹ ͫ f + ͩ2B + 1 ͪ␧ͬ ͑ ͒ ⑀ O,z ␮ 2 i ␬ In Eq. 86 , I is the unit tensor, p is the pressure, and is the 0K Vf 1+⍀˜ 2 alternating unit tensor. Further, C is the surface couple stress f 2 tensor representing the diffusive transport of internal angular + O͑␧ ͒. ͑94͒ 18 momentum between contiguous material elements with where B and G are given by Eqs. ͑71͒ and ͑72͒, respectively. constitutive form The expression for torque given in Eq. ͑94͒ takes into ac- .C = ␩Јٌ͓␻ + ٌ␻T͔ + ␭ٌ͑ · ␻͒I. ͑87͒ count contributions from antisymmetric and couple stresses In order to quantify the magnitude of antisymmetric stresses, In addition to the expected viscous contributions to the we take the limit of Eq. ͑94͒ when ␬→ϱ, obtaining torque, surface excess magnetic forces could also contribute to the measured torque. Surface excess forces arise as a con- ␹ ⍀˜ ͑7−⍀˜ 2 +2⍀˜ 4͒ lim ˜L =− i ͫ1− ␧ͬ + O͑␧2͒. sequence of the change in magnetization at the fluid-wall O,z ␬→ϱ 1+⍀˜ 2 20͑1+⍀˜ 2͒2 interface and can be determined by evaluating the jump in the Maxwell stress tensor across the wall-fluid interface, ͑95͒ ϫ ʈ Mʈ ͑ ͒ ͑ ͒ ͑ ͒ LM = r n · T . 88 Comparison of Eqs. 94 and 95 will show that contributions due to antisymmetric stresses dominate the torque in In this, r is the position vector, n is the unit outward normal a ferrofluid in spin-up flow. These antisymmetric stresses are vector of the surface, which in our case is i , and TM is the r the result of the spin of the particles on the wall of the cyl- Maxwell stress tensor, which for an incompressible ferrofluid inder, as a consequence of magnetic body couples. This is is given by Refs. 11, 24, and 39, verified below by comparison to the experimental torque M 1 ␮ 2 ͑ ͒ measurements. In addition, the sole estimated parameter in T = BH − 2 0H I. 89 Eq. ͑95͒ is the relaxation time in ⍀˜ ϵ⍀ ␶, suggesting that In Eq. ͑89͒, B=␮ ͑H+M͒. The azimuthal component of the f f 0 the agreement between the magnitude of the theoretical pre- surface excess magnetic force is relevant with respect to an dictions and experimental measurements of torque will con- axial torque and is given by firm that the relaxation time has been estimated correctly. ͉͑ ͉͒ ͉͑ ͉͒ ͑ ͒ f␪ = BrH␪ r R+ − BrH␪ r R− . 90 = O = O III. APPARATUS AND EXPERIMENTAL METHOD Using the continuity conditions for the normal magnetic flux and the tangential component of magnetic field intensity, one A. Ferrofluid characterization obtains that Water ͑EMG705គ0͒ and kerosene ͑EMG900គ0͒ based ͉͑ ͉ ͉ ͉ ͒ ͑ ͒ ͑ f␪ = Br H␪ r R+ − H␪ r R− =0, 91 ferrofluids obtained from Ferrotec Corporation Nashua, = O = O New Hampshire USA͒ were used for the torque and flow showing that the surface excess magnetic force does not con- measurements. In order to obtain fluids with different physi- tribute to the torque. Returning to Eq. ͑85͒, an expression for cal and magnetic properties, these ferrofluids were diluted the torque can be obtained by introducing the constitutive using de-ionized water for EMG705គ0 and a suitable solvent expressions for stress ͓Eq. ͑86͔͒ and surface couple stress obtained from Ferrotec for EMG900គ0. Table I shows results tensor ͓Eq. ͑87͔͒ in Eq. ͑85͒ obtaining for mass density obtained by using the gravimetric method l 2␲ and shear viscosity obtained by using a STRESSTECH HR d v␪ d ͫ␩ 2 ͩ ͪ ␨ ͑ ͒ ␨ ␻ L = ͵ ͵ r + rv␪ −2 r rheometer with double gap geometry. Measurements of shear O dr r dr z 0 0 viscosity versus shear rate between 10 and 110 s−1 confirm ␻ that these ferrofluids display Newtonian behavior under these d z + ␩Ј ͬrd␪dz, ͑92͒ conditions. Magnetic properties such as saturation magneti- dr zation and initial susceptibility were obtained from magneti- where l is the height of the cylinder. Introducing the scaled zation curves at 300 K using a MPMS-XL7 SQUID magne-

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TABLE I. Physical and magnetic properties at room temperature for water phase analysis of an echo generated by a mobile screw when and kerosene based ferrofluid. it is moved over a known distance. The sound velocities for ␩ ͑ −2͒ ␳ ͑ / 3͒ ␮ ͑ ͒ ␹ ␾ EMG705គ0 and EMG900គ2 were 1450 and 1150 m/s, Ferrofluid Nsm kg m 0Ms mT i respectively. EMG705គ02.5ϫ10−3 1220 21.9 4.99 0.039 −3 EMG705គ11.7ϫ10 1120 10.1 1.28 0.018 B. Torque measurements EMG900គ01.1ϫ10−2 1391 61.3 12.20 0.111 EMG900គ17.7ϫ10−3 1270 47.8 3.63 0.087 Torque measurements were made by using a rotational EMG900គ24.5ϫ10−3 1030 23.9 1.19 0.043 viscometer ͑Brookfield Engineering Laboratories, model LV-IIIϩ, Brookfield, MA USA͒ as a torque meter, with a full scale range of −6.73–67.3 ␮N m. The viscometer measures the torque needed to keep a spindle rotating at a constant ͑ ͒ tometer Quantum Design, San Diego CA USA . These angular velocity ͑including zero rotation͒. Because we are magnetization curves were corrected for the demagnetization concerned with torque measurements on the inner surface of effect, using a demagnetization factor of 0.5, which corre- 46 a cylindrical container, we replaced the standard stainless sponds to the cylindrical container geometry. The initial steel spindles with custom hollow polycarbonate spindles of susceptibility was obtained from the slope of the magnetiza- 12 and 28 ml volumes. The polycarbonate spindle filled with tion curves at low fields, whereas the saturation magnetiza- ferrofluid was centered in the gap of a two-pole, three-phase tion was obtained from the asymptotic value at high fields.11 induction motor stator winding. In all experiments, the angu- The magnetic volume fractions were estimated by using the ␾ / lar velocity of the inner spindle was set to zero. relation Ms = Md, and a value of 446 kA m for the domain magnetization of magnetite. A rotating magnetic field was obtained by exciting the The median particle diameter of the ferrofluids was de- three-phase winding of a magnetic induction motor using termined from direct measurements of particle diameter by balanced three-phase currents each with 120° phase differ- transmission electron microscopy ͑TEM͒ assuming that the ence from each other. We do this by grounding one phase, particle size distribution obeys a log-normal distribution exciting the remaining two phases with sinusoidal voltages at function. TEM measurements yield the number median di- Ϯ60° phase difference, and letting the neutral point float. ameter Dpg which is ideally related to the volume median This results in balanced three-phase currents in the windings diameter Dpgv by the expression to create clockwise or counterclockwise rotating magnetic ln D =lnD +3ln2 ␴ , ͑96͒ fields. The amplitude, frequency, and direction of the field pgv pg g can be controlled by using the signal generator or linear am- ␴ where g is the geometric deviation. plifier gain. The length and diameter of the stator winding In order to obtain a better comparison between theoreti- were 77.7 and 63.6 mm, respectively. cal results and experimental measurements, we obtained an Measurements of the magnetic fields produced by the effective Brownian relaxation time that takes into account stator, with and without ferrofluid in the cylindrical con- the effect of particle size polydispersity observed in ͑ 47 tainer, were made by using a gaussmeter Sypris Test & ferrofluids, Measurement, Orlando, FL USA͒ with a three-axis probe ϱ n ͑D ͒ −1 ␲␩ ͑model ZOA73͒. From these measurements, it was deter- ¯␶ ϵ ͫ͵ v p dD ͬ = o D3 exp͑−9/2ln2 ␴ ͒, B ␶ ͑ ͒ p pgv g mined that the stator produces a magnetic field of 0 B Dp 2kBT 4.54 mT rms/A rms of input current in the absence of the ͑ ͒ 97 ferrofluid. The radial field in homogeneity was determined to ͑ ͒ where nv Dp is the log-normal particle volume distribution be 2.2% from axis to outer container radius at midheight, function. In deriving Eq. ͑97͒, it was assumed that the par- with a maximum of 6% at the top of the container, while ϳ 3 ticle hydrodynamic diameter is the same as the particle core axial field in homogeneity was 4% from middle to 4 height diameter. Table II shows the values for the median particle and 21% to top, along the gap axis. diameter and effective Brownian relaxation times for our fer- In all experiments the direction of the rotating field was rofluids. set clockwise in order to obtain positive torque measure- Sound velocity was measured by using an attachment to ments and thereby utilize the full instrument range. Revers- ͑ the DOP 2000 ultrasound velocimeter Signal Processing, ing the field rotation direction merely reversed the direction ͒ Lausanne, Switzerland . This measurement is based on the of torque and flow. The torque required to restrain the spindle from rotating was measured for field frequencies from 25 to 500 Hz and TABLE II. Estimated magnetic particle diameters and effective relaxation times at room temperature for water and kerosene based ferrofluid. field amplitudes from 0 to 17.0 mT rms, for the two different volumes of fluid. This series of measurements was made for ͑ ͒ ␴ ␶ ͑ ͒ Ferrofluid Dpgv nm ln g ¯B s both ferrofluids and their dilutions. Figure 2 presents typical measurements of torque as a function of applied magnetic EMG705គ0, EMG705គ1 20.1 0.288 2.1ϫ10−6 គ EMG900គ0, EMG900គ1, EMG900គ2 14.2 0.410 1.9ϫ10−6 field amplitude and frequency for the EMG705 0 ferrofluid using a spindle with 12 ml of ferrofluid.

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FIG. 2. Torque measurements as a function of frequency and amplitude of the applied magnetic ﬁeld for EMG705គ0 using a spindle volume of 12 ml.

C. Velocity profile measurements We applied the UVP technique48,49 by using a DOP 2000 ultrasound velocimeter ͑Signal Processing, Lausanne, Switzerland͒ to obtain bulk velocity profiles. This technique allows measurement of velocity profiles in opaque liquids, such as ferrofluids. An ultrasonic pulse is periodically emit- ted from a transducer and sent through the fluid. Echoes due to tracer particles are recorded by the same transducer. The equipment measures the component of the velocity vector FIG. 4. Velocity profiles for ͑a͒ EMG705គ0 and ͑b͒ EMG900គ2 ferrofluids parallel to the direction of beam propagation, with the spatial showing the dependence of flow with frequency for an applied magnetic location being determined from the time delay between emit- field of amplitude 12.3 mT rms. ted burst and recorded echo, while the velocity is obtained from the corresponding Doppler frequency shift. The velocity signal is positive when tracers are moving away from the ͱ 2 ␣ probe. Assuming that the flow is purely in the azimuthal Ro + xp −2Ro cos ␪ = ʈ . ͑99͒ v v ␣ direction, an assumption that is supported by the experiments Ro sin as shown below, the relation between parallel and azimuthal velocities is given by Minute amounts of polyamide powder were used as tracers, such that the estimated weight fraction of tracer particles ␸ ͑ ͒ vʈ = v␪ cos . 98 was less than 0.0002, in order to prevent phenomena such as 50–53 It can be shown that for our experimental geometry, the is found in the so-called inverse ferrofluids. These mi- croparticles had a mean diameter of 15–20 ␮m and a density azimuthal component v␪ of the velocity can be related to the parallel velocity by close to that of our ferrofluid, to avoid settling or buoying. In addition, these polyamide particles efficiently reflect ultrasound as their acoustical impedance is different from that of the ferrofluid.54 Our transducers had an emission frequency of 4 MHz, a diameter of 8 mm, and a length of 10 mm. We used a pulse repetition frequency of 200 Hz, taking 100 emissions per profile and averaging over 70 profiles to obtain our reported results. These conditions ensured a spatial resolution of 0.36 mm along the beam propagation direction and a maximum measurable parallel velocity of 18.12 mm/s. The spatial resolution transverse to beam propagation is set by the diameter of the emitter element and was approximately 3 mm in our experiments. Velocity profiles were measured for the ferrofluid in a cylindrical container made of polycarbonate and with dimen- sions of 49.4 mm inner diameter and 63.5 mm height. Our FIG. 3. Experimental velocity profiles for EMG705គ0 and EMG900គ2 ferrofluids. Ultrasonic transducers were placed at half the height of the con- apparatus design permitted simultaneous measurement of ve- tainer. Angles are with respect to the diagonal. locity profiles relative to ultrasonic beams propagating at

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FIG. 6. Velocity profile for ͑a͒ EMG705គ0 and ͑b͒ EMG900គ2 ferrofluids using probes placed at four different heights of the container with cover. FIG. 5. Velocity profiles for ͑a͒ EMG705គ0 and ͑b͒ EMG900គ2 ferrofluids, showing the dependence of flow with amplitude of the applied magnetic field for a frequency of 85 Hz. of frequencies and amplitude of the magnetic field studied, we always found corotation of field and fluid in the bulk of three distinct angles ͑using three ultrasonic transducers͒ rela- the fluid and a linear dependence of the velocity with radial tive to the diagonal. A cover was used to eliminate the free position ͑i.e., rigid-body-like motion͒ over most of the fluid. surface and thereby avoid interfacial effects which could In addition, the velocity profiles measured by each one of the confound our measurements. Although, the DOP 2000 does transducers superimpose, indicating that the flow is uniform not require calibration, we verified the correct operation of and azimuthal in the plane. Another important observation of our experimental setup by measuring the velocity profile for these experiments ͑Fig. 3͒, is that the velocity maximum glycerin and ferrofluid in a cylindrical Couette flow geom- occurs at different radial positions for the two fluids, even etry and found good agreement with theory. under identical experimental conditions. This is important as The rotating magnetic field was generated by using the the expression for flow obtained by Pshenichnikov et al.14 three-phase, two-pole magnetic induction motor stator wind- does not predict variation in the maximum position with fluid ing described in Sec. III B. Experiments were carried out for type. The results of similar experiments at various field fre- field frequencies between 20 and 120 Hz and amplitudes be- quencies and amplitudes are shown in Figs. 4 and 5, where a tween 8.3 and 14.3 mT rms. single probe angle is shown for simplicity as all three probes The first set of experiments was made in such a way as provide similar results. to allow us to verify the existence, direction, and uniformity Next, we measured the azimuthal velocity profiles at of the flow in the bulk of the fluid. To this end, we used three four different heights by using probes at h,3/4h,1/2h, and transducers at angles of 7.1°, 11.7°, and 14.2° placed at half 1/4h ͑h=63.5 mm being the container height͒, and the axial the height of a container with cover. In obtaining these velocity profile vz using probes placed on the cover of the / angles, we have taken into account the diffraction angle pre- container at r=0 and at r=1 2RO. In these experiments, the dicted by Snell’s law when the ultrasonic beam passes top probe was placed 4 mm below the cover ͑measured from through the container-fluid interface. In this experiment, the the probe center͒ to avoid reflections of the pulse. Represen- direction of field rotation was set counterclockwise so that a tative results are shown in Figs. 6 and 7, respectively. These positive velocity measurement implied corotation of field measurements show that the direction of flow is maintained and fluid. As is observed in Fig. 3 and for the whole interval throughout the container and that the axial velocity compo-

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FIG. 8. Startup of spin-up ﬂow for EMG705គ0.

of the startup of the flow by taking velocity profiles at intervals of 2 s for a period of 100 s. The results are presented in Fig. 8 where profiles are shown at 4 s intervals for better illustration. The poor quality of the velocity profiles of Fig. 8 is because these are instantaneous velocity measurements under transient conditions, whereas the other flow measurements reported herein correspond to steady flow for which we report the average of over 70 profiles. The transient experiment was carried out at three different frequencies. The measurements demonstrate that steady state flow is obtained ͑ ͒ គ ͑ ͒ FIG. 7. Axial velocity component measured for a EMG705 0 and b after approximately 30 s, with an immediate response to the EMG900គ2 ferrofluids using a probe placed on the fixed top cover ͑z-measured from the top͒. startup of magnetic field rotation. A third series of experiments was carried out under the same conditions but by using an open container. In these experiments, the top probe was placed 4 mm below the free nent is negligible in comparison to the azimuthal velocity surface, as measured from the probe center. The velocity ͑ / ͒ even for the small aspect ratio of the container l d=1.2 . profiles in Fig. 9 show a transition from corotation of field The observation of uniform flow throughout the bulk and and fluid in the bulk to counter-rotation close to the free negligible axial velocity is indicative that though the mag- surface. These observations are in agreement with the mea- netic field generated in the stator varies by as much as 21% surements of Rosensweig et al.,7 but they demonstrate that ͑ ͒ along the vertical axis 4% throughout most the fluid , this the bulk flow is not negligible in comparison to the surface variation does not induce appreciable flow in the axial direc- flow. Both the surface driven flow and the body couple in- tion. Furthermore, as noted before, the location of the veloc- duced flow coexist under our experimental conditions. These ity maximum is different for the two fluids, which is contrary to what would be expected if the flow was driven by a field inhomogeneity which is concentrated close to the container wall ͑i.e., close to the slots͒. Hence it is unlikely that our observations are the result of field inhomogeneity. Temperature measurements were made for field amplitudes of 10.4, 12.5, and 14.5 mT rms for the frequencies of 50, 65, 75, 85, 100, and 125 Hz during a time interval of 56 min. Three temperature sensors were placed in the middle height of the cylinder at r=0, r=0.5Ro, and r=Ro. We found a temperature increase of 4 °C over the duration of the experiment, and a maximum radial temperature variation of 0.2 °C. Clearly, radial temperature gradients are negligible, hence the mechanism explained by Pshenichnikov et al.14 is not at work in our experiments. Furthermore, if dissipative effects were responsible for the observed flow, one would FIG. 9. Velocity profiles for EMG705គ0 at different heights in a container expect a time delay between turning on the magnetic field without a cover. The negative velocities indicate counter-rotation of fluid and start of the flow. To verify this, we made measurements and field.

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experiments further show that surface velocity profile measurements are inappropriate in assessing theoretical analyses of the bulk flow as these do not represent the flow behavior in the bulk of the fluid.

IV. COMPARISON BETWEEN THEORETICAL PREDICTIONS AND EXPERIMENTAL MEASUREMENTS Glazov3 analyzed the phenomenon of spin-up flow using a set of equations similar to Eqs. ͑3͒–͑14͒, but setting ␩Ј =0. The result was that flow is not expected when ferrofluid in a cylindrical container is subjected to a uniform rotating magnetic field. Glazov then attributed the experimental observations of Moskowitz and Rosensweig1 to field inhomogeneity. To demonstrate this, Glazov obtained a general solution for flow using higher spatial harmonics, m1 in Eq. ͑2͒, to simulate the rotating magnetic field generated in a multipole magnetic induction machine stator for two asymptotic cases: ␣Ӷ1 and ␣ӷ1. Only a qualitative comparison of the predictions for the case of ␣Ӷ1 can be made with our experimental measurements, as our experiments were made at moderate amplitudes of the magnetic field ͑ ͒ ͑ ͒ ͑0.4Շ␣Շ1.25͒. The velocity profiles predicted by Glazov’s FIG. 10. Theoretical a translational and b spin velocity profile for zeroth, first, and composite asymptotic solution obtained from regular perturbation theory are of similar shape to the measured velocity profiles and with ␧=0.162 using ferrofluid properties for EMG705គ0. for EMG705គ0, however, Glazov’s theory cannot describe the boundary layer character of the velocity profiles for EMG900គ2. In addition, Glazov predicts that the flow direction reverses, becoming contrary to the field direction, when for ␬ based on our experiments. All figures of theoretical ␾ Ͼ ␾ predictions shown here assume ␥=0. Velocity profiles with h 0.12, with h the hydrodynamic volume fraction of particles. Velocity profiles measured by using EMG902គ1, with ␥=1 only show a small variation in the magnitude of the ␾ Ϸ flow especially for low ␬ values. The analysis predicts for h 0.18, showed that the fluid corotates with the magnetic field even for this hydrodynamic volume fraction of mag- zeroth order a rigid-body-like azimuthal velocity profile netic particles, hence Glazov’s theory does not explain our corotating with the field through most of the fluid and de- experimental observations. creasing to zero near the container wall in a thin layer of From our measurements, it is clear that the bulk fluid thickness which depends on the dimensionless parameter ␬ corotates with the magnetic field, therefore the results of and as a result on the spin viscosity value ␩Ј. The first order Kaloni10 for a slip boundary condition in the translational analysis predicts a flow contribution opposite to field rota- velocity do not agree to the experiments. Furthermore, the tion. However, it is likely that in contributions of higher- experimental results described above show that thermal ef- order, correction terms of even order yield corotation be- fects are not responsible for the observed flow, hence the tween flow and field whereas counter-rotation will be theories presented by Shliomis28 and Pshenichnikov et al.14 obtained for terms of odd order in similar fashion to the case do not explain the observed flow. Finally, as can be seen in studied in Ref. 45 Thus, our composite solution for flow Fig. 9, even though surface effects dominate at the free sur- predicts corotation with field in agreement with the experi- face, as observed by Rosensweig et al.,7 the bulk flow is not mental bulk velocity profiles. Based on the alternating char- negligible in comparison. These observations leave us with acter of higher-order contributions to the flow field, we ex- the original spin diffusion theory of Zaitsev and Shliomis2 as pect a decreasing dependence of the flow on the applied field the only published model in ͑at least qualitative͒ agreement magnitude. This is to be expected as the ferrofluid magneti- with our measurements. zation tends to a saturation value with increasing magnetic In order to compare our theoretical extension of the spin field. Thus, it can be expected that for conditions under diffusion theory with our experimental flow measurements, which ␣ϳ1, the flow is qualitatively similar to that of our the translational and spin velocity of our asymptotic analysis solution but with a smaller power dependence on the applied for zeroth, first, and composite solution are shown in Fig. 10. magnetic field amplitude. These were obtained by using the properties of the The analysis predicts a power of two dependence of flow EMG705គ0 ferrofluid given in Tables I and II. Because there on amplitude and linear dependence on dimensionless are no measurements or rigorous estimates for the spin vis- frequency of the applied magnetic field for zeroth order. cosity, we have arbitrarily chosen the value ␬=100 to illus- Log-log plots of the dimensionless zeroth order extrapolated ͑ ⍀˜ 2͒ /͑⍀ ͒ trate the general features of the solution. Below, we will wall velocity 1+ f Uw,o fRO versus the applied mag- consider this matter further and attempt to estimate a value netic field amplitude ͑made dimensionless by using the

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FIG. 11. Log-log plots of the experimental extrapolated wall velocity vs Langevin parameter for EMG705គ0 and EMG900គ2.

Langevin parameter͒ were obtained ͑Fig. 11͒ for EMG705គ0 and EMG900គ2 ferrofluids in order to determine the dependence of the measured flow on magnetic field amplitude. By using a power law analysis we determined a power of one dependence on magnetic field for the EMG705គ0 ferrofluid and a power of 1.6 dependence for the EMG900គ2 ferrofluid. As mentioned above, this difference between theoretical and experimental dependence on amplitude of applied magnetic field can be due to moderate saturation of the ferrofluid at the magnetic fields used in the experiments. Log-log plots were similarly used to determine the de- FIG. 12. Log-log plots of the experimental normalized torque vs Langevin parameter for ͑a͒ EMG705គ0 and ͑b͒ EMG900គ0 ferrofluids and all their pendence of the experimental torque measurements on the dilutions. amplitude of the applied magnetic field. The experimental torque data Lexp were normalized using the zeroth-order ͑ / ͒ torque expression Lexp Lth , gible. More recently, Feng et al.55 neglected ␩Ј arguing that ␹ ⍀˜ the functional form of the spin viscosity depends on the ␮ 2 i f O͑␧͒͑͒ Lth =− 0K Vf + 100 square of particle diameter, ⍀˜ 2 1+ f ␩Ј ϳ ␩ 2 ͑␾ ͒ ͑ ͒ odpf h , 101 and plotted ͑Fig. 12͒ versus the amplitude of the magnetic field, made dimensionless by using the Langevin parameter, suggesting that this effect is negligible for ferrofluids ͑ ϳ ͒ for both ferrofluids and their dilutions. We use the zeroth- dp 10 nm . However, the bulk flow measurements pre- order torque expression because the principal contribution to sented in the previous sections, which are in agreement with torque is due to antisymmetric stresses, as was discussed in the theoretical predictions obtained from the spin diffusion relation to Eq. ͑95͒. These plots show that data for the two theory, indicate that this phenomenon is important in driving ␬ fluid volumes and different volumetric particle fractions are the flow. In order to obtain an estimate for from our flow គ ͑ well described by the theoretical expression for torque, indi- experiments for the EMG900 2 ferrofluid as this shows the cating that Eq. ͑94͒ predicts the correct magnitude for the better agreement between theoretical predictions and experi- ͒ theoretical torque, especially for the EMG900 ferrofluid. mental measurements , we used the concept of the extrapo- ˜ From a power analysis, we determined that the dependence lated wall velocity Uw,o, given by the dimensionless zeroth- 45 of torque on applied magnetic field is 1.5 for the water based order vorticity evaluated at ˜r=0, ferrofluid, whereas for the kerosene based ferrofluid it is 1 ␨ 1+I ͑␬͒ power of 2, in agreement with the theoretical predictions. U˜ = ⍀˜ = ͭ1− 2 ͮ. ͑102͒ w,o z,0 ͑␬͒ As has been previously mentioned, definitive estimates 2 ␩*͑ ⍀˜ 2͒ I0 4 1+ f or experimental measurements of the spin viscosity ␩Ј or the dimensionless parameter ␬ are not available. The first esti- Introducing the translational velocity scale, the definition of ␹ ␣ mate obtained by Ref. 2 from dimensional analysis predicts i, and the definition of the Langevin parameter in Eq. that ␬ϳ106, implying that spin diffusion effects are negli- ͑102͒, we obtain

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͑1+⍀2␶2͒ ␾␶k T 1+I ͑␬͒ showed that the dependence on amplitude of magnetic field f U = B ͭ1− 2 ͮ␣2. ͑103͒ ⍀ R w,o 12V ␩* I ͑␬͒ agrees with theoretical predictions. Direct quantitative com- f o c 0 parison between theoretical results and experimental mea- ͑ ⍀2␶2͒ /͑⍀ ͒ Thus, from the slope of a plot of 1+ f Uw,o fRO surements is not possible as the perturbation parameter of versus ␣2 and by using Eq. ͑103͒, we obtained a value our asymptotic analysis is ␧ϳ1 for the range of amplitude of of ␬=33.7 for the EMG900គ2 ferrofluid. Using the esti- the applied magnetic field used in the experimental measure- mated values of ␨=3ϫ10−4 kg m−1 s−1 and ␶=10−6 s, ments. Still, we estimate from comparing the experimental this result for ␬ yields an estimated spin viscosity of measurements and predictions of the extrapolated wall veloc- ␩Ј=6ϫ10−10 kg m s−1. Allowing for order of magnitude er- ity of the asymptotic analysis a range of 10−8–10−12 kg m s−1 rors in the estimated values of ␨ and ␶ results in an estimated for the spin viscosity of the kerosene based ferrofluid. We range for the spin viscosity of 10−8–10−12 kg m s−1. These realize that this is a controversial value, since the majority of estimated values are several orders of magnitude higher than the current literature on the subject tacitly assumes this dy- expected by dimensional analysis. Note, however, that ex- namical parameter to be negligible. The result underscores pression ͑101͒ is a dilute limit result applicable for suspen- the need for a more thorough consideration of possible ori- sions of hard spheres in a Newtonian fluid and does not take gins for a couple stress in ferrofluids or alternate constitutive into account the fact that ferrofluids are typically stabilized equations capable of capturing the observed flow by surfactants layers and hence may have significant free phenomena. surfactant concentrations in solution. Also, for the oil based ferrofluid used in these experiments, the magnetic volume fraction is ␾=0.043, the average magnetic diameter ͑ob- ACKNOWLEDGMENTS tained from magnetization measurements͒ is D =10 nm, pm This work was supported by the U.S. NSF ͑Grant Nos. and the average physical diameter ͑obtained from TEM͒ is 0457359 and 0547150͒. Dp =14 nm, hence the estimated hydrodynamic volume fraction is ␾ Ϸ0.18, assuming a 1 nm surfactant layer thickness. h 1R. Moskowitz and R. E. Rosensweig, “Nonmechanical torque-driven flow At such hydrodynamic volume fractions, the particle-particle of a ferromagnetic fluid by an electromagnetic field,” Appl. Phys. Lett. 11, separation is less than the particle diameter, hence hydrody- 301 ͑1967͒. 2 namic interactions between particles, neglected in deriving V. M. Zaitsev and M. I. Shliomis, “Entrainment of ferromagnetic suspen- ͑ ͒ sion by a rotating field,” J. Appl. Mech. Tech. Phys. 10,696͑1969͒. Eq. 101 , become relevant. 3O. A. Glazov, “Motion of a ferrosuspension in rotating magnetic fields,” Magn. Gidrodin. 11,16͑1975͒. 4O. A. Glazov, “Role of higher harmonics in ferrosuspension motion in a V. CONCLUSIONS rotating magnetic field,” Magn. Gidrodin. 11,31͑1975͒. 5O. A. Glazov, “Velocity profiles for magnetic fluids in rotating magnetic 1 Since Moskowitz and Rosensweig first observed fields,” Magn. Gidrodin. 18,27͑1982͒. spin-up flow, only surface velocity profiles have been avail- 6R. E. Rosensweig and R. J. Johnston, in Continuum Mechanics and its ͑ able to evaluate the various theories describing the phenom- Applications, edited by C. Graham and S. Malik Hemisphere, New York, 1989͒, pp. 707–729. enon. However, as shown above, surface velocity profiles are 7R. E. Rosensweig, J. Popplewell, and R. J. Johnston, “Magnetic fluid not suitable to evaluate bulk theories as flow of the fluid near motion in rotating field,” J. Magn. Magn. Mater. 85, 171 ͑1990͒. the air-fluid interface is dominated by surface effects and is 8A. V. Lebedev and A. F. Pshenichnikov, “Motion of a magnetic fluid in a ͑ ͒ therefore not representative of the bulk flow. It is no surprise rotating magnetic field,” Magn. Gidrodin. 27,7 1991 . 9A. V. Lebedev and A. F. Pshenichnikov, “Rotational effect: The influence that surface velocity profile measurements have resulted in of free or solid moving boundaries,” J. Magn. Magn. Mater. 122,227 confusion in testing the spin-diffusion theory, as the theory ͑1993͒. 10 could not explain the observed counter-rotation between flow P. N. Kaloni, “Some remarks on the boundary-conditions for magnetic 7,15 fluids,” Int. J. Eng. Sci. 30, 1451 ͑1992͒. and field directions. In stark contrast, our bulk flow mea- 11 R. E. Rosensweig, Ferrohydrodynamics ͑Dover, Mineola, 1997͒. surements show that the flow corotates with the field in the 12R. E. Rosensweig, “Role of internal rotations in selected magnetic fluid bulk even when fluid near the air-fluid interface counter- applications,” Magn. Gidrodin. 36,241͑2000͒. 13 rotates with the field as a consequence of surface effects. A. F. Pshenichnikov and A. V. Lebedev, “Tangential stresses on the magnetic fluid boundary and the rotational effect,” Magnetohydrodynamics Critical comparison of our experimental observations and the ͑ ͒ ͑ ͒ 3 10 N.Y. 36, 254 2000 . model proposed by Glazov, Kaloni, and Pshenichnikov 14A. F. 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