Ind. Eng. Chem. Res. 2007, 46, 6113-6117
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Stress Boundary-Conditions in Ferrohydrodynamics Ronald E. Rosensweig* Consultant, 34 Gloucester Rd., Summit, New Jersey 07901
Although much study has been devoted to the fluid magnetic behavior (i.e., ferrohydrodynamics) of the ultrastable, colloidal, magnetizable fluids that are known as ferrofluids, the appropriate stress-boundary conditions for a magnetically unequilibrated medium are lesser known. In this review, the companion stressboundary conditions are derived in pedagogic detail from the total pressure-viscous-magnetic stress tensor. Phenomena of spin-up flow and other systems are noted as examples of the relationships. Introduction The pressure-viscous-magnetic stress tensor is key to constituting the governing equations of a magnetic fluid, and it offers an expeditious means for formulating associated stress boundary-conditions. This review develops the stress-boundary conditions pedagogically in a general form and indicates areas of their importance. This work is dedicated to Professor K. A. Smith for his longterm interest in the mechanics of magnetized flows, and for his contributions to phenomena that occur at fluid interfaces. Background
The stress vector tn acting on an oriented area of a medium is determined by tn ) n‚T, so that
tn,1 ) n1‚T1
(4a)
tn,2 ) n2‚T2
(4b)
where n1 is the outward facing unit normal to the interfacial area of medium 1 and n2 is the outward facing unit normal to the interfacial area of medium 2. Accordingly, for media that share a common interface of contact,
n1 + n2 ) 0
General Equations. The Cauchy unconstituted equations of linear momentum and internal angular momentum are as follows:1,2
Dv F ) Fg + ∇‚T Dt
(1)
Ds F ) FG + ∇‚C + vec T Dt
(2)
where D/Dt is the convective derivative (D/Dt ) ∂/∂t + v‚∇), t denotes the time, F is the mass density, v is the velocity, g is the body force transmitted from afar, s represents the spin angular momentum of particles, G is the body couple transmitted from afar, C is the couple stress tensor, and vec T ) -E:T, with E being the polyadic alternator E ) eiejekijk, where ei, ej, and ek each represent any of the unit Cartesian vectors. The last term of eq 2 represents body couple density that results from the conversion of angular momentum between external and internal types. Body force g most commonly denotes the gravitational force per unit mass, and G can result from gravitational dipoles3 or other sources but is not commonly treated. It is important to note that magnetic force and couple in this formulation are not part of g or G. In the past, G has been viewed in that context but logically results from magnetic field stresses that are local and inherent to the tensor T. The couple stress tensor C results from the subscale transport of angular momentum, e.g., Brownian diffusion of rotating magnetized particles. Equations 1 and 2 are augmented by the usual equation of mass conservation:
∂F + ∇‚(Fv) ) 0 ∂t
(3)
* To whom correspondence should be addressed. Tel: 908-277-2846. Fax: 908-277-1084. E-mail:
[email protected].
(5)
The condition of stress equilibrium is determined by
[tn] ) 2σH n1
(6)
where the square brackets ([‚]) denote the difference (‚)2 - (‚)1 on opposite sides of the interface, and, thus, [tn] ) tn,2 - tn,1. The parameter σ is the interfacial tension and H is the mean curvature of the interface, with its sign referred to medium 1.4 The difference in stress vectors across the interface balances the stress of interfacial tension. Constitutive Relationships. The dyadic T can be derived from Maxwell’s equations of the electromagnetic field, Newton’s equations of motion, and conservation of energy to satisfy positive entropy production of irreversible processes.5,6 Neglecting relativistic terms on the order of v/c and smaller, T is given by
T ) -pI + η(∇v + (∇v)T) + 2ζE‚(Ω - ω) + µ 0H 2 I + BH (7) λ(∇‚v)I 2 where η is the ordinary or first coefficient of viscosity, λ the second coefficient of viscosity, ζ the vortex viscosity, and Ω is half the vorticity and equals the fluid angular rate (Ω ) 1/2(∇ × v)). The symbol ω represents the particle angular rate, the unit dyadic is given as I ) δijeiej ) ii + jj + kk (where i, j, k are unit Cartesian vectors), H is the magnetic field (in units of amperes per meter (A/m)), and B (given in tesla) is the induction field with the defining equation, in SI units, given by B ) µ0H + M, wherein M is the magnetization (tesla) and µ0 is the permeability of free space (4π × 10-7 tesla m /A). Note that the defining equation used here differs from the more commonly written B ) µ0(H + M), where M has units of A/m. Force density terms that result from the spatial distribution of stress are determined as the divergence of T, using eq 7.
10.1021/ie060657e CCC: $37.00 © 2007 American Chemical Society Published on Web 01/18/2007
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∇‚T ) -∇p + η∇2v - 2ζ∇×(Ω - ω) + (η + λ)∇(∇‚v) + (∇H)‚M (8) The last term expands to (∇H)‚M ) M‚∇H + M×(∇×H), and, from Maxwell’s generalization of Ampere’s law, with an electrically nonconductive medium in the absence of a timevarying field of electric displacement, ∇×H ) 0. Under this condition, the force density of eq 8 reduces to
∇‚T ) -∇p + η∇2v - 2ζ∇×(Ω - ω) + (η + λ)∇(∇‚v) + M‚∇H (9) Equation 7 shows that, for a compressible or incompressible medium,
vec T ) 4ζ(Ω - ω) + M×H
(10)
The couple stress tensor accounts for the diffusive transfer of angular momentum. It is expressed as
C ) η′(∇ω + (∇ω)T) + λ′(∇‚ω)I
(12)
It has long been thought that couple stress is vanishingly small. Recent numerical experiments and theory give support to this viewpoint.7 Constituted Equations of Momentum. Combining eqs 1 and 9 yields the constituted equation for a compressible medium. For simplicity, we write just the form for an incompressible medium.
F
Dv ) Fg + -∇p + η∇2v - 2ζ∇×(Ω - ω) + M‚∇H Dt (13)
Combining eqs 2, 10, and 12 yields, for the equation of internal angular momentum,
Ds ) FG + (η′ + λ′)∇(∇‚ω) + η′∇2ω + Dt 4ζ(Ω - ω) + M×H (14)
F
Constituted Stress-Boundary Conditions Next, the form of constituted equations of boundary-stress equilibrium are developed, assuming that there are no surface or volume electrical currents. As a prerequisite, the stress vector tn, which corresponds to eq 7 (see the Appendix), is determined as
(
(
))
∂Vn ∂Vt ∂Vn n+ + t + λ(∇‚v)n + ∂xn ∂xn ∂xt µ0 2ζn×(Ω - ω) - H2n + BnH (15) 2
tn ) -pn + η 2
[
]
∂Vn µ0 - λ(∇‚v) + M2n + 2σH ) 0 ∂xn 2
p - 2η
where Vn is the component of velocity along the direction of the normal at a point on the interface, Vt the tangential component of velocity along the interface, xn the distance along the direction of the normal, and xt the distance along the tangential direction. Computation of the stress difference across an interface between fluids in eq 6 then yields the following expression for the normal component of the stress-boundary condition at the
(16)
Note that the magnetic final two terms of eq 15 yield a normal component but no tangential component of stress difference.1 Magnetic field boundary conditions that express continuity across the interface of the normal induction field Bn,1 ) Bn,2 and continuity of the tangential magnetic field Ht1 ) Ht2 are used to obtain that result. The latter condition assumes that there are no surface electrical currents. The stress balance for the tangential direction at an interface between fluids can be obtained using the following formulation, which subtracts the normal stress difference from the total stress difference:
[tn]‚(I - nn) ) 0
(11)
where η′ is the first or ordinary coefficient of spin viscosity and λ′ is the second coefficient of spin viscosity.
∇‚C ) (η′ + λ′)∇(∇‚ω) + η′∇2ω
interface, where, as defined previously, brackets denote a difference.
(17)
Interfacial tension is absent from this expression, because its orientation is purely normal to the interface, and only the ordinary viscosity and the vortex viscosity terms of eq 15 contribute terms to the tangential balance. The result is
[( η
)
]
∂Vn ∂Vt + + 2ζ(Ω sin(R) - ω sin(β)) ) 0 (18) ∂xt ∂xn
where R is the angle between the unit normal n and the vorticity Ω, and β is the angle between the unit normal n and the angular velocity ω. At an interface between the magnetic fluid and a solid, the terms in the bracket equate to the elastic stress in the solid. The stress balances are written in a form that is valid in the local coordinate system. This representation highlights the phenomenological origin of the stresses. Simplifications Many of the preceding relationships are simplified for a magnetic fluid in contact with the atmosphere or another gaseous environment that has a uniform pressure p0 and negligible viscous forces. For example, the normal direction stress balance of eq 16 simplifies to
∂Vn µ0 2 + M - p0 ) 2σK p - 2η ∂xn 2 n
(19)
Angular acceleration (Ds/Dt) of the typically 10-nm magnetic particles of a magnetic fluid will almost always be negligible in eq 14, compared to the magnetic and viscous couples. It has long been thought, and recent numerical study7 indicates, that C is negligible. In the absence of a body force transmitted from afar (G ) 0), which is the usual case, and with negligible couple stress (C ) 0), the equation of internal angular momentum reduces to
4ζ(Ω - ω) + M×H ) 0
(20)
Under the conditions of eq 20, the term 2ζn×(Ω - ω) in the expression for tn of eq 15 may be replaced by
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6115 Table 1. Interfacial Stress Relationships for Nonconductive Magnetized Fluidsa description
conditions
relationship
Normal Stress compressible
M1, M2 > 0; η1, η2 > 0
incompressible
∇‚v ) 0; M1, M2 > 0; η1, η2 > 0
incompressible, one-phase-nonmagnetic with negligible viscosity
∇‚v ) 0; M2 ) 0; η2 ) 0 Tangential Stress
compressible or incompressible
M1, M2 > 0; η1, η2 > 0
compressible or incompressible, negligible spin angular acceleration or diffusion
∂s/∂t ) 0; C ) 0; M1, M2 > 0; η1, η2 > 0
same as preceding case and one-phasenonmagnetic with negligible viscosity
∂s/∂t ) 0; C ) 0; M2 ) 0; η2 ) 0
a
]
∂Vn µ0 - λ(∇‚v) + Mn2 + 2σH ) 0 ∂xn 2 ∂Vn µ0 2 p - 2η + M + 2σH ) 0 ∂xn 2 n ∂Vn µ0 2 p - 2η + M ) p0 + 2σK ∂xn 2 n p - 2η
]
[( [( (
) )] )
]
∂Vn ∂Vt + + 2ζ(Ω sin(R) - ω sin(β)) ) 0 ∂xt ∂xn ∂Vn ∂Vt 1 η + - [MH sin(γ) sin(θ)] ) 0 ∂xt ∂xn 2 ∂Vn ∂Vt 1 η + - (MH sin(γ) sin(θ)) ) 0 ∂xt ∂xn 2 η
Subscripts “2” and “1” denote above and below the interface, respectively. M1 t M, η1 t η. All other symbols are defined in the Nomenclature section.
-1/2n×(M×H), giving
(
( ( )
))
∂Vn ∂Vt ∂Vn n+ + t + λ(∇‚v)n + ∂xn ∂xn ∂xt µ0 2 1 - n×(M×H) H n + BnH (21) 2 2
tn ) -pn + η 2
()
Substituting from eq 20 for the vortex viscosity term in eq 13 yields a well-known form of the linear momentum equation that highlights the role of magnetization:
F
[ [
1 Dv ) Fg + -∇p + η∇2v + ∇×(M×H) + M‚∇H (22) Dt 2
In an unequilibrated magnetic fluid, the magnetization direction M and magnitude M generally have values other than their equilibrium values in the instantaneous field H. Accordingly, an additional relationship (a relaxation equation), is needed to determine the relationship between these field vectors. A relatively simple relaxation equation has the form8
1 DM + M(∇‚v) ) ω×M + (M0 - M) Dt τ
(23)
where M0 is the equilibrium magnetization in field H. For an incompressible medium under the conditions of eq 20, this relationship may be expressed in a form that eliminates ω.
1 1 DM ) Ω×M M×(M×H) + (M0 - M) (24) Dt 4ζ τ
( )
The tangential stress balance corresponding to the simplified eq 20 is determined next. Again, the term 2σn×(Ω - ω) in eq 15 for the stress vector tn is replaced by -1/2n×(M×H), which is a vector lying in the tangential plane of the interface. From the property of the cross product, the term can be written as 1/ MH sin(γ) sin(θ)t, where γ is the angle between M and H, 2 and θ is the angle between M×H/MH and the normal vector n. The projection of this term along the tangential direction, i.e., t‚(n×(M×H)), is just its magnitude. Thus, the tangential stress balance takes the following form and is correct for compressible or incompressible media:
[( η
)]
∂Vn ∂Vt + ∂xt ∂xn
1 - [MH sin(γ) sin(θ)] ) 0 2
(25)
If the interface separates magnetic fluid from a typical gas, the brackets may be dropped. Table 1 summarizes various general and specialized forms of the boundary stress relationships. Applications A uniform magnetic field rotated in the horizontal plane induces ferrofluid contained in a beaker into a spin-up flow, as observed at the free surface. Surprisingly, with a concaveupward meniscus, the direction of the flow is counter to that of the field, and with a convex-upward meniscus, the flow corotates with the field.9 It can be understood in reference to eq 15, and was shown in ref 9, with ω vertical and the unit normal n subtending the angle β, the 2ζn×ω term produces a surface tangential stress over the area of the meniscus that has a tendency to drive the flow in the observed directions. Also interesting is the fact that the ferrofluid rotation rate is faster in smaller, cylindrical vessels. This response is understood by the noting that the meniscus then covers a larger fraction of the surface area and, hence, contributes a relatively larger spin-up couple. Compared to the spin-up flow described previously, magnetic surface stress can be configured in a more efficient configuration that is suitable for the transport of fluid in open channels.10 This is modeled by the same authors, using their eq 2. The source produces a rotating magnetic field, with the axis of rotation oriented azimuthally everywhere around a circular duct. Hence, the particle angular rate ω is oriented in the radial direction and is everywhere perpendicular to the surface normal n. As a result, the tangential stress is operative over the entire free surface. For the circular duct, vn ) 0, so that ∂Vn/∂t ) 0, and the angle θ is 90°; therefore, in eq 25, sin θ ) 1. The equilibrium shapes of rotating bodies are of interest in various fields of physics, such as the stationary form of planets and stars, and the geometry of atomic nuclei. In the laboratory, drops may be spun up by mechanical means. For drops where a magnetic fluid is involved, it is convenient to use magnetic fields. The rotation of ferrofluid drops is an area in which surface tangential and normal stresses have an important role. The first investigation studied small droplets (∼10 µm in size) formed as a product of phase separation and having a tiny amount of interfacial tension, subjected to a
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rapidly rotating magnetic field.11 The study produced shapes resembling starfish, worms, loops, and other configurations. Shapes of macroscopic rotating ferrofluid drops have been studied experimentally and theoretically with predictions made of bifurcative behavior.12 Macroscopic drops confined between the closely spaced parallel walls of a Hele-Shaw cell exhibit very novel shapes in which patterns, produced in steady magnetic field and vice versa, are subjected to a superposed rotating field.13 In all these cases, a complete analysis of the dynamics must consider the complete relationships for surface normal and tangential stresses.
parameters Vn and Vt are not necessarily the components normal and tangential to the interface at other points on the curved interfacesonly at the point in question. The Cartesian coordinate xn is also the distance along the constant direction of the normal to the interface at the point in question and xt is the distance along the constant tangential direction at the point in question. They are not the normal and tangential distances at other points on the interface, which generally change along the interface. Thus, after the point in question is identified, derivatives with respect to xn and to xt of Vn and Vt are not derivatives, with respect to coordinates that change direction to follow interfacial normal and tangential vectors. Adding the aforementioned contributions gives
Conclusion The stress-boundary relationships that have been available are rather incompletely elaborated and scattered in the literature.14 This work utilizes known governing equations for the motion of magnetic fluids to assist in the formulation and compilation of general, interfacial stress-boundary conditions. It is hoped that the results will be useful as tools for researchers and others in the analysis and understanding of these free surface flows.
(
)
∂Vn ∂Vt ∂Vn n+ + t ∂xn ∂xn ∂xt
n‚(∇v + (∇v)T) ) 2
(A.4)
The vortex viscosity term 2ζ‚(Ω - ω) is transformed using the expansion of the polyadic alternator and identifying e1 ) i, e2 ) j, and e3 ) k.
E ) ijkeiejek ) ijk + jki + kij - ikj - jik - kji (A.5) Defining (Ω - ω) ) A ) Axi +Ayj +Azk, it follows that
Appendix Form of the Stress Vector. The stress vector tn is obtained as the scalar product tn ) n‚T, where T is given by eq 7, which is repeated here (renamed as eq A.1).
T ) -pI + η(∇v + (∇v)T) + 2ζ‚(Ω - ω) + µ0H2 λ(∇‚v)I I + BH (A.1) 2 The various terms of this relationship will be transformed individually. Recalling that I ) δijeiej ) ii + jj + kk and expressing n ) nxi + nyj + nzk, it is readily observed that n‚I ) (nxi + nyj + nzk) ) n, which illustrates the general result that the scalar product of any vector with the idem factor I yields the vector itself. This fact serves to transform the first, fourth, and fifth terms on the right-hand side of eq A.1. At any point on the interface, a stress vector tn that is due to the ordinary viscous stresses has a component in the normal direction to the interface and a component in the tangential plane. Denoting t as the unit vector in the direction of the tangential component, a two-dimensional representation may be used. With n the unit normal vector,
(
n‚∇v ) n‚ nn
)
∂Vn ∂Vt ∂Vn ∂Vt + nt + tn + tt ∂xn ∂xn ∂xt ∂xt
∂Vn ∂Vt +t ∂xn ∂xn
)n
(
n‚(∇v)T ) n‚ nn
)
∂Vn ∂Vt ∂Vn ∂Vt + tn + nt + tt ∂xn ∂xn ∂xt ∂xt
∂Vn ∂Vn +t ∂xn ∂xt
)n
(A.2)
(A.3)
In the above, Vn and Vt are components of the velocity that are respectively referenced to the fixed Cartesian axes normal and tangential to the interface at the fixed point in question, which also serves as the coordinate system origin. The
E‚A ) Ax(jk - kj) + Ay(ki - ik) + Az(ij - ji) (A.6) With n ) nxi + nyj + nzk, the scalar product with E‚A is easily determined from eq A.6 as
n‚(E‚A) ) (nzAy - nyAz)i + (nxAz - nzAx)j + (nyAx - nxAy)k (A.7) where the right-hand side can be recognized as the vector product -n×A; hence,
n‚(‚(Ω - ω)) ) n×(Ω - ω)
(A.8)
Substituting the aforementioned results into eq A.1 yields the relationship that appears as eq 15 in the text.
(
(
))
∂Vn ∂Vt ∂Vn n+ + t + λ(∇‚v)n + ∂xn ∂xn ∂xt µ0 2ζn×(Ω - ω) - H2n + BnH (A.9) 2
tn ) -pn + η 2
Nomenclature B ) vector magnetic induction (tesla) C ) dyadic couple stress tensor (N/m) G ) vector body couple per unit mass transmitted from afar (N m/kg) ei ) unit vector in the ith direction ej ) unit vector in the jth direction ek ) unit vector in the kth direction g ) vector acceleration due to gravity (N/kg) H ) vector magnetic field (A/m) H ) curvature of interface (m-1) M ) vector magnetization (tesla) Ms ) vector saturation magnetization (tesla) n ) unit normal vector p ) pressure (N/m2) s ) vector spin angular momentum per unit mass (m2/s) t ) time (s) t ) unit tangent vector tn ) stress vector (N/m2)
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6117
T ) stress dyadic (N/m2) v ) fluid velocity (m/s) xn ) coordinate distance along the normal direction (m) xt ) coordinate distance along the tangential direction (m) Greek Symbols R ) angle subtended between the vorticity and the interfacial unit normal (radian) β ) angle subtended between the particle angular rate and the interfacial unit normal (rad) γ ) angle subtended between the vector magnetization and the vector magnetic field ζ ) vortex viscosity (N s/m2) η ) ordinary or first coefficient of viscosity (kg m-1 s-1) η′ ) ordinary or first coefficient of spin viscosity (N s) θ ) angle subtended between the unit vectors M×H/MH and n λ ) second coefficient of viscosity (kg m-1 s-1) λ′ ) second coefficient of spin viscosity (N s) µ0 ) permeability of free space (tesla m/A) F ) mass density (kg/m3) σ ) interfacial tension (N/m) τ ) relaxation time constant (s) ω ) vector angular velocity of particles (rad/s) Ω ) vector vorticity of the fluid medium (rad/s) Note Added after ASAP Publication. The version of this paper that was published on the web 1/18/2007 had errors involving some of the symbols used in the paper. The corrected version was published on 2/2/2007. Literature Cited (1) See, for example: Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: New York, 1985. (Reprinted with minor corrections and updates by Dover Publications, Mineola, NY, 1997.) (2) The dyadic notation is used, wherein ∇‚T ) ej(∂Tij/∂xi), with subscript i denoting the orientation of the surface upon which stress component of orientation j acts. Other definitions are stated where introduced in the text. (3) Brenner, H. Antisymmetric stresses induced by the rigid-body rotation of dipolar suspensions. Int. J. Eng. Sci. (Oxford, U.K.) 1984, 22, 645. (4) 2H ) ∇‚n ) R1-1 + R2-1 where R1 and R2 are the radii of surface curvature in two planes at right angles. A radius of curvature is positive within the medium, and negative outside the medium. (5) (a) In addition to the usual, symmetric, pressure-viscous stress terms, eq 7 incorporates effects that result from particle rotation in the viscous carrier liquid and particle interaction with the magnetic field. The magnetic field terms (last two terms) have the same form as that for equilibrated flow. A recent review by the investigator initially responsible for formulating unequilibrated dynamics of magnetic fluids is given in: Shliomis, M. In Ferrofluids: Magnetically Controllable Fluids and Their Applications; Odenbach, S., Ed.; Lecture Notes in Physics, Vol. 594; Springer: Berlin, 2002, p 85. (b) Proof of the applicability of the magnetic field terms in unequilibrated systems is developed in: Rosensweig, R. E. Continuum
equations for magnetic and dielectric fluids with internal rotations. J. Chem. Phys. 2004, 121, 1228, and less completely in: Felderhof, B. U.; Kroh, H. J. Hydrodynamics of magnetic and dielectric fluids in interaction with the electromagnetic field. J. Chem. Phys. 1999, 110, 7403. The vortex viscosity terms are determined to ensure positive rates of entropy production. The third term on the right of eq 7 and the last term are asymmetric, whereas their sum is symmetric under the conditions of eq 20, see ref 6. (c) In preparing the present paper, the author was unaware of the Shliomis formulation of boundary stress relationships in his chapter cited above. Although the Shliomis results are not quite as general as those presented in this work, and apparently are not intended to be, the chapter is highly recommended as an alternative resource, with the formulation developed independently using a somewhat different methodology. (6) The total stress dyadic contains two antisymmetric terms. One term is that which has the vortex viscosity ζ as a coefficient, as seen from eq A.1 in the Appendix. Expanding BH ) 1/2(MH - HM) + 1/2(BH + HB) in eq 7 shows that 1/2(MH - HM) is the other. When particle angular acceleration and couple stress are negligible, the antisymmetric terms combine to yield a symmetric total stress dyadic. (7) Feng, S.; Graham, A. L.; Abbot, J. R.; Brenner, H. Antisymmetric stresses in suspensions. Vortex viscosity and energy dissipation. Phys. Fluids 2006, 563, 97-122. (8) Except for the ∇‚v term, which comes into play for compressible fluids, this is the well-known Shliomis relaxation equation. Originally proposed on phenomenological grounds, the relationship is derived with the aid of irreversible thermodynamics in the citations to Rosensweig and Felderhof noted in ref 5. A microscopic derivation of a relaxation equation thought to be more accurate at large deviations from equilibrium is discussed in: Shliomis, M. I. Ferrohydrodynamics: Retrospective and Issues. In Ferrofluids; Odenbach, S., Ed.: Springer: Berlin, 2002; pp 85-111. (9) Rosensweig, R. E.; Popplewell, J.; Johnston, R. J. Magnetic fluid motion in rotating field. J. Magn. Magn. Mater. 1990, 85, 171. (10) Krauss, R.; Liu, M.; Reimann, B.; Richter R.; Rehberg, I. Pumping fluid by magnetic surface stress. New J. Phys. 2006, 8, 18. (11) Bacri, J.-C.; Cebers, A. O.; Perzynski, R. Behavior of a magnetic microdrop in a rotating magnetic field. Phys. ReV. Lett. 1994, 72, 2705. (12) Lebedev, A. V.; Enge, A.; Morozov, K. I.; Bauke H. Ferrofluid drops in rotating magnetic fields. New J. Phys. 2003, 5, 57. (13) Rhodes, S.; Perez, J.; Elborai, S.; Lee, S-H.; Zahn, M. Ferrofluid spiral formations and continuous-discrete phase transitions under simultaneously applied DC axial and AC in-plane rotating magnetic fields. J. Magn. Magn. Mater. 2005, 289, 353. (14) As examples: Stress boundary conditions in unequilibrated magnetic fluid are not treated in ref 1. A formula for the magnetic component of tangential antisymmetric stress is given in Section 5.2 (Blums, E.; Cebers, A.; Mairov, M. M. Magnetic Fluids; Walter de Gruyter: Berlin and New York, 1997) and relates to the early work of Cebers (Cebers, A. Interphase pressure in the hydrodynamics of liquid with internal rotation. Magnetohydrodynamics 1975, 11 (1), 63). Reference 9 assumes that the form of the equilibrium stress tensor can be used for the unequilibrated case but writes only one of the tangential stress terms. In an earlier work (Rosenthal, A. D.; Rinaldi, C.; Franklin, T.; Zahn, M. Torque measurements in spin-up flow of ferrofluids. J. Fluids Eng. 2004, 126, 198), the authors applied the stress tensor methodology to find wall torque without explicitly writing the stress difference formula. Other examples could be given.
ReceiVed for reView May 24, 2006 ReVised manuscript receiVed November 21, 2006 Accepted November 22, 2006 IE060657E
