onthemagnetogravitationalinstabilityofaferromagneticdust ...eﬀect of nonuniform rotation on the...

International Scholarly Research NetworkISRN Astronomy and AstrophysicsVolume 2012, Article ID 104941, 6 pagesdoi:10.5402/2012/104941

Research Article

On the Magnetogravitational Instability of a Ferromagnetic DustCloud in the Presence of Nonuniform Rotation

Joginder S. Dhiman and Rekha Dadwal

Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla 171005, India

Correspondence should be addressed to Joginder S. Dhiman, [email protected]

Received 10 April 2012; Accepted 6 June 2012

Academic Editors: S. Bogovalov, C. W. Engelbracht, and R. Hawkes

Copyright © 2012 J. S. Dhiman and R. Dadwal. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The effect of a nonuniform magnetic field on the gravitational instability of a nonuniformly rotating infinitely extendingaxisymmetric cylinder in a homogenous ferromagnetic medium has been studied. The propagation of the wave is allowed alongradial direction. A general dispersion relation, using the normal mode analysis method on the perturbation equations of theproblem, is obtained. It is found that Bel and Schatzman criterion determines the gravitational instability of this general problem.Thus, it appears that the effect of non-uniform magnetic field on the gravitational instability as discussed by (Dhiman and Dadwal,2010) is marginalized by the magnetic polarizability of ferrofluid.

1. Introduction

The gravitational instability (GI) or the Jeans instabilitycauses the collapse of interstellar gas clouds and subsequentstar formation. It occurs when the internal gas pressureis not strong enough to prevent gravitational collapse ofa region filled with matter. GI plays a dominant role indetermining the star formation properties of galaxies. Theproblem of instability of self-gravitating large gas clouds wasfirst considered by Jeans [1] and underlined the importanceof self-gravitating instabilities in astrophysics because oftheir crucial role in understanding collapse, formation, andevolution of interstellar molecular clouds, star formation,galactic structure and its evolution, and so forth. For a latestand broader review on the subject of gravitational instability,one may refer to [1–8] and references therein.

Theoretical studies have shown that the magnetic fieldplays a vital role in self-gravitating star-forming regions andin the evolution of interstellar clouds into self-gravitatingstar forming regions. The problem of magnetogravitationalinstability of interstellar rotating medium is of considerableimportance in connection with the protostar and starformation in magnetic dust clouds. In astrophysics, theproblems that are considered generally assume magneticfield/rotation to be uniform: however, this idealization of

the uniform character in theoretical investigations is validonly for laboratory purposes (cf. Larson [8] and referenceswithin), because in the interstellar interior and atmosphere,the magnetic field/rotation may be variable and may togetheralter the nature of the instability. Limited efforts have beenput to investigate the GI by considering the magnetic fieldor rotation to be variable. Bel and Schatzman [9] studied theeffect of nonuniform rotation on the onset of gravitationalinstability and obtained a modified expression for the Jeanscriterion, which is now known as the Bel and Schatzmancriterion. Recently, Dhiman and Dadwal [10, 11] also studiedthe simultaneous effect of nonuniform magnetic field androtation on the gravitational instability of gaseous mediumand obtained some general qualitative results.

In astronomy, the interstellar medium (ISM) is thematter that exists in the space between the star systemsin a galaxy. This matter includes gas in ionic, atomic,and molecular form, dust, and cosmic rays. It fills inter-stellar space and blends smoothly into the surroundingintergalactic space. The dust, which is composed of smallsolid particles by segregation in the interstellar clouds, is avery important component of the ISM. In the recent past,there have been dramatic changes in the conception of theinterstellar medium, and recent observational and numericalworks have suggested that interstellar medium (ISM) plays

2 ISRN Astronomy and Astrophysics

an important role in the star formation. Spitzer [12] hasshown that solid particles in space including dust grainsare eclectically charged, relative to surrounding plasma.Further, the efficiency and timescale of stellar birth inGalactic molecular clouds strongly depend on the propertiesof interstellar medium (ISM).

Ferrofluids are colloidal liquids made of nanoscaleferromagnetic, or ferrimagnetic, particles suspended in acarrier fluid (usually an organic solvent or water). Aferrofluid flow in the presence of an applied magnetic fieldis accompanied by an intertwinement of hydrodynamicand magnetic interactions (Rosenwieg [13]). A ferrofluidis stable and does not solidify even in the presence ofa high magnetic field and is commonly characterized byits strong tendency to magnetize in the direction of themagnetic field. Jones and Spitzer [14] provided a modelfor the existence of gas-dust interstellar mediums with ahighly pronounced property of magnetic polarizability. Thiscan be assumed due to a super paramagnetic dispersionof the fine ferromagnetic grains suspended in a gaseouscloud of molecular hydrogen. Mamun and Shukla [15]observed the usual Jeans instability in a self-gravitating darkinterstellar molecular cloud containing ferromagnetic dustgrains and baryonic gas clouds and supported the existenceof ferromagnetic dust particles in a magnetically supporteddark interstellar self-gravitating interstellar molecular cloud.

The present study is primarily motivated by the investiga-tions of Mamun and Shukla [15] regarding the instability of aself-gravitating dark interstellar molecular cloud containingferromagnetic dust grains. Our aim here is to investigatethe gravitational instability of an infinite axisymmetriccylinder of a homogenous nonuniformly rotating mediumcontaining the ferromagnetic dust particles in the presenceof nonuniform magnetic field. The mathematical analysisfollowed in this paper is precisely the same as that of [9,10]. Since the system of linearized perturbation equationgoverning the problem contains variable coefficients, the suf-ficient condition for GI is derived using the local instabilityapproach as adopted in [9].

2. Mathematical Model and Basic Equations

Consider an infinite homogeneous, interstellar self-gravitating molecular cloud containing ferromagnetic dustparticles. The system is permeated with the simultaneousaction of a nonuniform rotation and a nonuniformmagnetic field. In a rectangular coordinate system, let

�u = (ux,uy ,uz) be the velocity, �M = (Mx,My ,Mz) bethe ferrofluid magnetization under the magnetic field�H = (Hx,Hy ,Hz). For the treatment of this physicalconfiguration for gravitational instability, the conventionalset of ferrohydrodynamic equations are given by (cf. [16, 17])

d�udt= −grad p + ρ grad φ + μ0

(�M · ∇

)�H , (1)

∂ρ

∂t+(�u · grad

)ρ + ρ

(∇.�u) = 0, (2)

∇2φ = −4πG(ρ − ρ0

). (3)

The ferrofluid magnetization �M satisfies Shliomis’ [16, 17]equation of magnetization, which is given by

d �Mdt

= 12

(∇× �u)× �M − α(�M − �M0

)− β �M ×

(�M × �H

).

(4)

In the above equations p, φ, μ0, ρ, and G, respectively, denotethe pressure, gravitational potential, the inverse Browniantime constant for particle diffusion in the ferrofluid, themagnetic permeability, density, the gravitational constant,α = 3ηV/kbT is the Brownian time of rotational particlediffusion and β = 1/6ηϕ, where η is the dynamic viscosityof the carrier fluid and ϕ = nV is the volume fractionof magnetic grains in the liquid. Here, n is the numberdensity and V the volume of a single particle. Also, ρ0 and�M0 = (0, 0,M0) are the equilibrium values of density and

magnetization and M0 the equilibrium magnetization offerrofluid is related to the equilibrium magnetic felid H0 by

M0 = nM

(coth ψ − 1

ψ

)H0

‖H0‖ , (5)

where M is the magnetic moment of single ferromagneticparticle and the nondimensional quantity ψ is given by ψ =μ0(M‖H0‖/kbTb), where kb is the Boltzman constant andTb is temperature of the ferrofluid. Further, the Poisson’sequation (3) is considered so as to avoid the “Jeans Swindle”(cf. Speigel and Thiffeault [7]).

In Chu formulation of electrodynamics (see Penfield and

Haus [18]), the magnetic field �H , magnetization �M, and

magnetic induction �B are related by

�B = μ0

(�H + �M

). (6)

We know that the magnetic field �H satisfies the followingMaxwell’s equations:

∇× �H = 0, (7)

∇ · �B = 0. (8)

Equation (6), using (8) yields

∇ · �H = −∇ · �M. (9)

In the present analysis, we shall consider an infiniteaxisymmetric cylinder of homogeneous, infinitely conduct-ing self-gravitating ferromagnetic dust cloud under thesimultaneous effect of a nonuniform rotation and a nonuni-form magnetic field, therefore, transforming the abovefundamental equations in cylindrical coordinates (r, θ, z).

Denoting �u = (ur ,uθ ,uz) and �M = (Mr ,Mθ ,Mz) by the

components of velocity �u and ferrofluid magnetization �Malong the radial r, the transverse θ, and the axial z directions,respectively. The cylinder is assumed to be rotating aboutits axis (z-axis) with nonuniform angular velocity ω. Thepropagation of wave is taken along the radial direction “r”

ISRN Astronomy and Astrophysics 3

of the cylinder, hence ∂/∂r is the only nonzero component ofthe gradient.

Following Dhiman and Dadwal [10], the basic equations(1)–(4), (7) and (9) under these assumptions take thefollowing forms in cylindrical polar coordinates

ρ

(∂ur∂t

+ ur∂ur∂r

− uθ2

r

)

= ρ∂φ

∂r− ∂p

∂r+ μ0

[Mr

∂Hr

∂r− MθHθ

r

],

ρ(∂uθ∂t

+ ur∂

∂ruθ +

uruθr

)

= μ0

2

[Mr

∂Hθ

∂r+Hr

∂Mθ

∂r+Mθ

∂Hr

∂r

−Hθ∂Mr

∂r+

2MθHr

r

],

ρ(∂uz∂t

+ ur∂uz∂r

)

= μ0

2

[Mr

∂Hz

∂r+Hr

∂Mz

∂r+Mz

∂Hr

∂r−Hz

∂Mr

∂r

+(MzHr −HzMr

r

)],

∂Mr

∂t+ ur

∂Mr

∂r+

12

(Mθ

(∂uθ∂r

+uθr

)+Mz

∂uz∂r

)

+ α(Mr −M0) + β(Mθ[MrHθ −HrMθ]

−Mz [HrMz −MrHz]) = 0,

∂Mθ

∂t+ ur

∂Mθ

∂r− 1

2

(Mr

(∂uθ∂r

+uθr

))

+ α(Mθ −M0) + β(Mz[MθHz −HθMz]

−Mr [MrHθ −HrMθ]) = 0,

∂Mz

∂t+ ur

∂Mz

∂r− 1

2

(Mr

∂uz∂r

)+ α(Mz −M0)

+ β(Mr[HrMz −MrHz]

−Mθ [MθHZ −HθMz]) = 0,

∂Hz

∂r= 0,

(∂Hθ

∂r+Hθ

r

)= 0,

(∂Hr

∂r+Hr

r

)= −

(∂Mr

∂r+Mr

r

),

∂ρ

∂t+(�u · grad

)ρ + ρ

(∇ · �u) = 0,

∇2φ = −4πG(ρ − ρ0

).

(10)

In these equations, the operators ∇ and ∇2 now are respec-tive operators in cylindrical coordinates.

3. Equilibrium State andPerturbation Equations

Following the physical models of Mamun and Shukla [15]and Kumar et al. [19], the magnetization is taken along r andz directions.

The equilibrium state under discussion is clearly charac-terized as follows:

�u = (0, rω, 0); �H = (0, 0,Hz); �M = (Mr , 0,Mz)

p = p0; φ = φ0; ρ = ρ0

(11)

Using (11) in (10) and using the fact that propagation ofwave is along r direction and thus ∂/∂r is the only nonzerocomponent, we obtain the following basic solution:

∂p0

∂r= ρ0

(∂φ0

∂r+ rω2

)(12)

Mr∂Hz

∂r−Hz

(∂Mr

∂r+Mr

r

)= 0 (13)

α(Mr −M0) + βMzMrHz = 0 (14)

−12Mr

(∂uθ∂r

+uθr

)+ αM0 = 0 (15)

α(Mz −M0)− βM2r Hz = 0 (16)

∂Hz

∂r= 0 (17)

(∂Mr

∂r+Mr

r

)= 0 (18)

1r

(∂

∂r

(r∂φ0

∂r

))= 0 (19)

Equation (17) implies that

Hz = Constant (20)

Since pressure and density are uniform initially, therefore itis clear from (12) that the gravitational potential is balancedby centrifugal force.

Let us allow the small perturbation in the initial statedescribed by (11) as following;

�u′ =(u′r , rω + u′θ ,u′z

); �H′ =

(h′r ,h

′θ ,Hz + h′z

);

p′ = p0 + δp �M =(Mr +m′

r ,m′θ ,Mz +m′

z

);

φ′ = φ0 + δφ; ρ′ = ρ0 + δρ.

(21)


Using (21) in (10), ignoring the terms of second and higherorders in the perturbations, and using (12)–(20), we have thefollowing linearized perturbed equations:

ρ0

(∂ur∂t− 2ωuθ

)= ρ0

∂δφ

∂r− ∂δp

∂r+ μ0

(Mr

∂hr∂r

), (22)

ρ0

(∂uθ∂t

+ ur∂

∂r(rω) + urω

)= μ0

2

(Mr

∂hθ∂r

− hθ ∂Mr

∂r

),

(23)

ρ0∂uz∂t

= μ0

2

(hr∂Mz

∂r+Mz

1r

∂

∂r(rhr)−Hz

1r

∂

∂r(rmr)

),

(24)

∂mr

∂t+ ur

∂Mr

∂r+

12

(Mz

∂uz∂r

+mθ

(∂

∂r(rω) + ω

))+ αmr

+ β(−M2

z hr +HzMzmr +MzMrhz +HzMrmz) = 0,

(25)

∂mθ

∂t− 1

2

(mr

(∂

∂r(rω) + ω

)+Mr

1r

∂

∂r(ruθ)

)+ αmθ

+ β(HzMzmθ −M2

z hθ −M2r hθ

) = 0,(26)

∂mz

∂t+ ur

∂Mz

∂r− 1

2Mr

∂uz∂r

+ αmz

+ β(MrMzhr − 2MrHzmr −M2

r hz) = 0,

(27)

∂hz∂r

= 0, (28)

(∂hθ∂r

+hθr

)= 0, (29)

1r

∂

∂r(rmr) = −1

r

∂

∂r(rhr), (30)

∂

∂tδρ + ρ0

(∂ur∂r

+urr

)= 0, (31)

1r

(∂

∂r

(r∂δφ

∂r

))= −4πGδρ, (32)

where (u′r ,u′θ ,u′z), (h′r ,h

′θ ,h′z), (m′

r ,m′θ ,m′

z), δp, δρ, andδφ are the respective perturbations from basic state invelocity, magnetic field, magnetization, pressure, density,and gravitational potential vector. In the above equations, thedashes have been dropped for convenience in writing.

Since we have considered that the fluctuations in pressureand density take place adiabatically, therefore

δp = c2δρ. (33)

Using this equation of state (33) for adiabatic medium, (22)reduces to

ρ0

(∂ur∂t− 2ωuθ

)= ρ0

∂δφ

∂r− c2 ∂δρ

∂r+ μ0

(Mr

∂hr∂r

).

(34)

4. Gravitational Instability

In order to investigate the stability of the forgoing stationarystate, we shall consider the dependence of the perturbationon r and t of the form

ψ∗(r) exp(σt), (35)

where σ is frequency of the perturbation.For this type of dependence of perturbation on r and t,

we have

∂

∂t≡ σ ,

∂

∂rf (r) = d

drf (r). (36)

Using the above dependence, the perturbations equations(34) and (23)–(32) assume the following forms:

σur − 2ωuθ − dδφ

dr+c2

ρ0

dδρ

dr− μ0

ρ0

(Mr

dhrdr

)= 0, (37)

σuθ + urd

dr(rω) + urω − 1

2μ0

ρ0

(Mr

dhθdr

− hθ dMr

dr

)= 0,

(38)

σuz − 12μ0

ρ0

(hrdMz

dr+Mz

1r

d

dr(rhr)−Hz

1r

d

dr(rmr)

)= 0,

(39)

(σ + α)mr + urdMr

dr+

12

(Mz

duzdr

+mθ

(d

dr(rω) + ω

))

+ β(−M2

z hr +HzMzmr +MzMrhz +HzMrmz) = 0,

(40)

(σ + α)mθ − 12

(mr

(d

dr(rω) + ω

)+Mr

1r

d

dr(ruθ)

)

+ β(HzMzmθ −M2

z hθ −M2r hθ

) = 0,(41)

(σ + α)mz + urdMz

dr− 1

2Mr

duzdr

+ β(MrMzhr − 2MrHzmr −M2

r hz) = 0,

(42)

dhzdr

= 0, (43)

1r

d

dr(rhθ) = 0, (44)

1r

d

dr(rmr) = −1

r

d

dr(rhr), (45)

σδρ +ρ0

r

d

dr(rur) = 0, (46)

1r

(d

dr

(rdδφ

dr

))= −4πGδρ. (47)

Since, the above equations involve the variable coeffi-cients, we shall therefore investigate the local stability of the

ISRN Astronomy and Astrophysics 5

above system in the neighborhood of r = r0. For this, let usassume that the perturbations have a periodic form in theneighborhood of r = r0, as

f exp(−ikr), (48)where k is the wave number. For this type of dependence, wehave

d

dr≡ −ik. (49)

Equations (43)–(45) upon using (49) yield

hz = 0, hθ = 0, hr = −mr. (50)

Now, substituting (49) in (37)–(42), (46), (47) and using(50), we obtain a system of algebraic equations (withcoefficients, in the vicinity of r = r0) for amplitudes, markedwith bars, which can be put in the matrix notation asfollowing;

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1σ

(σ2 +Q

) −2ω 0 −ik μ0

ρ0Mr 0 0

P σ 0 0 0 0

0 0 σμ0

2ρ0

[dMz

dr− ikBz

]0 0

dMz

dr0 − ikMz

2σ +N

P

2βHzMz

0 − ikMr

20

P

2σ + R 0

dMz

dr0 − ikMr

2T 0 σ + α

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎣

uruθuzmr

mθ

mz

⎤⎥⎥⎥⎥⎥⎥⎥⎦= 0. (51)

Here, Bz = (Mz + Hz), Q = k2c2 − 4πGρ0, N = α + β(Mz2 +

HzMz), R = (α + βMzHz), and T = (MrMz − 2MrHz).For the nontrivial solution of the system (51), the

determinant of the matrix of coefficients should vanish.Thus, on expanding the coefficient matrix and equatingthe real part equal to zero, we get the following dispersionrelation:

σ6 + B1σ5 + B2σ

4 + B3σ3 + B4σ

2 + B5σ + B6 = 0, (52)

where,

B1 = (α +N + R),

B2 =(

8Pω + 4Q − P2 + 4NR + 4αN + 4αR

+k2 μ0

ρ0BzMz − 4βTHzMr

),

B3 =(μoρ0

4NQ+4QR + 4αQ − αP2 + 4NR + 8ωNP + 8ωPR

+ 8αωP + k2 μ0

ρ0RBzMz + αk2 μo

ρ0RBzMz

+k2βμoρ0RBzMzM

2r − 4βRTMrHz

),

B4 =(− P2Q − 2ωP3 + 4ρ0NQR + 4αNQ + 4αρ0QR

+ 8ωNPR + 8αωNP + 8αωPR− 4βQTMrHz

− 8ωβPTMrHz + k2 μ0

ρ0P2M2

r + k2 μ0

ρ0QBzMz

+αk2 μ0

ρ0RBzMz+2ωk2 μ0

ρ0PBzMz+βk2 μ0

ρ0RHzM

2r

),

B5 =(− 2αωP3 − αP2Q + 4αRNQ + 8αωNPR

− 4βQRTMrHz−8ωβPRTMrHz + k2 μ0

ρ0QRBzMz

+ αk2 μ0

ρ0QBzMz + αk2 μ0

ρ0P2M2

r

+ βk2 μ0

ρ0QBzHzM

2r + 2ωk2 μ0

ρ0PRBzMz

+2αωk2 μ0

ρ0PBzMz − 2ωβk2 μ0

ρ0PBzHzM

2r

),

B6 = μ0k2

4ρ0RBz

(αMz + βHzM

2r

)(Q + P).

(53)

Equation (52) is of sixth degree in σ with all the coefficientsreal and the coefficients of σ6 and σ5 are clearly positive whilethe coefficients of σ4, σ3, σ2, σ , and the constant term maybe negative depending on B2, B3, B4, B5, and B6. As perthe criteria of Guillemin [20] for the signs of the roots, weget that if B6 is negative, then the constant term of (52) isnegative, which is sufficient to give the required condition ofinstability. Therefore, we obtain the following condition forthe onset of gravitational instability for the present problem:

k2c2 + 2ωd

dr

(ωr2) < (4πGρ0

), (54)

which is the same result as obtained by Bel and Schatzman[9].


5. Conclusions

In the present analysis, we have investigated the effect offerromagnetization in the presence of nonuniform magneticfield on the gravitational instability of a homogenous,ferromagnetic dust cloud in the presence of nonuniformrotation. The inequality (54) implies that the Bel andSchatzman criterion determines the gravitational instabilityof this general problem, thus we can conclude that theferromagnetism in the presence of magnetic field has noeffect on the gravitational instability on the self-gravitatingcloud. Comparing this result with result obtained by Dhimanand Dadwal [10] for the gravitational instability of a homo-geneous gaseous medium in the presence of nonuniformmagnetic field for nonferromagnetic case, it appears thatthe effect of magnetic field on the criterion for instabilityhas been marginalized by the magnetic polarizability offerrofluid. Further, from inequality (54) for ω = 0, we candeduce that gravitational instability is governed by the Jeanscriterion, in the absence of nonuniform rotation.

References

[1] J. H. Jeans, “The stability of spherical nebulae,” PhilosophicalTransactions of the Royal Society, vol. 199, pp. 1–53, 1902.

[2] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,Oxford University Press, 1961.

[3] T. C. Mouschovias, “Magnetic braking, ambipolar diffusion,cloud cores, and star formation—natural length scales andprotostellar masses,” Astrophysical Journal, vol. 373, pp. 169–186, 1991.

[4] L. Mestel and L. Spitzer, “Star formation in magnetic dustclouds,” Monthly Notices of the Royal Astronomical Society, vol.116, no. 5, pp. 503–514, 1956.

[5] Y. Li, M. M. Mac Low, and R. S. Klessen, “Star formationin isolated disk galaxies. I. Models and characteristics ofnonlinear gravitational collapse,” Astrophysical Journal, vol.626, no. 2, pp. 823–843, 2005.

[6] C. F. McKee and E. C. Ostriker, “Theory of star formation,”Annual Review of Astronomy and Astrophysics, vol. 45, pp. 565–687, 2007.

[7] E. A. Spiegel and J. L. Thiffeault, “Continuum equationsfor stellar dynamics:,” in Proceedings of the Chateau deMons Meeting in Honour of Douglas Gough’s 60th Birthday,Cambridge University Press, 2003.

[8] R. B. Larson, “The physics of star formation,” Reports onProgress in Physics, vol. 66, no. 10, p. 1651, 2003.

[9] N. Bel and E. Schatzman, “On the gravitational instability of amedium in nonuniform rotation,” Reviews of Modern Physics,vol. 30, no. 3, pp. 1015–1016, 1958.

[10] J. S. Dhiman and R. Dadwal, “On the gravitational instabilityof a medium in non-uniform rotation and magnetic field,”Astrophysics and Space Science, vol. 325, no. 2, pp. 195–200,2010.

[11] J. S. Dhiman and R. Dadwal, “The gravitational instabilityof a non-uniformly rotating heat conducting medium in thepresence of non-uniform magnetic field,” Astrophysics andSpace Science, vol. 332, no. 2, pp. 373–378, 2011.

[12] L. Spitzer, Physics of Fully Ionized Gases, Inter Science, NewYork, NY, USA, 1962.

[13] R. E. Rosenwieg, Ferrohydrodynamics, Cambridge UniversityPress, 1985.

[14] R. V. Jones and L. Spitzer, “Magnetic alignment of interstellargrains,” Astrophysical Journal, vol. 146, p. 943, 1967.

[15] A. A. Mamun and P. K. Shukla, “Waves and instabilities indark interstellar molecular clouds containing ferromagneticdust grains,” JETP Letters, vol. 75, no. 5, pp. 213–216, 2002.

[16] M. I. Shliomis, “Effective viscosity of magnetic suspensions,”Soviet Journal of Experimental and Theoretical Physics, vol. 34,pp. 1291–1294, 1972.

[17] M. I. Shliomis, “Comment on “magnetoviscosity and relax-ation in ferrofluids”,” Physical Review E, vol. 64, Article ID063501, 2001.

[18] P. Penfield and H. A. Haus, Electrodynamics of Moving Media,MIT Press, Cambridge, Mass, USA, 1967.

[19] D. Kumar, P. Sinha, and P. Chandra, “Ferrofluid squeeze filmfor spherical and conical bearings,” International Journal ofEngineering Science, vol. 30, no. 5, pp. 645–656, 1992.

[20] E. A. Guillemin, The Mathematics of Circuit Analysis, JohnWiley & Sons, New York, NY, USA, 1950.

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