ARL-TR-8058 ● July 2017

US Army Research Laboratory

Equilibrium Shape of Ferrofluid in the Uniform 

External Field

by Michael Grinfeld and Pavel Grinfeld

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ARL-TR-8058 ● JULY 2017

US Army Research Laboratory

Equilibrium Shape of Ferrofluid Systems in the 

Uniform External Field

Michael Grinfeld

Weapons and Materials Research Directorate, ARL

Pavel Grinfeld

Department of Mathematics, Drexel University, Philadelphia, PA

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Equilibrium Shape of Ferrofluid in the Uniform External Field

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Michael Grinfeld and Pavel Grinfeld

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14. ABSTRACT

It is well known that the morphology of magnetic or electric ferrofluids is extremely sensitive to the applied external 

electromagnetic fields. Even in the static regimes, they demonstrate a variety of qualitative and quantitative transformations 

often mentioned as free-surface instabilities. That makes their computational modeling rather challenging. For the sake of 

validation and verification, there is a need for having exact solutions. Modelers noticed that quite often isolated ferrofluids 

take on the shape of a spheroid co-axial with the applied quasi-uniform field. There were several claims that a consistent 

theory allows researchers to prove that an isolated drop of ferrofluid exposed to a uniform field, indeed, has a shape of a 

spheroid. This report will analyze the computational algorithm that allows us to resolve this and similar problems numerically.

15. SUBJECT TERMS

ellipsoidal solutions, equilibrium shapes, Maxwell stress tensor, thermodynamics, variational computational algorithms

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Michael Grinfeld

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Contents

List of Figures iv

Acknowledgment v

1. Introduction 1

2. The Boundary Value Problem to be Solved 2

3. A 2-Step Iteration Algorithm 4

4. Toward Verification of the Ellipsoidal Shape 5

5. Conclusion 6

6. References 7

Distribution List 8
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List of Figures

Fig. 1 The equilibrium shape of a ferrofluid inside the Helmholtz coil takes 

on a shape close to an elongated spheroid co-axial with the field........ 1
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Acknowledgment

The authors are grateful to Dr John Niederhaus from Sandia National Laboratories 

(Albuquerque, New Mexico) for helpful discussions.
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1. Introduction

Consider a finite chunk of ferrofluid exposed to a uniform magnetic field as shown 

in Fig. 1.The shape is very close to the shape of a spheroid. Several authors claimed 

that this fact can be established on the basis of a rigorous theory, like it was done 

for the equilibrium shape of rotating self-gravitating uniform incompressible liquid 

(see Chandrasekhar’s classics1 and multiple references therein). To learn more 

about the magnetic ferrofluids in the uniform magnetic fields, interested readers 

can find the approaches and further references in the monograph.

2 Other references, 

relating to ellipsoidal solutions in the problems of electromagnetism, can be found 

in the works of Stratton,3 Landau and Lifshitz,4 and Akhiezer et al.

5

Fig. 1 The equilibrium shape of a ferrofluid inside the Helmholtz coil takes on a shape close 

to an elongated spheroid co-axial with the field

Despite all these remarkable efforts and progress, the problem of equilibrium shape 

of ferrofluids is far from being exhaustively explored. There are several reasons for 

further mandatory improvements. For instance, the master systems, describing the 

dynamics of ferrofluid, face various objections.

6,7 The problem of equations of 

state—which will be the object of hot debates for a long time—also needs further 

study. From that standpoint it seems reasonable to limit ourselves to static 

(thermodynamic) approaches, especially when the problem under study is of a static 

(equilibrium) nature up front.

There are several fundamental problems relating to the static and thermodynamic 

aspects of the ponderomotive forces in substances with polar interaction: the 

authors’ vision of those problems has been presented in the works of Michael and 

Pavel Grinfeld.

8–11 Despite all the existing controversies, the “static” approaches 

are much older and face less objections than the “dynamics” of electromagnetic 

media. This report will analyze the problem of equilibrium shape by directly using 
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the static approach (i.e., not deriving the static system from the more questionable 

dynamic system of magnetohydrodynamics). When considering the ellipsoidal 

configurations of ferrofluid we follow the method suggested by Eshelby in the 

1950s,

12 instead of the classical approaches of Chandrasekhar,1 Bashtovoy et al.,2

Stratton,3 Landau and Lifshitz,4 or Akhiezer et al.5

2. The Boundary Value Problem to be Solved

We assume that the ferrofluid is maintained at a fixed temperature, and let ψ be 

the free-energy density at this temperature. Because the fluid is assumed 

incompressible, the free-energy density is the function of the polarization vector

P



, onlyψ ψ= (P)

 . 

Our suggested approach to this problem is based on the minimization of the total 

energy of the system Wtotal . The total energy includes 3 ingredients: the total free 

energy total Ψ , the total magnetic energy Emag , and the total surface energy Esurf

W EE total total mag surf =Ψ + + , (1)

where the energy ingredients are chosen to be the following:

( )

( )

2

, , / 8

.

total mag Body Space

surf

d ME dH

E dN

ψ π

σ

Ξ

Ψ= Ω = Ω ∫ ∫

= Ω ∫

 

 (2)

In Eq. 2, H

 is the magnetic field, M is the magnetization vector, and σ (N)  is the 

dependence of the surface energy density upon the orientation of the unit normal

unit.

In the beginning we postulate the classical equations of magnetostatics (when there 

is no current of free charges, the system of electrostatics and of magnetostatics are 

the same4

).

The variational problem under study is the symbiosis of 2 classical problems: the 

first is the classical problem of magnetostatics and the second is the problem of the 

equilibrium shape of liquid or crystal. For our situation it is essential that the 

magnetostatic problem with the given shape of the boundary is essentially linear if 

the constitutive relationship between the field and inductance is linear. At the same 

time, the second problem is essentially nonlinear, even when the surface energy σ
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is just a constant. The linear electrostatics and magnetostatics can be addressed by 

various algorithms that have already been developed. Because of its nonlinearity, 

the problem of the equilibrium shape is treated in the suggested approach as the 

more difficult. This fact dictates our iterative approach, and we concentrate on this 

problem in the following paragraphs. 

In the first step, we fix the current position of the boundary Ξ and solve the 

magnetostatic problem for the fixed current domain. This allows us to determine 

all the current fields P H, ,

  and B



. Afterwards, we transition to solving the second 

problem. Namely, we slightly update the current location of the boundary Ξ in such 

a way that the total energy Wtotal in the updated domain is smaller than in the current 

domain. In Eqs. 7–10 we formulate explicitly what should be done to that end. 

Then, for the updated boundary we return to the first step and repeat the procedure 

iteratively. 

The linear boundary value in the first problem can be solved differently (see the 

variational approach in Bossavit and Mayergoyz’s work13). Analytically, it reduces 

to solving the following system of linear equations and boundary conditions: within 

the fluid (Eq. 3), within vacuum (Eq.4), at the interface liquid-vacuum (Eq.5) at 

infinity (Eq. 6), or at a given boundary. 

4 , 0, M BH M B H = + ∇⋅ = = π ψ

     , (3)

∇⋅ = H 0

 , (4)

[φ] 0, B N 0

+ + = ⋅= − −    

  , (5)

H H → ∞

  . (6)

In the combined system (Eqs. 3–6), φ is the scalar field potential and B

 is the 

induction vector, respectively. The symbolic thermodynamic relationship M H =ψ



in Eq. 3 has the following index counterpart: ( ) / k i H MM i =∂ ∂ ψ . 

In fact, solving the system (Eqs. 3–6) is equivalent to minimizing the energy Wtotal

at fixed boundary Ξ with respect to the distribution of the magnetization 

vector M

 .

The problem of minimizing the total energy Wtotal possess one more essential degree 

of freedom: it is the position of the boundaryΞ . Theoretical analysis of this 
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variational problem leads to the boundary equation (presented in the component 

form)

ij T NNi j R

σ     ψ  

+

+ − =Λ − , (7)

where R is the mean curvature of the boundary, [a]

+

− symbolizes a jump of the 

function a across the interface, Λ is an unknown constant (the Lagrange multiplier, 

which can be determined from the condition of a given total mass of the ferrofluid), 

and the tensor ij T is defined as follows: 

1 1

8 4

ij kki ij j T H H H B BH k k δ

π π

=− +

      . (8)

In the empty space, the tensor ij T reduces to the celebrated Maxwell stress tensor.

In the absence of the polarization associated terms, the boundary in Eq. 6 reduces 

to the classical Laplace condition of a constant mean curvature of the liquid drop 

under the action of surface tension. 

3. A 2-Step Iteration Algorithm

The following statements, established in Ferroelectrics,

9 play the key role in the 

suggested computational algorithm. To formulate them explicitly, consider the 

current distribution of the magnetization vector M

 satisfying the bulk and boundary

equations of magnetostatics (Eqs. 3–6). If we insert this field into the left-hand side 

of the relationship in Eq. 7 we get a certain function χ , defined on the current 

surface Ξ . Generally speaking, this function is not constant. If we move each point 

of the surface in the direction of the local unit normal N



on the distance ∆ such that

( ) mean ∆=− − τχ χ , (9)

then τ is a sufficiently small positive number and mean χ is the mean value of the 

of the function χ over the surface Ξ

1

mean area χ d

Ξ Ξ

= ∫ Ω . (10)

We call this new surface an upgraded surface * Ξ and the domain * B inside it an 

upgraded domain. Let us solve the magnetostatic problem (Eqs. 3–6) for the 
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upgraded domain and calculate the upgraded total energy *

total W . The following 2 

relationships are valid:

*

total W W total < (11)

and

* B ≅ B , (12)

where the symbol ≅ means the equality to within the first order of smallness in τ .

4. Toward Verification of the Ellipsoidal Shape

The solution for ellipsoid is of importance for 2 reasons. First, theorists claim2 that 

the isolated chunk of ferrofluid in a uniform (at infinity) field takes on the shape of 

an ellipsoid. Second, this solution being very simple inside the ellipsoid is a 

convenient tool for quick verification of the computational code. Following

Eshelby’s research,

12 we will get the presentation of the Cartesian component of 

the magnetic field outside the ellipsoid as

k HA H i ik i ω ∞ =− ∇∇ + , (13)

whereω is the Newtonian potential of the same ellipsoid having the mass density 

equal to 1, and k A has to be determined by the boundary conditions.

The Newtonian potential inside the ellipsoid is very simple:

0

1

2

i j

ij ω = − C Yzz , (14)

where 0 C and ij Y are the constant scalar and symmetric tensor satisfying the 

conditions

.

. 4 ii Y = π . (15)

Using Eq. 13, we arrive at the very simple relationship of the magnetostatic field 

inside the ellipsoid as the following:

k

i ik i H AY H ∞ = + . (16)
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The field outside the ellipsoid is much more complex. Tensor ij Y can be easily 

expressed in terms of the ellipsoid’s semi-axis and orientation and vice versa. When 

the tensor ij Y is known we can determine the geometry of the ellipsoid.

5. Conclusion

Ferrofluids keep finding more and more applications in engineering practice and 

academic research. From a scientific point of view, ferrofluids raise many novel 

and exciting problems in mathematical, theoretical, computational, and applied 

physics. Many simple fundamental problems of the discipline remain unanswered. 

Among those is the particularly appealing problem of the equilibrium shape of an 

isolated chunk of ferrofluid exposed to a static uniform magnetic field. In the 

suggested approach, this problem is treated as the problem of multidimensional 

calculus of variation with free (i.e., unknown) boundary. The presence of the 

unknown boundary makes the problem deeply nonlinear even when the constitutive 

relationships between the magnetic field and induction are linear. 

To solve this and similar problems computationally, we suggested an algorithm 

similar to the method of steepest descent. The algorithm is based on splitting the 

solution into 2 steps. The first step consists of solving the linear problem of 

magnetostatics within the a priori known domain. The second step reduces to 

updating the current domain in such a way that the total volume of the domain 

remains unchanged and, at the same time, the total accumulated energy becomes 

smaller. This step is described by the relationships in Eqs. 7–10. The 

implementation of this algorithm is currently under development.
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6. References

1. Chandrasekhar S. Ellipsoidal figures of equilibrium—an historical account.

Chicago (IL): University of Chicago; 1969.

2. Bashtovoy VG, Berkovsky BM, Vislovich AN. Introduction to 

thermomechanics of magnetic fluids. New York (NY): Hemisphere Publishing 

Corporation; 1988. 

3. Stratton JA. Electromagnetic theory. New York (NY): McGraw-Hill; 1941. 

4. Landau LD, Lifshitz EM. Electrodynamics of continuous media. New York

(NY): Pergamon; 1963.

5. Akhiezer AI, Bar'yakhtar VG, Peletminsky SV. Spin waves. New York (NY): 

Interscience; 1967. 

6. Rosenzweig RE. Introduction to ferro-hydrodynamics, New York (NY): 

Cambridge University Press; 1967. 

7. Kovetz A. The principles of electromagnetic theory, Oxford (United 

Kingdom): Oxford University Press; 2000. 

8. Grinfeld P. Morphological instability of liquid metallic nuclei condensing on 

charged inhomogeneities. Phys Rev Lett. 2001;87(9):095701. 

9. Grinfeld P, Grinfeld M. Thermodynamic aspects of equilibrium shape and 

growth of crystalline films with electromechanical interaction. Ferroelectrics.

2006;342:89–100. 

10. Grinfeld P. Morphological instability of the dielectric Thomson nuclei. Phys 

Rev B. 2010;81:184110. 

11. Grinfeld M, Grinfeld P. A variational approach to electrostatics of polarizable 

heterogeneous substances. Adv Math Phys. 2015; Article ID 659127.

12. Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, 

and related problems. Proceedings of the Royal Society A. 1957;241(1226): 

376–396. 

13. Bossavit,A. Mayergoyz I. Computational electromagnetism. Cambridge (MA):

Academic Press; 1998.
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