A numerical solution strategy for calculating equilibrium free surfaces of a ferrofluid drop under the action of uniform magnetic fields is proposed. Based on this strategy, drop shapes of nonlinear magnetizable fluids are obtained numerically in a wide range of field intensities and compared with existing theoretical result

Lavrova, O.

Polevikov, V.

Tobiska, L.

English

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PAMM Proc. Appl. Math. Mech,5, 837-838 (2005) / DOI 10 1
Equilibrium shapes of a ferrofluid drop
O. Lavroya*r, V. Polevikov**2, and L. Tobiska ...1
I Institut fijr Analysis und Numerik, Otto-von-Guericke-Universitiit Magdeburg, PF4l20, 39016 Magdeburg, Germany
2 Department of Computational Mathematics, Belarusian State University, F Skaryna Ave 4' 220050 Minsk' Belarus
A numerical solutlon strategy for calculating equilibrium free surfaces of a ferrofluid drop under the action of uniform mag-
netic fields is proposed. Based on this strategy, drop shapes of nonlinear magnetisable fluids are obtained numetically in a
wide range of field intensities and compared with existing theoretical results
o 2OOi WILEY_VCH v*14 CmbH & Co KC.A, Wcinhcib
I Introduction
The problem for calculating equilibrium free surfaces of a ferrofluid drop is described by a coupled system of the Maxwell's
equationsandthemagneticallyaugmentedYoung-Laplaceequation[4]'TheMaxwell 'sequationsareformulatedinsideof
the bounded fluid domain Or with an a-priori u-nknown boundary f and outside in the air domain Oz 
: lR3 \ Ql The
boundary position I will be determined as a solution of the Young-Laplace equation. Both the magnetic field and drop
surface have to be found simultaneously. one possible way to handle numerically the coupled problem is to.split it into two
subproblerns: solving the Maxwell's equations for the given drop shape and solving the Young-Laplace quation for the 
given
field dis*ibution on ihe interface f. Iterations between two subproblems will result in the equilibrium shape. Discretisation
of the Maxwell's equations is realised by the coupled boundary-element/finite-element method and fol the Young-Laplace
equation by the finite-difference method.
2 Mathematicalformulation
under the assumption of the axial symmetry for the dlop shape we formulate the mathematical model 
in cylindrical coordinates
(r, z ). Taking radius fio of rhe initial circut* d.op shap" as a characteristic length and intensity of the 
applied field 116 as a
characteristic field strength we write the two-dimensional Maxwell's equations complemented by the boundary 
conditions in
dimensionless form
- V  ( P ( l V u l l ) V u l )  : 0  i n  Q r '
, 0 q  ) u z
u r  -  u 2 ,  l r ( l v u r L )  A "  :  0 "
u2 : z1 \r '  z) + (F;
,  ,  e .  { c o t h l t  -  t l l ' t r ) )  , :  3 l H o .p \ r )  :  t  r r Y - - - - - - t - . ,  -  M "
Here u1 and u2 denote the scalar potentials in Q1 and Q2i x is the initial susceptibility, M, the magnetlc satulatlon
We describe the equilibrium shape of the fenofluid drop by parametric functions r : r(s), z : z(s), where s denotes the
arc length of the equiilbrium tine l. Using the approach in [3] the Young-LaPlace quation can be wdtten as
z " : r ' F ,  r " : - z ' 4  o < s < 1 ;  F : f  - / + c ;
r ( 0 )  : 0 ,  z ' ( 0 )  : 0 ,  r ' ( 1 )  : 0 '  z ( 1 )  : 0 ;
T ^  ' ( r H )  
,  ( l ^ . , . - .  1 \ H o \ ' l  s : u o M l v ' / 3 .r : -wL l i ' " *# r+ ( ( co r r ra  ;E ) i )  )  2o
Thepo in ts :0 l iesonthezax isandthepo in ts : lon thep lanez : .0 'Fordeta i l so fca lcu la t ingcons tan tsCand 'see
[3]. Here 11 : lvt,l J is the magnetic field intensity' I/n : lut l0n' po d"notes the magnetic permeability' 
V : 4r RSlz the
droD volume, o the surface tension coeffrcient'
'Conespondingauthor:e-mail :olga|avrova@mathematikuni-magdeburgd€'Phone:+493916?12633'Fax:+4939167180?3'
" e-maii: poteii*ov@usu.by, Ph;ner +375 l? 209 55 32, Fax:. +3'5 '1}?1:1:1".."
*V (Vur) : 0 in g22;
on f ,
.."";;il1;;ti;6---jir'"."tii rni-tugdeburg.de, Phone; +4e 3el 6? I8650' Fax: {4e 3el 67 I8073'
( l  )
(2)
(3)
(4)
(s)
e2005 WtLEY.VCH vcrhSCnbH & Ca KC:A Weinh'rm
Section 25
3 Discretisation of the problem
To discretise equations (l )-(3) we use the idea of the coupled collocation boundary-element method (BEM) and the Galerkin
Rnite-element method (FEM), analysed in [5]. We apply the BEM in the unbounded omain c)2 to fulfill exactly the boundary
condition for the magnetic field at the infinity (3). The FEM is used in the bounded domain Or, where nonlinearities of
the Maxwell's equations occur. We reformulate equations (l)-(3) as the nonlocal boundary value problem for ul in Or and
\uyl0n onl. Piecewise linear and piecewise constant functions are used for the approximation of uy and 0u1f0n, respec-
tively. The fluid-air interface I is fixed during the process of solving equations (1)-(3) with a piecewise linear approximation
ofthe interface in the FEM discretisation and a cubic spline approximation in the BEM discretisation being used. The resulting
nonlinea.r discrete system is solved by a Rxed-point iteration method with application of the Gaussian elimination method for
the linearised systems.
We use a finite-difference scheme of the second order approximation to discretise equations (4)-(5), for details see [3]. Grid
adaptation based on the information about the surface curyature is applied [3]. A two-layer iterative scheme is constructed
with ridiagonal matrices of the linearised systems.
4 Numerical results
The current work extents results of [2], where only linear magnetisable fluids were considered. The linear magnetisation is
a reasonable assumption in the region of weak fields [4], whereas nonlinear magnetisation is a necessary requirement for the
problem modeling in a wide range of field intensities.
Fig, 1 The dependence of the drop elongation versus the dimensionless magnetic field. Calculations wer€ made for the fefiofluid with
paramete6rs X = 1.9,W :62.
If we assume that the drop shape is spheroid then we can determinethe drop elongation for every field intensity by applying
a theoretical approach, so-called the virial method [1]. The corresponding dependence is dmwn by the solid line. The curves
with markers present numerical results, From the picture we see that for 1 € [0,50] theoretical and numerical results nearly
coincide. It follows that for the considered ferrofluid the drop shape is rather close to spheroid in a wide range of field
intensities. For,l > 50 the numerical results show a qualitative difference with the theory: a larger applied field produces less
elongated shape. Such a behavior of numerical results should be further studied.
Acknowledgements The authors would like to thank the State Sachsen-Anhalt nd DFc-Project TO 143/4-2 for supporting the research.
References
lll E. Blums, A. Cebers, M. Maiorov, Magnetic fluids (De Gruyter, Berlin, 1997).
[2] O. Lavrova, G. Matthies, V Poleviko\ L. Tobiska, Proc. Appl. Math. Mech. 4, 704-?05 (2004).
[3] V K. Polevikov, Comput. Methods in Applied Math. 4, No. 1, 66-93 (2004).
[4] R. E. Rosensweig, Ferrohydrodynamics (Dover Pubns, New York, 1998).
[5] W. L. Wendland, in Finite element and boundary element techniques from mathematical and engineering point of view, edited by E.
Stein (Springer, Wien, 1988), p. 273.
838
O 2005 wI LEY'VCH V.rla8 CnbH & Co. KG.A. W.rn in


Equilibrium shapes of a ferrofluid drop

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Equilibrium shapes of a ferrofluid drop

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