Total magnetic force on a ferrofluid droplet in microgravity

Abstract

The formulation of the total force exerted by magnetic fields on ferrofluids has historically been a subject of intense debate and controversy. Although the theoretical foundations of this problem can now be considered to be well established, significant confusion still remains regarding the implementation of the associated expressions. However, the development of future applications in low-gravity environments is highly dependent on the correct modeling of this force. This paper presents a contextualized analysis of different proposed calculation procedures and validation in a space-like environment. Kinematic measurements of the movement of a ferrofluid droplet subjected to an inhomogeneous magnetic field in microgravity are compared with numerical predictions from a simplified physical model. Theoretical results are consistent with the assumptions of the model and show an excellent agreement with the experiment. The Kelvin force predictions are included in the discussion to exemplify how an incomplete modeling of the magnetic force leads to significant errors in the absence of gravity.

Full text

Total magnetic force on a ferrofluid droplet in microgravity
´
Alvaro Romero-Calvoa,b,∗
, Gabriel Cano-G´omezc, Tim H. J. Hermansd, Lidia Parrilla Ben´ıtezb, Miguel
´
Angel Herrada Guti´errezb, Elena Castro-Hern´andezb
aDepartment of Aerospace Engineering Sciences, University of Colorado Boulder, 429 UCB, 80303, Boulder, CO, US
b´
Area de Mec´anica de Fluidos, Departamento de Ingenier´ıa Aeroespacial y Mec´anica de Fluidos, Universidad de Sevilla,
Avenida de los Descubrimientos s/n 41092, Sevilla, Spain
cDepartamento de F´ısica Aplicada III, Universidad de Sevilla, Avenida de los Descubrimientos s/n 41092, Sevilla, Spain
dAstrodynamics & Space Missions, Delft University of Technology, Delft, The Netherlands
Abstract
The formulation of the total force exerted by magnetic fields on ferrofluids has historically been a subject of
intense debate and controversy. Although the theoretical foundations of this problem can now be considered
to be well established, significant confusion still remains regarding the implementation of the associated ex-
pressions. However, the development of future applications in low-gravity environments is highly dependent
on the correct modeling of this force. This paper presents a contextualized analysis of different proposed
calculation procedures and validation in a space-like environment. Kinematic measurements of the move-
ment of a ferrofluid droplet subjected to an inhomogeneous magnetic field in microgravity are compared
with numerical predictions from a simplified physical model. Theoretical results are consistent with the
assumptions of the model and show an excellent agreement with the experiment. The Kelvin force predic-
tions are included in the discussion to exemplify how an incomplete modeling of the magnetic force leads to
significant errors in the absence of gravity.
Keywords: Total magnetic force, ferrofluids, microgravity, magnetic mass transfer, space engineering
1. Introduction
The calculation of the total force exerted by elec-
tromagnetic fields on electrically or magnetically
polarizable continuous media is a relevant problem
of electromagnetism. Different procedures can be
used to face this problem and, although the valid-
ity and equivalence between them is generally ac-
cepted, their associated physical models have be-
come a subject of debate in the past.
The discussion has been particularly intense in
what refers to the distribution of forces within a
magnetic fluid. While in [1] it is considered that the
action of an external field produces forces through-
out the medium, in [2] it is concluded that only
the surface exhibits a force distribution with true
physical meaning. Further considerations on how
magnetic fields contribute to the energy variation
result in different formulations of this interaction
[2–7].
∗alvaro.romerocalvo@colorado.edu
A larger consensus exists in the different formula-
tions of the total force. The subject was revisited in
2001 when the validity of the Kelvin force expres-
sion in ferrofluids was contested [3] and a significant
response was produced [2; 4; 6; 8; 9]. Although this
debate can be regarded as outdated, a review of
recent publications unveils that the total force is
not always formulated in a fully rigorous manner.
Common simplifying assumptions are (i) the iden-
tification of the applied H0and internal Hmag-
netic fields, (ii) the elimination of the surface force
components related to the normal or tangential dis-
continuities of the magnetization field, or (iii) the
removal of the demagnetization field. For instance,
in [10] an approximate model that neglects the con-
tribution of magnetized liquid oxygen to the mag-
netic field is employed, while surface forces arising
from the discontinuity in the magnetization field
are directly ignored. Those surface components are
also neglected in [11; 12], where only the Kelvin
volume force density is considered. In [13; 14] the
Preprint submitted to Experimental Thermal and Fluid Science July 8, 2022
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

small size of the ferrofluid medium justifies the cal-
culation of the volume force density by means of
simplified expressions in terms of the Bfield, but
without mentioning the corresponding surface com-
ponents. These simplifications may be appropriate
under normal-gravity conditions, but may result in
significant errors in the presence of a strong mag-
netic force. Unfortunately, most contributions do
not explicitly refer to the simplifying assumptions
behind their formulations, hence becoming a source
of confusion for non-experts.
The correct formulation and implementation of
the total magnetic force becomes particularly rele-
vant for space engineering applications dealing with
ferrofluids. Ferrofluids are colloidal suspensions of
magnetic nanoparticles in a carrier liquid. Their
invention aimed to enable an artificially imposed
gravity environment in microgravity, hence eas-
ing the management of liquid propellants in space
[15]. The basic equations governing the dynamics
of ferrofluids were first presented by Neuringer and
Rosensweig in 1964 [16], giving rise to the field of
Ferrohydrodynamics [17]. A non-exhaustive list of
applications in space includes thermomagnetic con-
vection [18; 19], mass transfer [20], micropropulsion
[21; 22] and magnetic liquid positioning [11; 23; 24].
Due to their enhanced magnetic susceptibility and
the absence of gravity, the magnetic force acquires
a major role and the impact of the aforementioned
simplifications is hence increased.
In this paper, kinematic measurements of the
translation of a ferrofluid droplet subjected to an
inhomogeneous magnetic field in microgravity are
compared with numerical computations from dif-
ferent models. The total force formulations are de-
picted together with the incomplete result that the
Kelvin force alone would produce. The droplet was
generated during a series of microgravity experi-
ments performed at ZARM’s drop tower in Bremen
as part of the Drop Your Thesis! 2017 programme
run by ESA Academy.
This work aims to provide: (i) a comprehensive
review of the existing formulations for the total
magnetic force, (ii) further evidence on their reli-
ability and proper implementation, and (iii) an ex-
perimental example of magnetic mass transfer in
microgravity in the midterm between typical labo-
ratory setups and practical applications. The the-
oretical problem is analyzed in Section 2, Section
3 presents the microgravity experiment, Section 4
shows and discusses the results and Section 5 sum-
marizes the main conclusions of the work.
2. Theoretical Background
The total force exerted by an electric or magnetic
field on a polarizable medium has been formulated
and solved by analyzing the variation of energy of
the microscopic dipoles composing the material [25–
28]. This general approach can be developed by fol-
lowing different methods [2; 8]: (i) calculation of the
total force in terms of the electromagnetic traction
on a surrounding boundary surface; (ii) calculation
of the resultant of the system of distributed forces,
and (iii) evaluation of the energy balance of the
system by means of the principle of virtual works.
These procedures are equivalent and, if correctly
applied, must lead to the same result. From the
physical viewpoint, however, they rely on different
models of the system of distributed forces acting on
the medium.
2.1. Total magnetic force derived from the Maxwell
stress tensor
The aforementioned procedures (i) and (ii) em-
ploy the Maxwell stress tensor TM, which can be
regarded as a powerful tool for describing the elec-
trodynamic and electromagnetic forces acting on
a continuous medium. In the general framework
of electromagnetism, the Maxwell stress tensor has
been introduced in several ways with different in-
terpretations of its physical or merely mathematical
meaning [17; 25–27; 29; 30].
In the case of electrically and/or magneti-
cally polarized fluid media, where dissipative and
electro/magnetostriction processes are considered,
Maxwell equations and dynamic balance relation-
ships (including linear momentum, angular momen-
tum, energy and entropy production rate) lead to a
general pressure-viscous-electromagnetic stress ten-
sor scheme for TM[2; 31].
In the present study, the material medium un-
der analysis is electrically neutral and remains in
thermodynamic equilibrium with constant density,
temperature and chemical potentials. The stress
tensor can be then formulated as
TM=−pI+Tm,(1)
where Tmis the magnetic stress tensor, and the
unit dyadic I= (δij) is multiplied by a pressure-
like variable p. The canonical form of the mag-
netic work per unit volume done to magnetize the
2
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

medium, H·δB, leads to the magnetic stress tensor
Tm=BH −µ0
2H2I,(2)
with the first term being the dyadic resulting from
the tensor product of the induction Band mag-
netic Hfields, related through B=µ0(H+M),
and where Mis the magnetization field. The mag-
netic stress tensor expressed in Eq. (2) has been
widely used in classical and recent bibliography on
electromagnetism to obtain the magnetic force on
a magnetizable body [1; 8; 25; 29; 31]. Here it has
been defined by grouping in Eq. (1) those terms
of the magnetic work that exclusively contribute to
the variation of the magnetic energy density. The
corresponding pressure-like variable pis
p(υ, T , H) = p0(υ, T ) + πm(H),(3)
where p0(υ, T ) is the thermodynamic pressure in
the absence of magnetic fields and υ=ρ−1is the
specific volume of the ferrofluid. The term
πm(H) = µ0ZH
0
∂[υM ]
∂υ dH0,(4)
reflects the pressure change due to the interaction
between magnetic dipoles.
An identical formulation of the Maxwell stress
tensor for a magnetized medium in equilibrium is
obtained in [2; 17] by applying a balance of mag-
netic work and free energy. However, it is arranged
so that the above magnetic pressure term πm(H)
is considered to be part of the magnetic stress ten-
sor Tmfrom which the magnetic force distributions
are later derived. Therefore, only the zero–field
pressure p0(υ, T ) constitutes the pressure–like vari-
able. These two definitions of Tmlead to differ-
ent magnetic force distributions but, as shown in
Subsection 2.1.3 and Appendix A, result in iden-
tical expression for the total force. In Ref. 17,
Rosensweig emphasizes that the grouping of terms
is arbitrary and may lead to confusion. Since in
magnetically diluted ferrofluids it is commonly as-
sumed that M∼ρ, the contribution of πm(H) to
the magnetic stress tensor is in any case negligible.
2.1.1. Resultant of the magnetic traction on the
boundary
In this study, a volume Vof a magnetizable and
electrically neutral medium, immersed in a non-
magnetic environment (M=0), is subjected to
the action of an applied magnetostatic field H0in
microgravity. The resulting translation movement
is due to the force distributions derived from the
magnetic stress tensor Tm. Assuming that the mag-
netized medium is in thermodynamic equilibrium,
the total magnetic force can be computed by ap-
plying
FT
m=I∂V
dS(n· T +
m) (5)
in the external surface ∂V +of the body, with n
being the external normal vector [1; 2; 8; 25]. If
the polarized body is surrounded by a non-magnetic
and non-electric material, ∂V 0can be taken as any
closed surface around it [2; 8]. In this framework,
the term of magnetic pressure π+
mis strictly null
in the surrounding medium. Therefore, Eq. (5) is
not affected by its inclusion in the magnetic stress
tensor.
Although the practical utility of Eq. (5) is rec-
ognized for the calculation of the total magnetic
force, different interpretations of its physical mean-
ing have been suggested. In [2] it is proposed that
the actual forces acting on the polarized medium
are located at the body surface. In contrast, the
general validity of this procedure is discussed in
[1], where no physical meaning is attributed to the
Maxwell stress tensor. That is, it is not considered
to be a true Cauchy stress tensor, then leading to
a system of fictitious tensions. Finally, in [31] the
total pressure-viscous-electromagnetic stress tensor
is used to predict the existence of volume forces due
to coupled polarization, and its experimental veri-
fication is proposed.
2.1.2. Resultant of the magnetic force distributions
The local effect of electromagnetic fields on con-
tinuous media can be formulated in terms of a body
force determined by the divergence of the Maxwell
stress tensor [17; 29; 30]. In the case of a magne-
tizable medium subjected to a magnetic field, the
magnetic body force density is
fm,V =∇·Tm∀P∈V, (6)
where Pis a spatial position. This force density
would then depend on the local magnetic field.
Since the tensor is discontinuous between the in-
ternal (∂V −) and external (∂V +) faces, a surface
force density fm,S arises at ∂V [1; 2; 8; 17]
fm,S =n· T +
m−n· T −
m∀P∈∂V. (7)
3
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

Therefore, the general expression of the total mag-
netic force calculated from this physical model is
[1; 2; 5; 7–9].
Fm=ZV
dVfm,V +I∂V
dSfm,S ,(8)
where fm,V and fm,S are given by Eqs. (6) and (7),
respectively.
The mathematical equivalence between Eqs. (8)
and (5) is immediate: the magnetic field and Tmare
continuous in V, so the divergence theorem leads to
ZV
dV(∇ · Tm) = I∂V
dS(n· T −
m),(9)
where the second term is evaluated in the internal
face of ∂V . If this identity is applied to Eq. (8)
considering Eqs. (6) and (7), Eq. (5) is obtained.
Only the magnetic contribution of the Maxwell
stress tensor formulated in Eq. (1) is employed for
the calculation of the magnetic force densities given
by Eqs. (6) and (7), or the corresponding surface
density in Eq. (5). However, the pressure term must
be considered in the determination of the equilib-
rium surface ∂V , and hence in the resultant forces
shown in Eqs. (5) or (8).
2.1.3. Expressions for the resultant of the distribu-
tions of forces
The magnetic fluid in equilibrium exhibits the
behavior of a soft magnetic material, so that lo-
cal magnetic fields are collinear. The natural con-
stitutive relation that macroscopically characterizes
the magnetized fluid would then be M=χ(H)H,
where χis the magnetic susceptibility.
The application of Eq. (6) to the canonical form
in Eq. (1), together with some elementary vector-
dyadic identities, results in the corresponding mag-
netic body-force density
fH
m,V =µ0(M· ∇)H=µ0M∇H, (10)
where, due to the absence of electric currents in the
volume of the magnetized medium, ∇ × H=0in
V. The above expression is usually known as Kelvin
force.
The corresponding surface-force density fH
m,S in
∂V is obtained by applying Eq. (7). It should
be noted that, according to Gauss’s law, the nor-
mal component of the induction field is continuous
through the interface (B+
n=B−
n=Bn). In addi-
tion, the absence of electrical surface currents in ∂V
results in the continuity of the tangential magnetic
field component (H+
t=H−
t=Ht). The result is
fH
m,S =µ0
2M2
nn,(11)
being Mnthe normal magnetization component at
∂V −. By making use of Eq. (8), the total magnetic
force becomes
FH
m=µ0ZV
dV M∇H+µ0
2I∂V
dS M2
nn,(12)
This expression has been applied in [7; 9], showing
that the volume term corresponding to the Kelvin
force must be completed by the surface term. It has
also been proposed in [2] as one of the equivalent
forms of the total magnetic force acting on a mag-
netized fluid in vacuum, but considering that the
volume term belongs to a force distribution physi-
cally located at the surface.
The magnetic and pressure stress tensors given
by Eqs. (2) and (3) can be defined with the Cowley-
Rosensweig formulation, that includes the integral
term πm(H) in the magnetic tensor [17]. The Ap-
pendix A shows how this leads to different volume
and surface force distributions and the same total
force expression.
Eqs. (2) and (3) are not, however, the only pro-
posed form of the total stress tensor presented in
Eq. (1). In [3; 32] the magnetic body-force den-
sity given by Eq. (15) is derived assuming a con-
stitutive relation in terms of the induction field
M= ˜χ(B)B/µ0, where ˜χis an alternative def-
inition of the magnetic susceptibility. The total
stress tensor corresponding to this volume distri-
bution is obtained in Ref. 2 assuming that the dif-
ferential contribution of the magnetic work to the
local energy distribution in the polarizable medium
is −B·δH. Dual expressions are then obtained for
the magnetic and pressure stress tensors [2; 5]
Tm=BH −µ0
2[H2−M2]I,(13)
p(υ, T , B) = p0(υ, T ) + ZB
0
∂[υM ]
∂υ dB0,(14)
where, as in Eq. (2), the magnetic tensor is built
only from terms which are directly related to the
variation of the magnetic energy density in the
medium. The divergence of Eq. (13) in Vand its
discontinuity in ∂V produce the volume and sur-
face densities associated with this dual form. Ap-
plying Eq. (6) and Eq. (7) and considering again
4
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

the absence of electric currents in Vand ∂V , the
magnetic body force fB
m,V and surface force fB
m,S
densities are
fB
m,V = (M· ∇)B+µ0M×(∇ × M) = M∇B,
(15)
fB
m,S =−µ0
2M2
tn.(16)
with Mtbeing the tangential magnetization com-
ponent at ∂V −. The expression for the resulting
magnetic force derived from the dual form of the
magnetic stress tensor is then
FB
m=ZV
dV M∇B−µ0
2I∂V
dS M2
tn.(17)
This dual formulation gives rise to different dis-
tributions of forces with respect to the canonical
form. However, in Appendix B it is shown that
both approaches are equivalent in terms of the total
magnetic force acting on a volume Vof magnetized
fluid, and then also to the resultant of the magnetic
traction in the boundary given by Eq. (5). To the
best knowledge of the authors, this demonstration
has not been presented in previous works.
The magnetic term in Eq. (14), ˜πm(B) =
RB
0∂[νM ]/∂νdB0, may be included in the magnetic
tensor to obtain the corresponding force distribu-
tions. Following the procedure developed in the
Appendix A, it is found that although the result-
ing magnetic force densities differ from Eqs. (15)
and (16), they lead to the same total magnetic force
expressed in Eq. (17).
2.2. Energy balance and virtual works scheme
The free energy variation δF of a magnetizable
medium due to changes in the applied magnetic
field H0is given by [25]
δF =−µ0ZV
dVM·δH0.(18)
If this variation is due to the displacement of the
body in a non-magnetic environment with respect
to the applied magnetic field H0, then δH0=
(δr· ∇)H0. Assuming that the displacement is not
caused by other interactions, the application of the
principle of virtual works results in the total mag-
netic force [4; 8; 25]
F0
m=µ0ZV
dV(M· ∇)H0.(19)
It should be noted that this formulation does not
add information about the stresses in the body.
If the thermodynamic equilibrium condition is
verified for a system involving magnetized fluids
in a non-magnetizable environment and subjected
to an external magnetic field, the energy-balance-
based total force expression given by Eq. (19) must
give the same result as Eq. (5) and, therefore, as
Eq. (12) or Eq. (17) [2; 4; 8]. The demonstration of
this equivalence is given in Section 3.3 of Ref. 2.
The procedure formulated in Eq. (19) can be con-
sidered as the most straightforward method to com-
pute the total force on a magnetically polarized
body. Its application to magnetized fluids is re-
stricted to the condition of thermodynamic equilib-
rium, discarding any process characterized by sig-
nificant magnetostrictive or magnetodisipative ef-
fects. For non-equilibrium processes, procedures in-
volving the Maxwell stress tensor are recommended
instead [8].
In order to set a correct validity range, it is con-
venient to develop a criteria to verify the magneto-
static condition, and hence the equivalence between
the proposed procedures.
2.3. Magnetostatic Conditions
When the applied magnetic field varies, the mi-
croscopic dipoles inside the ferrofluid shift to a new
equilibrium position. This process occurs by means
of two mechanisms: the Brownian relaxation in-
volving a physical rotation of the dipoles, and the
N´eel relaxation associated with a rotation of the
magnetic dipoles within the particles. Each process
is respectively characterized by a relaxation time τB
and τNwith an effective relaxation time τgiven by
[33] 1
τ=1
τB
+1
τN
.(20)
Any process satisfying the magnetostatic hypoth-
esis must have a characteristic time significantly
larger than τto allow an instantaneous reorienta-
tion of the dipoles. If the ferrofluid is moving in
an inhomogeneous and static magnetic field, the
characteristic time of change of the field may be
computed as
τf=Href
vf∇H,(21)
where Href is a reference value of the magnetic field
intensity and vfis the velocity of the ferrofluid vol-
ume. If the ratio τf/τ is greater than 1, the mag-
netostatic hypothesis can be assumed as valid.
5
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

2.4. Summary of total force formulations
For the sake of clarity, the equivalent formula-
tions of the total magnetic force on ferrofluids de-
rived in this section are summarized in Table 1. The
volume term of FH
m, named Kelvin force and iden-
tified by FH
m,V , is computed separately in Section
4 to exemplify how incomplete formulations lead
to wrong predictions even for low-susceptibility fer-
rofluids.
3. Microgravity experiment
Once the equivalence between Eq. (5), Eq. (12),
Eq. (17) and Eq. (19) has been verified, it is conve-
nient to address their applicability and numerical
implementation. A comparison with experimental
results offers an insight into the validity and robust-
ness of those procedures under reasonable simplifi-
cations.
Measurements of the total force on ferrofluids
have been traditionally obtained by means of long
pendulum setups where the magnetic force was
compensated with the action of gravity [3; 9]. In
the case under analysis, however, the generation of
a floating ferrofluid droplet in microgravity is used
for that purpose. This has the potential advantage
of enabling a high-quality three-dimensional obser-
vation of the kinematics of the ferrofluid volume
while subjected to an arbitrary magnetic field.
3.1. Experiment Setup
The experiment here presented is framed in the
ESA Drop Your Thesis! 2017 campaign, that stud-
ied the free axisymmetric oscillations of a ferrofluid
solution in a cylindrical vessel in microgravity [34].
The drop tower setup, represented in Figure 1, was
designed to impose a vertical percussion to the fer-
rofluid container and measure the deformation of
the free surface. 9.3 seconds of microgravity condi-
tions were achieved by making use of ZARM’s drop
tower catapult mode.
Table 1: Equivalent total magnetic force expressions
Symbol Volume term Surface term
FT
m-n· T +
m
FH
mµ0M∇H µ0M2
nn/2
FB
mM∇B−µ0M2
tn/2
F0
mµ0(M· ∇)H0-
Figure 1: Experiment setup. The fixed and sliding assem-
blies are labeled in blue and red, respectively. The red arrow
shows the movement induced by the percussion mechanism
on the sliding assembly.
The experiment setup is composed of a Plexiglas
vessel that contains a ferrofluid, a circular copper
coil located at its base and a pair of cameras. This
assembly is duplicated 368 mm below. The Plex-
iglas vessels have an inner diameter of 110 m, a
height of 200 mm and are filled up to 50 mm by
475 ml of the EMG-700 ferofluid solution described
in Section 3.3. The coils are connected in series
and have 200 windings of a 1.8 mm diameter wire,
an inner diameter of 160 mm and a width of 31
mm. A constant intensity power source working at
16.1 A feeds the coils during the experiment. Two
GoPro Hero 5 Session cameras are located at ap-
proximately opposite sides of the vessel and work
at 60 fps, 1920×1080 px2resolution and wide FOV.
In order to start a free surface oscillation, a stepper
engine imposes a vertical percussion to the setup
4.5 s after launch, as sketched in Figure 1. Further
details on the experimental setup can be found in
Refs. 34 and 35.
During the fourth drop of the ESA Drop Your
Thesis! 2017 campaign, the vertical percussion pro-
duced by the stepper engine generated a ferrofluid
jet and a floating droplet of 11 mm diameter in the
upper assembly. This effect was not observed in the
other four drops and is a consequence of the desta-
bilization of the free surface. From the classical
low-gravity sloshing theory perspective, the desta-
6
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

Figure 2: Sequence captured by Camera 1 showing the fer-
rofluid droplet formation and evolution after the application
of the vertical percussion. a) t=-2.5 s; b) t=-1 s; c) t=-0.5 s;
d) t=0 s; e) t=0.5 s; f) t=1 s; g) t=1.5 s; h) t=2 s; i) t=2.5
s.
bilization is produced when the critical acceleration
(or critical Bond number) is reached [36]. Figure 2
represents the formation and breakup of the fer-
rofluid jet, that generated several droplets of differ-
ent sizes. Only one of them, surrounded by a red
circle, could be tracked with reasonable accuracy.
Surface oscillations were induced by the percussion,
but, as shown in Appendix C, their impact on the
magnetic field can be considered negligible.
3.2. Droplet tracking system
The position of the droplet is triangulated by
making use of the two lateral cameras of the upper
assembly (see Figure 1). After correcting the intrin-
sic image deformation by means of Scaramuzza’s
camera model [37] and synchronizing both video
signals, a circle is manually fitted to the contour
of the droplet to obtain the approximate position
{x0
i, z0
i}of the center of mass in the image axes. The
process is repeated for each video frame. These
measurements are then converted to their corre-
sponding spherical angles, which are referred to the
optical axis. Employing a pinhole camera model,
the spherical angles are
βx
i=x0
i
Rx
·F OVx, βz
i=z0
i
Rz
·F OVz,(22)
where i= 1,2 identifies the camera, Rjis the image
resolution, and F OVjis the field of view. Finally,
a5(z) and 4 (x) degree polynomial interpolation
is employed to filter the manual measurements.
A right Cartesian reference system {ˆx,ˆy,ˆz}lo-
cated at the base of the container and represented
in Figure 3 defines the world coordinates of the
droplet. The ˆzaxis is coincident with the axis of
symmetry, while the ˆxand ˆyaxes are aligned with
the square platform shown in Figure 1. It is con-
venient to first determine the coordinate y, which
based on Figure 3 is given by
tan(βz
1−αz
1)(d1+y)−tan(βz
2−αz
2)(d2−y)+∆h= 0,
(23)
where αz
iis the vertical inclination error, diis the
distance between the optical center and the mean
plane, ∆h=hz
1−hz
2, and hz
iis the height with
respect to the base of the container. Due to the
uncertainty in the geometric parameters, the error
in the determination of yincreases when the droplet
gets closer to the line of sight that connects both
cameras. However, artifacts are minimized if both
videos are correctly synchronized.
Once the depth is known, the zcoordinate is
given by
z=tan(βz
1−αz
1)·(d1+y) + hz
1.(24)
The process is analogous for the xcoordinate,
where
x=tan(βx
1−αx
1)·(d1+y) + hx
1,(25)
and hx
1is the lateral displacement of the camera in
the xaxis. The droplet acceleration values are ob-
tained by applying a central finite difference scheme
to the filtered position values.
7
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

Figure 3: Upper vessel and sketch of the visualization system
The parameters of the droplet tracking system
and their tolerances are given in Table 2. The lasts
are a conservative estimation based on the mea-
surement instruments and, in the case of the tilting
angles, on indirect image analysis procedures. The
tolerance in sensible parameters is increased to ac-
count for the relaxation of the experiment setup in
the absence of gravity.
A Monte Carlo simulation is carried out to esti-
mate the error of the visualization system. Each
geometric parameter listed in Table 2 is perturbed
with an additive white Gaussian noise by identify-
ing the measurement tolerance with the 3σinterval.
The fitting error in βx
iand βz
i, whose standard devi-
ation ranges from 0.04◦to 0.06◦, is also considered.
This set of perturbed parameters and variables is
fed into Eqs. (22) to (24) to obtain the droplet po-
sition and acceleration. The droplet tracking sys-
tem error is finally determined by computing the
standard deviation of the full Monte Carlo dataset
for the desired variables. 2500 simulations are per-
formed, that converge to the standard deviation of
the position with a rate of less than a 0.05%.
The uncertainties in the experimental droplet po-
sition also define the standard deviation of the total
force expressions listed in Table 1. The magnetic
field is determined with a standard deviation error
of ±3% and contributes to this effect.
3.3. Ferrofluid properties
The commercial Ferrotec EMG-700 water-based
ferrofluid is diluted in a 1:10 volume solution of
demineralized water (0.58% volume concentration).
Density ρ= 1.020 ±0.003 g/ml, viscosity µ=
1.448±0.007 cP and surface tension σ= 61.70±0.95
Table 2: Droplet tracking system parameters
Symbol Value Tolerance Units
Rx1080 - px
Rz1920 - px
F OVx58.7 ±0.5 deg
F OVz96.2 ±0.5 deg
αx
1-1.7 ±2.2*deg
αz
1-1.3 ±1.3*deg
αz
2-1.3 ±1.3*deg
d1199 ±2 mm
d2201 ±2 mm
hz
1143 ±3*mm
hz
2146 ±3*mm
hx
10±2 mm
*Increased to account for structure relaxation.
mN/m are measured. The magnetization curve,
measured with MicroSense EZ-9 Vibrating Sample
Magnetometer, is represented in Figure 4 and shows
an initial susceptibility χ= 0.181 and saturation
magnetization Ms= 3160 A/m.
The 10 nm nanoparticles of the ferrofluid solu-
tion are made of magnetite (F e3O4) and are coated
with an anionic surfactant. Their relaxation time
is estimated to be τ≈5·10−9s [33].
4. Results & Discussion
The magnetic field H0applied to the ferrofluid
droplet is produced by the coils and the magne-
tized ferrofluid volume that remains at the base of
the vessel. This system is approximated by means
of a 2D axisymmetric Comsol Multiphysics finite
M [A/m]
0
500
1000
1500
2000
2500
3000
3500
H [106 A/m]
0 0.2 0.4 0.6 0.8 1
χ=0.181
M [A/m]
0
250
500
750
1000
1250
1500
H [104 A/m]
0 0.2 0.4 0.6 0.8 1
Figure 4: Measured magnetization curve of the 1:10 EMG-
700 ferrofluid solution.
8
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

elements model. The internal fields Hand Mare
computed from H0by assuming that (i) the droplet
is spherical with a demagnetization factor D= 1/3,
(ii) the fields H,Mand H0are collinear, and (iii)
the internal dipoles are reoriented in a thermody-
namic quasi-equilibrium process. The different to-
tal force formulations can be easily integrated in
the droplet volume from these results. A detailed
explanation of the numerical model can be found in
the Appendix C.
The droplet projections in the cameras axes and
their corresponding interpolations are first pre-
sented in Figure 5. The measurement error is more
relevant in the xicomponent, as the movement of
the droplet is mainly produced in the ˆzaxis.
Figure 6 represents the position of the droplet as
a function of time in the Cartesian reference sys-
tem. The proximity to the vertical mid plane of
the image and the lower accuracy associated to the
lateral geometric parameters result in an uncertain
measurement of xin Figure 6(a). The depth yis
not affected by those factors, since it is estimated
from βiin Eq. (23). However, the errors in βihave
a significant impact on y, as shown in Figure 6(b).
Three axis acceleration values are derived from
the position of the droplet with the aforementioned
central finite differences scheme. The result is de-
a)
b)
σ=0.04º
σ=0.06º
β1
z
β1
z,fit
β2
z
β2
z,fit
βi
z(deg)
0
5
10
15
σ=0.05º
σ=0.06º
β1
x
β1
x,fit
β2
x
β2
x,fit
βi
x (deg)
−4
−2
0
2
4
6
t (s)
012
Figure 5: Droplet angles in cameras 1 and 2. Polynomials
with degree 5 and 4 are employed to filter the (a) βz
iand (b)
βx
iangles, respectively. The measurement error is quantified
with the standard deviation σ.
c)
a)
b)
x (m)
-0.014
-0.012
-0.01
-0.008
-0.006
y (m)
-0.02
-0.015
-0.01
-0.005
0
z (m)
0.13
0.14
0.15
0.16
0.17
0.18
t (s)
0 1 2
Figure 6: Position of the ferrofluid droplet as a function of
time. The error bands represent the standard deviation. (a)
x component, (b) y component, (c) z component
picted in Figure 7 and detailed in Table 3 with pre-
dictions of the total magnetic force per unit mass
given by the different formulations: FT
mand F0
m
denote the values given by Eq. (5) and Eq. (19), re-
spectively; Eq. (12) and Eq. (17) produce the same
output, which is marked with the label “FH
mor
FB
m”. Theoretical predictions are presented with
an error band determined by the uncertainty in the
droplet position and magnetic field model, which
shows standard deviations of a 3% of the H0mod-
ule when compared with actual measurements. The
high relative error in the lateral measurements re-
sults in an unreliable estimation of the lateral com-
ponents. Higher errors are clearly observed around
t= 2.4 s corresponding to the singularity in the line
of sight between the cameras. Large uncertainties
are also depicted at the beginning of the flight due
to the mathematical derivation of the polynomials
in the Monte Carlo analysis. Vertical accelerations,
however, agree well with the theory. This is con-
sistent with the relative uniformity of the vertical
magnetic field component near the axis of symme-
try, being less affected by radial uncertainties.
Since Eq. (5), Eq. (12), Eq. (17) and Eq. (19)
are equivalent, the numerical discrepancies between
them should be attributed to the simplifications in-
9
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

Table 3: Theoretical and experimental vertical acceleration values (in mm/s2) according to the formulations given by Eq. (5),
Eq. (10), Eq. (12), Eq. (17) and Eq. (19) as a function of time and the applied magnetic field in the center of the droplet. The
error bands are represented by the standard deviation.
t (s) 0 0.5 1 1.5 2 2.5
H0(A/m) 2445 ±73 2345 ±70 2395 ±72 2607 ±78 3078 ±92 4114 ±123
FT
m/md-17.5 ±1.4 -16.1 ±1.3 -16.8 ±1.3 -20.1 ±1.6 -28.3 ±2.2 -50.2 ±3.6
FH
m,V /md-16.8 ±1.4 -15.4 ±1.2 -16.1 ±1.3 -19.3 ±1.6 -27.5 ±2.2 -50.4 ±3.9
FH
m/md-17.9 ±1.4 -16.4 ±1.3 -17.2 ±1.4 -20.6 ±1.6 -29.3 ±2.3 -53.3 ±4.1
FB
m/md-17.9 ±1.4 -16.4 ±1.3 -17.2 ±1.4 -20.6 ±1.6 -29.3 ±2.3 -53.3 ±4.1
F0
m/md-17.8 ±1.4 -16.3 ±1.3 -17.0 ±1.4 -20.4 ±1.6 -29.0 ±2.3 -53.0 ±4.0
Exp. -17.5 ±2.4 -17.4 ±0.3 -17.9 ±0.3 -21.5 ±0.3 -31.0 ±0.3 -50.4 ±2.0
troduced in the physical model: (i) the assimila-
tion of the droplet geometry to a sphere, and (ii)
the approximation of Hgiven by Eq. (C.8), where
a constant demagnetization factor Dis assumed.
As indicated in the Appendix C and illustrated by
Figure C.10, the size of the droplet and the smooth-
c)
b)
a)
Fx/md (m/s2)
−0.01
0
0.01
0.02
Fy/md (m/s2)
−0.01
0
0.01
0.02
0.03
Fm
0
Fm
B or Fm
H
Fm
T
Kelvin Force (Fm,V
H)
Experimental
Fz/md (m/s2)
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
t (s)
0 1 2
Figure 7: Experimental acceleration and theoretical forces
per unit mass as a function of time. Theoretical values were
computed based on Eq. (5), Eq. (12), Eq. (17) and Eq. (19)
for a spherical droplet geometry. The error bands represent
the standard deviation. (a) x component, (b) y component,
(c) z component
ness of the magnetic field in the region under anal-
ysis justify those assumptions. The first has a sim-
ilar impact on every calculation procedure, but the
effects of the second depend on the formulation.
While Eq. (12), Eq. (17) and Eq. (19) compute
the fields in the magnetized volume, Eq. (5) only
employs the value of Mat the surface. Similarly,
Eq. (5), Eq. (12) and Eq. (17) make use of the fields
M,Hand B, but Eq. (19) only requires M. A
different effect of approximation (ii) on the total
force values from each formulation should then be
expected for different sets of parameters, like initial
susceptibility or droplet size. As observed in Fig-
ure 7, however, the discrepancies are not significant
due to the reduced size of the droplet. It should
be noted that the expression for p=p(υ, T , H) in
Eq. (3) or p=p(υ, T , B) in Eq. (14), reflects the
contribution of the magnetic field to the shape of
the magnetic fluid volume, which will have an in-
fluence on the total force acting on the medium.
Although the aforementioned simplifications may
contribute to the discrepancy between theoretical
and experimental values, the uncertainties associ-
ated to the droplet tracking system or the electro-
magnetic parameters play a more significant role,
as shown by the error bands. In terms of the slope,
however, it can be observed how experimental and
theoretical values have a similar behavior in the first
1.4 seconds of flight. When the droplet is subjected
to a stronger magnetic field (t > 1.4 s), the slopes
predicted by Eq. (12), Eq. (17) and Eq. (19) diverge
from the experimental ones. The results given by
Eq. (5), on the contrary, show a better agreement.
Again, this may be related with the impact of the
approximation (ii) on each expression.
The integration of the Kelvin force density given
by Eq. (10) in V, denoted by FH
m,V and correspond-
ing to the volume part of Eq. (12), is also repre-
10
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

sented for comparison. Although the differences be-
tween the predictions from that term and the total
force in Eq. (12) may seem insignificant in Figure 7
or Table 3, their long-term effect is not. The the-
oretical acceleration profiles are integrated starting
from the initial position and velocity of the droplet
and compared with experimental measurements in
Figure 8. The error band in the theoretical predic-
tions is due to the uncertainty in the initial posi-
tion and magnetic field model. The initial velocity
is assumed to be known. The laws of motion z(t)
predicted by Eq. (5), Eq. (19), Eq. (12) or Eq. (17)
show the best agreement with the experimental re-
sults and deviate from the prediction given by the
Kelvin force FH
m,V , that falls outside the experimen-
tal error band.
5. Conclusions
This work presents the formulations of the to-
tal magnetic force on magnetic liquids and demon-
strates their mathematical equivalence. Their im-
plementation and performance are addressed by an-
alyzing the movement of a ferrofluid droplet sub-
jected to a static magnetic field in microgravity.
As expected, the formulations based on the
Maxwell stress tensor in Eqs. (5), (12) and (17) or
on the free energy balance given by Eq. (19) cor-
rectly approximate the kinematics of the ferrofluid
droplet, while the Kelvin force density predictions
deviate significantly. The similarity between these
results is a consequence of the low susceptibility of
Fm
0
Fm
B or Fm
H
Fm
T
Kelvin Force (Fm,V
H)
Experimental
z (m)
0.13
0.14
0.15
0.16
0.17
0.18
t (s)
012
Figure 8: Measured and simulated height of the droplet as a
function of time for the formulations given in 12, 17 and 19.
The error bands represent the standard deviation.
the ferrofluid solution (χ= 0.181), close to that
of paramagnetic and diamagnetic liquids. Greater
divergences may be observed with highly concen-
trated ferrofluids, where χ > 1 [7; 9]. The in-
discriminate use of Kelvin’s force (or its dual ex-
pression) without the corresponding surface term
may consequently lead to large errors when com-
puting the total force. This assumption, common
in normal-gravity research [12–14], has a critical im-
pact in space applications, where the magnetic force
acquires an overwhelming role due to the absence
of gravity. Special care should then be taken when
modeling, implementing and describing the mag-
netic interaction.
The small differences between the numerical val-
ues given by Eq. (5), Eq. (12) or Eq. (17) and
Eq. (19) are due to the assumptions in the calcula-
tion of H. The results converge if the droplet size
is small with respect to the characteristic length of
variation of the magnetic field. Otherwise, the equi-
librium shape of the droplet will play an important
role in the calculation of the total magnetic force.
Future work could investigate how those differences
evolve for larger droplets if the previous assump-
tions are kept.
Besides the validation of the theoretical frame-
work, this experiment presents an example of mag-
netic mass transport in space-like environments.
The dynamics of the droplet are predicted with high
accuracy despite the inherent complexity of the
magnetic setup, conceptually close to real imple-
mentations. Therefore, applications dealing with
the control of magnetic liquids in low-gravity envi-
ronments may benefit from this contribution.
6. Competing Interests
The authors declare no competing interests.
7. Funding Sources
This work was supported by the European Space
Agency Education Office and the Center of Applied
Space Technology and Microgravity [ESA Drop
Your Thesis! 2017 Programme]; the University of
Seville [VI Plan Propio de Investigaci´on y Transfer-
encia]; and the research groups TEP-219 and TEP-
956 of ETSI-Sevilla.
11
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

8. Acknowledgements
The authors thank the ESA Education Office for
its financial, administrative, and academic support,
ZARM for its technical assistance, the University
of Seville for its academic supervision and financial
contribution in the context of the VI Plan Propio de
Investigaci´on y Transferencia, the research groups
TEP-219 and TEP-956 of ETSI-Sevilla for their
financial support, the Aerospace Engineering De-
partment and the Applied Physics III Department
of ETSI-Seville for granting access to their facili-
ties, TU Delft for its academic supervision, IGUS
for its financial and technical support, and ´
Alava In-
genieros for lending the visualization material. The
authors would like to express their gratitude to the
individuals of the aforementioned institutions who
supported the ESA DYT 2017 The Ferros project.
Appendix A. Cowley-Rosensweig formula-
tion
The magnetic stress tensor can be defined to in-
clude all the magnetic terms in Eq. (1), so that the
pressure–like variable is identified with the zero–
field pressure [17]. The expressions of Eqs. (2) and
(3) are then reformulated as:
TCR
m=BH −hπm(H) + µ0
2H2Ii,(A.1)
pCR (υ, T ) = p0(υ, T ).(A.2)
By applying Eqs. (6) and (7), the new volume
and surface force densities are
fCR
m,V =µ0M∇H− ∇πm,(A.3)
fCR
m,S =µ0
2M2
nn+π−
mn(A.4)
These force densities are different from those ex-
pressed in Eqs. (10) and (11). If the resultant of
these distributions is calculated according to the
general expression Eq. (8) and the corollary of the
divergence theorem for the gradient of a scalar field
ZV
dV(∇πm) = I∂V
dS(π−
mn) (A.5)
is applied, the expression of the total force given
by Eq. 12 is obtained. Ultimately, this demon-
strates that the magnetic pressure term πm(H) is
associated to the dipole interaction forces; that is, a
distributed system of internal forces that alters the
shape of the liquid interface, but does not produce
a net force.
Appendix B. Equivalence of total magnetic
forces
The dual expression in Eq. (15) for the magnetic
body force and Kelvin force given by Eq. (10) differ
in Vby
fB
m,V −fH
m,V =M∇B−µ0M∇H=µ0
2∇(M2)6=0,
(B.1)
which is, in general, a non-zero term. The surface
distributions are also locally different because
fB
m,S −fH
m,S =−µ0
2(M2
t+M2
n)n=−µ0
2M2n6=0.
(B.2)
This result leads to a clear discrepancy in the lo-
cal magnetic interaction. However, the symmetri-
cal roles played by Hand Bin thermodynamics
should be reflected in the magnetic force expres-
sion, so an a priori choice of one of the two forms
would be unjustified [5].
If Eq.(B.1) and (B.2) are integrated respectively
in the volume and the surface of the magnetized
medium and then added, the equivalence of the to-
tal magnetic force derived from the canonical (12)
and dual (17) formulations is obtained
FB
m−FH
m=µ0
2ZV
dV∇(M2)−I∂V
dSM 2n=0.
(B.3)
Appendix C. Numerical Model
The applied magnetic field H0to which the
droplet is subjected is computed in Comsol Mul-
tiphysics by solving the stationary Maxwell equa-
tions
∇ × H0=J0,(C.1)
B0=∇ × A0,(C.2)
J0=σE0,(C.3)
where J0is the current field, A0is the magnetic
vector potential produced by the magnetized mate-
rials and E0is the electric displacement field. The
constitutive relation
B0=µ0µrH0(C.4)
is applied to the aluminum plates (µAl
r= 1.000022),
surrounding air (µair
r= 1) and copper coils (µC u
r=
1). Within the ferrofluid volume, the constitutive
12
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The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

relation is defined by the magnetization curve M=
f(H) depicted in Figure 4, that results in
B0=µ01 + f(H0)
H0H0,(C.5)
where H0is the module of the magnetic field H0.
The ferrofluid volume is modeled as a cylinder of
110 mm diameter and 50 mm height, neglecting
the contribution of the oscillating free surface. This
approximation is consistent with the low suscepti-
bility of the ferrofluid solution and its small impact
on the magnetic field H0. Finally, the current field
is computed through
J0=NI
Aecoil,(C.6)
with N= 200 being the number of turns, I= 16.11
A the current intensity flowing through each wire,
A= 509 mm2the coils cross section and ecoil the
circumferential vector.
The simulation domain is a rectangular 1×3 m
region enclosing the assemblies. An axisymmet-
ric boundary condition is applied to the symme-
try axis, while the tangential magnetic potential is
imposed at the external faces through n×A0=
n×Ad.Adis the dipole term of the magnetic vec-
tor potential generated by the magnetization fields
of the coils and ferrofluid volumes. Consequently,
Adis computed as the potential vector generated by
four point dipoles applied at the centers of the mag-
netization distributions and whose moments are
those of said distributions. While the dipoles as-
sociated to the coils can be calculated beforehand,
the ferrofluid dipoles need to be approximated iter-
atively by integrating Min the ferrofluid volume.
The relative error in the magnetic vector potential
due the dipole approximation is estimated to be
below 1.0% at the boundary of the domain with
respect to the exact value generated by equivalent
circular loops.
The mesh is composed by 3225100 irregular tri-
angular elements shown in Figure C.9. Mean and
minimum condition numbers of 0.955 and 0.644 are
measured.
The fields H0(shown in Figure C.10), B0and
M0represent the main output from the previous
model. In order to compute the total forces in
the droplet volume and surface, given by Eq. (12),
Eq. (17) and Eq. (19), the internal fields Hand M
have to be first obtained. The droplet is considered
to be spherical, since the magnetic Bond number,
Figure C.9: Mesh of the magnetic field FEM model. An
identical assembly is implemented 368 mm below. The sim-
ulation domain is a 1×3 m rectangular region enclosing both
elements.
defined by
Bom=µ0H2R
σ,(C.7)
is approximately equal to 0.4. Consequently, sur-
face tension dominates over the magnetic compo-
nent, giving rise to the quasi-spherical equilibrium
surface observed in Figure 2.
Within the magnetized medium, the applied and
internal magnetic fields are related through H=
H0+Hd, where the demagnetization field Hdis
approximated as Hd=−DM,Dbeing the de-
magnetization factor. Considering the linearity of
the fields H,Mand H0inside the droplet and the
magnetization curve M=f(H), the relation
H+D f(H) = H0(C.8)
has to be satisfied. This expression allows comput-
ing the fields Hand Minside the droplet and then
the total forces in the droplet volume and surface,
given by Eq. (12), Eq. (17) and Eq. (19).
An exact value of Dcan only be obtained for
ellipsoidal geometries subjected to uniform fields,
being necessary to perform analytical approxima-
tions or numerical simulations for other cases [38].
However, given the small size of the droplet with re-
spect to the system and the smooth variation of H0
observed in Figure C.10, a demagnetization factor
D= 1/3, corresponding to a sphere, is assumed.
Based on Eq. (21), the characteristic time of
change of the magnetic field is estimated to be 6
s for a reference field of 2500 A/m. Since the ratio
13
Accepted Manuscript
The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

Figure C.10: Representation of the magnetic field H0lines
and intensity computed with Comsol Multiphysics at the up-
per assembly and path of the droplet. The color map shows
the magnetic field intensity H0, while the black contours rep-
resent the ferrofluid volume, coil and aluminum platform.
τf/τ ≈109, the system is subjected to a magneto-
static process. Finally, the thermodynamic quasi-
equilibrium condition can be assumed due the ab-
sence of relevant magnetodisipative effects.
Eq. (5) can be implemented by considering that
the normal component of Band the tangential com-
ponent of Hare continuous in the droplet interface,
and that the external medium is non-magnetic.
Then, the stress vector t+
n, integrand of Eq. (5),
results to be
t+
n=n· T +
m=BnH−−µ0
2hH−2−M2
nin
(C.9)
and can be computed by assuming the relation
given by Eq. (C.8). This is valid for the canoni-
cal and dual formulations of Tmshown in Eq. (2)
and Eq. (13).
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