Content uploaded by Álvaro Romero-Calvo

Author content

All content in this area was uploaded by Álvaro Romero-Calvo on Jul 08, 2022

Content may be subject to copyright.

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

2ZV

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

Accepted Manuscript

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=µ01 + f(H0)

H0H0,(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

2hH−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).

References

[1] C. Rinaldi, H. Brenner, Body versus surface forces in

continuum mechanics: Is the maxwell stress tensor a

physically objective cauchy stress?, Phys. Rev. E 65

(2002) 036615.

[2] M. Liu, K. Stierstadt, Colloidal Magnetic Fluids, 2009.

[3] S. Odenbach, M.Liu, Invalidation of the kelvin force in

ferroﬂuids, Physical Review Letters 86 (2001) 328.

[4] A. Engel, Comment on “invalidation of the kelvin force

in ferroﬂuids”, Phys. Rev. Lett. 86 (2001) 4978–4978.

[5] M. Liu, Liu replies, Phys. Rev. Lett. 86 (2001) 4979.

[6] A. Lange, Kelvin force in a layer of magnetic ﬂuid,

Journal of Magnetism and Magnetic Materials 241 (2)

(2002) 327 – 329.

[7] M. Petit, A. Kedous-Lebouc, Y. Avenas, M. Tawk,

E. Arteaga, Calculation and analysis of local magnetic

forces in ferroﬂuids, Przeglad Elektrotechniczny (Elec-

trical Review) 87 (2011) 115–119.

[8] A. Engel, R. Friedrichs, On the electromagnetic force on

a polarizable body, American Journal of Physics 70 (4)

(2002) 428–432.

[9] A. F. Bakuzis, K. Chen, W. Luo, H. Zhuang, Magnetic

body force, International Journal of Modern Physics B

19 (07n09) (2005) 1205–1208.

[10] J. C. Boulware, H. Ban, S. Jensen, S. Wassom, Inﬂuence

of geometry on liquid oxygen magnetohydrodynamics,

Experimental Thermal and Fluid Science 34 (8) (2010)

1182 – 1193.

[11] J. Martin, J. Holt, Magnetically actuated propel-

lant orientation experiment, controlling ﬂuid motion

with magnetic ﬁelds in a low-gravity environment,

NASA/TM-2000-210129, M-975, NAS 1.15:210129.

[12] D. Shi, Q. Bi, Y. He, R. Zhou, Experimental inves-

tigation on falling ferroﬂuid droplets in vertical mag-

netic ﬁelds, Experimental Thermal and Fluid Science

54 (April) (2014) 313 – 320.

[13] P. Poesio, E. Wang, Resonance induced wetting state

transition of a ferroﬂuid droplet on superhydrophobic

surfaces, Experimental Thermal and Fluid Science 57

(2014) 353 – 357.

[14] M. E. Moghadam, M. B. Shaﬁi, E. A. Dehkordi, Hydro-

magnetic micropump and ﬂow controller. part a: Ex-

periments with nickel particles added to the water, Ex-

perimental Thermal and Fluid Science 33 (2009) 1021

– 1028.

[15] S. Papell, Low viscosity magnetic ﬂuid obtained by the

colloidal suspension of magnetic particles, US Patent

3215572.

[16] J. L. Neuringer, R. E. Rosensweig, Ferrohydrodynam-

ics, The Physics of Fluids 7 (12) (1964) 1927–1937.

[17] R. E. Rosensweig, Ferrohydrodynamics, Dover Publica-

tions, 1997.

[18] D. Ludovisi, S. S. Cha, N. Ramachandran, W. M.

Worek, Heat transfer of thermocapillary convection in a

two-layered ﬂuid system under the inﬂuence of magnetic

ﬁeld, Acta Astronautica 64 (11) (2009) 1066 – 1079.

[19] A. Bozhko, G. Putin, Thermomagnetic convection as

a tool for heat and mass transfer control in nanosize

materials under microgravity conditions, Microgravity

Science and Technology 21 (1) (2009) 89–93.

[20] A. Causevica, P. Sahli, F. Hild, K. Grunwald, M. Ehres-

mann, G. Herdrich, Papell: Interaction study of fer-

roﬂuid with electromagnets of an experiment on the in-

ternational space station, in: Proceedings of the 69th

International Astronautical Congress, 2018.

[21] B. A. Jackson, K. J. Terhune, L. B. King, Ionic liquid

ferroﬂuid interface deformation and spray onset under

electric and magnetic stresses, Physics of Fluids 29 (6)

(2017) 064105.

[22] K. Lemmer, Propulsion for cubesats, Acta Astronautica

134 (2017) 231 – 243.

14

Accepted Manuscript

The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124

[23] J. G. Marchetta, A. P. Winter, Simulation of magnetic

positive positioning for space based ﬂuid management

systems, Mathematical and Computer Modelling 51 (9)

(2010) 1202 – 1212.

[24] A. Romero-Calvo, G. Cano G´omez, E. Castro-

Hern´andez, F. Maggi, Free and Forced Oscillations of

Magnetic Liquids Under Low-Gravity Conditions, Jour-

nal of Applied Mechanics 87 (2), 021010.

[25] L. Landau, E. Lifshitz, Electrodynamics of Continuous

Media, Pergamon Press, 1960.

[26] W. Panofsky, M. Phillips, Classical Electricity and

Magnetism, Addison–Wesley Publishing Co., 1962.

[27] J. Jackson, Classical Electrodynamics, John Wiley,

1998.

[28] D. Griﬃths, Introduction to Electrodynamics,

Prentice–Hall, 1999.

[29] J. A. Stratton, Electromagnetic Theory, McGraw-Hill,

1941.

[30] J. Melcher, Continuum Electromechnanics, MIT Press,

1981.

[31] R. E. Rosensweig, Continuum equations for magnetic

and dielectric ﬂuids with internal rotations, The Journal

of Chemical Physics 121 (3) (2004) 1228–1242.

[32] M. Liu, Range of validity for the kelvin force, Phys.

Rev. Lett. 84 (2000) 2762.

[33] I. Torres-D´ıaz, C. Rinaldi, Recent progress in ferroﬂu-

ids research: novel applications of magnetically control-

lable and tunable ﬂuids, Soft Matter 10 (2014) 8584–

8602.

[34] A. Romero-Calvo, T. H. Hermans, L. P. Ben´ıtez,

E. Castro-Hern´andez, Drop Your Thesis! 2017 Exper-

iment Report - Ferroﬂuids Dynamics in Microgravity

Conditions, European Space Agency - Erasmus Exper-

iment Archive, 2018.

[35] A. Romero-Calvo, T. Hermans, G. Cano G´omez, L. Par-

rilla Ben´ıtez, M. A. Herrada Guti´errez, E. Castro-

Hern´andez, Ferroﬂuid dynamics in microgravity condi-

tions, in: Proceedings of the 2nd Symposion on Space

Educational Activities, 2018.

[36] A. Myshkis, R. Wadhwa, Low-gravity ﬂuid mechanics:

mathematical theory of capillary phenomena, Springer,

1987.

[37] D. Scaramuzza, A. Martinelli, R. Siegwart, A tool-

box for easily calibrating omnidirectional cameras, in:

2006 IEEE/RSJ International Conference on Intelligent

Robots and Systems, Benjing, China, 2006, pp. 5695–

5701.

[38] B. Pugh, D. Kramer, C. Chen, Demagnetizing factors

for various geometries precisely determined using 3-d

electromagnetic ﬁeld simulation, IEEE Transactions on

Magnetics 47 (10) (2011) 4100–4103.

15

Accepted Manuscript

The final version of this paper can be found in https://doi.org/10.1016/j.expthermflusci.2020.110124