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IAC-20,A2,2,1,x59279

Advanced numerical simulation of magnetic liquid sloshing in microgravity

´

A. Romero-Calvo1, M.A. Herrada 2, G. Cano-G´

omez3, H. Schaub4

The term sloshing refers to the movement of liquids in partially ﬁlled containers. Low-gravity sloshing plays an important role in

the conﬁguration of space vehicles, as it affects their dynamics and complicates the propellant management system design. Magnetic

forces can be used to position a susceptible ﬂuid, tune its natural frequencies, and increase its damping ratios in low-gravity. However,

prior work shows that the analysis of this phenomenon, named magnetic liquid sloshing, requires advanced modeling capabilities. This

paper introduces a coupled magnetohydrodynamic (MHD) model for the study of low-gravity axisymmetric magnetic liquid oscillations.

The incompressible, viscous mass and momentum balances are solved together with the steady-state Maxwell equations by following a

monolithic solution scheme. The method is fully implicit, allowing to reach a steady-state solution in a single time step. Five regions are

used to discretize the simulation domain, that combines non-singular mappings and a meshfree approach. The steady-state solution (basic

ﬂow) is veriﬁed with equivalent computations from Comsol Multiphysics assuming the geometry and physical properties of the ESA Drop

Your Thesis! 2017 drop tower experiment. Future steps include the study of linear oscillations, free surface stability properties, and lateral

oscillations, among others.

1 Introduction

Propellant sloshing represents a major concern for space engineers

due to its impact on the dynamics of space vehicles. During launch,

the uncontrolled movement of liquids may lead to a total or par-

tial mission failure [1]. In microgravity, sloshing becomes highly

stochastic, resulting in a complicated propellant management and

attitude control system design [2]. Propellant Management Devices

(PMD) are commonly employed to ensure a gas-free expulsion of

propellant, ﬁx the center of mass of the liquid, and increase its slosh-

ing frequencies and damping ratios. However, they also increase the

inert mass of the vehicle and complicate numerical analysis [3, 4].

A gravity-equivalent force may be generated by means of elec-

tromagnetic ﬁelds as an alternative to classical PMDs and active

settling methods. The use of dielectrophoresis, a phenomenon on

which an electric force is exerted on dielectric materials, was ex-

plored by the US Air Force with suitable propellants in 1963. The

study unveils a high risk of arcing inside the tanks and highlights

the need for large, heavy and noisy power sources [5]. The mag-

netic equivalent, named Magnetic Positive Positioning (MP2), has

also been suggested to exploit the inherent properties of paramag-

netic, diamagnetic, and ferromagnetic liquids [6–8].

MP2devices must deal with the rapid decay of magnetic ﬁelds

with distance, that limits their reachability to relatively small re-

gions. This difﬁculty may be compensated by employing highly

susceptible liquids, such as ferroﬂuids. Ferroﬂuids are colloidal sus-

pensions of magnetic nanoparticles developed in the early 1960s to

enhance the controllability of rocket propellants [7]. Despite hav-

ing numerous applications on Earth, contributions addressing their

original purpose are still scarce. Normal-gravity works have ex-

plored fundamental aspects of the dynamics of magnetic liquids,

such as the natural frequency shifts due to the magnetic interac-

1Department of Aerospace Engineering Sciences, University of Colorado

Boulder, CO, United States; alvaro.romerocalvo@colorado.edu

2Departamento de Ingenier´

ıa Aeroespacial y Mec´

anica de Fluidos, Univer-

sidad de Sevilla, Sevilla, Spain; herrada@us.es

3Departamento de F´

ısica Aplicada III, Universidad de Sevilla, Sevilla,

Spain; gabriel@us.es

4Department of Aerospace Engineering Sciences, University of Colorado

Boulder, CO, United States; hanspeter.schaub@colorado.edu

tion [9], axisymmetric sloshing [10, 11], two-layer sloshing [12],

liquid swirling [13] or the development of tuned magnetic liquid

dampers [14, 15]. Low-gravity contributions include the gravity

compensation experiments performed by Dodge in 1972, that in-

directly addressed the low-gravity sloshing of ferroﬂuids subjected

to quasi-uniform magnetic forces [16]. Motivated by the advent of

stronger permanent magnets and high-temperature superconductors,

the NASA MAPO experiment validated the magnetic positioning of

liquid oxygen in a series of parabolic ﬂights in 2001 [6]. Subse-

quent works present reﬁned numerical models and results of tech-

nical relevance [17–25]. The axisymmetric sloshing of water-based

ferroﬂuids was characterized in microgravity when subjected to an

inhomogeneous magnetic ﬁeld as part of the ESA Drop Your The-

sis! 2017 campaign [26–29]. As a follow-up, the lateral sloshing of

ferroﬂuids was studied in the framework of the UNOOSA DropTES

Programme 2019 [30, 31].

Most existing works assume that the ﬂuid-magnetic problem de-

scribed by the MP2concept can be studied with a set of uncoupled

ﬂuid-magnetic equations [6, 16–25]. This is appropriate for low-

susceptibility ﬂuids, such as liquid oxygen or liquid hydrogen. The

development of coupled magnetohydrodynamic simulation frame-

works has however been identiﬁed as a key step towards the de-

sign of novel magnetic liquid sloshing devices [8]. Since the po-

sition of a highly-susceptible ﬂuid modiﬁes the magnetic ﬁeld dis-

tribution, such numerical models should simultaneously solve the

Navier-Stokes (ﬂuid-dynamic) and Maxwell (magnetic) equations.

Desired results include, but are not limited to, sloshing modes and

frequencies, free surface stability properties, viscous damping coef-

ﬁcients, and time-dependent simulations.

This ongoing project describes the implementation of the cap-

illary ﬂuid modeling methodology developed by Herrada and Mon-

tanero [32] and includes the magnetostatic Maxwell equations and

associated ﬂuid-magnetic interactions to study the axisymmetric

magnetic sloshing problem. Unlike the quasi-analytical coupled

magnetic sloshing model developed by the authors in Ref. 33, the

ﬂuid-magnetic equations are solved simultaneously (not iteratively)

and a viscous ﬂuid is assumed. The numerical method is described

in Sec. 3, and the veriﬁcation of the steady-state solution is per-

formed in Sec. 4. Section 5 summarizes the main conclusions and

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Fig. 1: Problem formulation

future steps of the work.

2 Problem formulation

The system under analysis, represented in Fig. 1, consists on a par-

tially ﬁlled cylindrical tank subjected to the inﬂuence of a coil or

magnet. The vessel has radius R, height h, and holds a volume V

of an incompressible, Newtonian, magnetic liquid with density ρ,

speciﬁc volume ν=ρ−1, shear coefﬁcient of viscosity η, and sur-

face tension σ. The static contact angle between the liquid and the

wall is θc. A non-reactive inviscid gas at pressure pgﬁlls the free

space. The system is subjected to an inertial acceleration galong

the vertical axis and to an inhomogeneous, axisymmetric magnetic

ﬁeld imposed by either a coil or a magnet.

The magnetohydrodynamic model here developed is designed

to provide the axisymmetric (i) meniscus proﬁle, (ii) free surface os-

cillation frequencies, (iii) free surface stability properties, and (iv)

time evolution of the system. Although the model can be applied

to different tank geometries, liquids, and magnetic conﬁgurations,

the experimental setup implemented for the ESA Drop Your The-

sis! 2017 experiment [26–28] has been chosen to conduct the ver-

iﬁcation of results. Such experiment studies the axisymmetric free

surface oscillations frequencies of a ferroﬂuid solution subject to an

inhomogeneous magnetic ﬁeld in microgravity, and is hence appro-

priate for the validation of the numerical model presented in this

manuscript.

2.1 Stress tensor and force distributions

The magnetodynamic state of an incompressible continuous

medium can be described by means of the viscous Maxwell stress

tensor, given by [34–37]

T=Tp+Tν+Tm,(1)

where the pressure, viscous, and magnetic terms are

Tp=−p∗I,(2a)

Tν=ηh∇v+ (∇v)Ti,(2b)

Tm=−µ0

2H2I+BH ,(2c)

and where

p∗=p(ν, T ) + µ0ZH

0

∂

∂ν [νM]dH0(3)

is the composite pressure including the hydrostatic p(ν, T )and mag-

netopolarization terms. In these expressions, B=µ0(H+M),

H, and Mare the ﬂux density, magnetic, and magnetization ﬁelds,

µ0is the permeability of vacuum, I=δij eiejis the unit dyadic

in the Cartessian eireference system, and vis the ﬂuid velocity.

For soft magnetic materials, the magnetization ﬁeld is aligned with

the magnetic ﬁeld and follows the relation M=χvol(H)H, with

χvol(H)being the volume magnetic susceptibility. Applications in-

volving unequilibrated ferroﬂuid solutions, for which M×H6= 0,

should incorporate the effects resulting from particle rotation. An

additional term must be added to the viscous stress tensor Tν, and

the angular momentum and magnetic relaxation equations also have

to be considered [36, 37].

The forces per unit volume exerted on the medium in the ab-

sence of electric ﬁelds can be computed as the divergence of the

stress tensor given by Eq. 1, resulting in [37]

f=fp+fν+fm,(4)

with

fp=∇ · Tp=−∇p∗,(5a)

fν=∇ · Tν=∇ · ηh∇v+ (∇v)Ti,(5b)

fm=∇ · Tm=µ0M∇H=µ0(M· ∇)H(5c)

If the viscosity coefﬁcient ηis constant, the viscous term reduces to

fν=η∇2v.(6)

However, ferroﬂuids exhibit a nonlinear dependence of the shear

coefﬁcient of viscosity ηwith Hthrough [38, 39]

∆η

ζ=µ0M0Hτ

4ζ+µ0M0Hτ sin2β , (7)

where ζis the vortex viscosity,τis the Brownian relaxation time,

M0is the unperturbed magnetization value, and βis the angle be-

tween the magnetic ﬁeld Hand the vorticity vector Ω. For dilute

ferroﬂuids, ζ= (3/2)ηφ, with φbeing the volume fraction of solids

in the ferroﬂuid [38]. This effect is negligible for most applications,

but may be relevant for concentrated ferroﬂuids subjected to strong

magnetic ﬁelds.

Surface forces appear in the gas-liquid interface as a conse-

quence of the discontinuity in the stress tensor. Those forces are

balanced according to the ferrohydrodynamic incompressible vis-

cous boundary condition. Assuming a contact between a ferroﬂuid

and a non-magnetic, inviscid gas, the condition is expressed in nor-

mal (n) and tangential (t) components as [37]

n:p∗−2ηδvn

δxn

+pn−p0= 2σH,(8a)

t:η∂vn

∂xt

+∂vt

∂xn= 0,(8b)

with nbeing the external normal vector, pn=µ0M2

n/2the mag-

netic normal traction,Hthe mean curvature of the interface, vnand

vtthe normal and tangential velocity components, and xnand xtthe

distances along the normal and tangential directions, respectively.

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2.2 Governing equations for a magnetic, viscous, incompressible

ﬂuid

The magnetohydrodynamic mass and momentum conservation

equations deriving from the stress tensor given by Eq. 1 are [37]

∇ · v= 0,(9a)

ρDv

Dt =ρg+fp+fν+fm,(9b)

with Ddenoting the material derivative, and tthe time. This system

of equations is subjected to an appropriate set of boundary condi-

tions. Particular attention should be payed to those described by Eq.

8 for magnetic systems.

The terms fpand fmare deﬁned by the ﬁelds H,B, and M,

which have to be computed at each time step by solving the mag-

netic problem. Assuming a static magnetic conﬁguration without

surface currents and electric ﬁelds, the steady-state Maxwell’s equa-

tions are given by

∇ · B= 0,(10a)

∇ × H=Je,(10b)

where Jeis the volume density of electric current. If no surface

currents are applied to the system, the magnetic boundary conditions

become

n·(B2−B1) = 0,(11a)

n×(H2−H1) = 0,(11b)

with nbeing the external normal vector. Therefore, the normal com-

ponent of Band the tangential component of Hare continuous

across the interface.

Fig. 2: Numerical simulation domain and labels for the mapped

(A,B,C,E) and meshless (D) regions and interfaces. The ex-

ternal boundary of region D is a cylinder of 0.6 m radius and

1.2 m height.

3 Numerical method

The theoretical framework described in the previous section is here

implemented adopting the cylindrical reference system {er,eφ,ez}

shown in Fig. 1. The simulation domain is divided into ﬁve regions

and their corresponding interfaces, as shown in Fig. 2: A (liquid

domain), B (air domain inside the container), C (air domain over

the container), D (surrounding air environment), and E (the coil or

magnet domains).

3.1 Axisymmetric Navier-Stokes equations

The mass and momentum conservation equations deﬁned by Eq. 9

should be expressed in the cylindrical reference system after consid-

ering the axisymmetry of the problem. This results in

∂(ru)

∂r +∂(rw)

∂r = 0,(12a)

ρ∂u

∂t +u∂u

∂r +w∂u

∂z =−∂p∗

∂r +

η∂2u

∂r2+∂(u/r)

∂r +∂2u

∂z2+µ0Mr

∂Hr

∂r +Mz

∂Hr

∂z ,

(12b)

ρ∂w

∂t +u∂w

∂r +w∂w

∂z =−∂p∗

∂z +

η∂2w

∂r2+1

r

∂w

∂r +∂2w

∂z2+µ0Mr

∂Hz

∂r +Mz

∂Hz

∂z ,

(12c)

where r(z) is the radial (axial) coordinate, and u(w) is the radial

(axial) velocity component. The axisymmetry of the magnetic prob-

lem has been taken into account in the previous expressions, so that

Jehas only azimuthal components, and Mand Hlack from them,

resulting in H=Hrer+Hzez, and M=Mrer+Mzez. When a

magnet is implemented, its magnetization only consists on a vertical

component, so the axisymmetry properties are maintained.

3.2 Magnetic potentials formulation

Equations (10a) and (10b) can be rewritten as a function of H, re-

sulting in

∇ · H=−∇ · M,(13a)

∇ × H=Je.(13b)

Therefore, Hhas scalar sources in the magnetized region and vector

sources in the coil. According to Helmholtz’s theorem, Hcan be

expressed in terms of scalar and vector magnetic potentials. Taking

into account the axisymmetry of the problem,

Hr=−1

r

∂Ψ

∂z −∂Φ

∂r ;Hz=1

r

∂Ψ

∂r −∂Φ

∂z ,(14)

where Φis the scalar potential generated by scalar sources, and the

stream-like function Ψ = rAe(r, z)/µ0is directly related to the

azimuthal component Ae(r, z)of the vector magnetic potential cre-

ated by the electric current. The magnetic problem is then formu-

lated and solved in terms of Φand Ψby noting that

∇ · H=−∂2Φ

∂z2+∂2Φ

∂r2+1

r

∂Φ

∂r ,(15)

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∇ × H=−1

r∂2Ψ

∂r2+∂2Ψ

∂z2−1

r

∂Ψ

∂r uφ.(16)

In the domain A, M=χvol(H)Hand Eq. (13a) becomes

∂2Φ

∂z2+∂2Φ

∂r2+1

r

∂Φ

∂r =1

1 + χvol(H)Hr

∂χvol

∂r +Hz

∂χvol

∂z ,

(17)

where χvol also depends on Ψand Φthrough the magnetization law

χvol =aM

arctan(cMH)

H+bM

arctan(dMH)

H+eM.(18)

For domains B-E, Eq. (13a) is simply ∇ · H= 0 due to the ab-

sence of inhomogeneous magnetization ﬁelds (since the magnet is

uniformly magnetized). When the domain Eis occupied by a coil,

Je=NI

Sc

eφ,(19)

with Nbeing the number of wire turns, Ithe current ﬂowing

through each of them, and Scthe cross section of the coil. Con-

sequently, Eq. (13b) adopts the form

−1

r∂2Ψ

∂r2+∂2Ψ

∂z2−1

r

∂Ψ

∂r =N I

Sc

.(20)

For domains A-D (and E, when it represents a magnet), Je= 0,

and the previous expression simpliﬁes to ∇ × H=0. If a magnet

with uniform, vertical magnetization ﬁeld occupies the region E, the

condition M=Mmuzis imposed and the stream-like function has

the trivial solution Ψ = 0.

3.3 Boundary conditions

An axisymmetric boundary condition is applied at the axis of sym-

metry for both ﬂuid and magnetic problems, while the wall-liquid

interaction is described by the non-penetration boundary condition.

This results in

3 : u= 0,∂w

∂r = 0,(21)

9b:u=w= 0,(22)

9r:u=w= 0.(23)

In order to compute the viscous damping parameters of the problem,

the mesh has to be reﬁned at the walls 9b and 9r. The interfacial

conditions described by Eq. 8 are particularized at the free ﬂuid

surface 11 by following a parametrization of the form zin =G(s, t)

and rin =F(s, t), resulting in the normal balance

p∗+pn−p0=σ

F∂F

∂s −1∂

∂s

F∂G

∂s

q∂F

∂s 2+∂ G

∂s 2

+ 2η

∂u

∂r ∂ G

∂s 2+∂ w

∂z ∂ F

∂s 2−∂ w

∂r +∂ u

∂z ∂ F

∂s

∂G

∂s

∂G

∂s 2+∂ F

∂s 2,(24a)

and the kinematic and geometric compatibility equations

u−∂F

∂t ∂G

∂s −w−∂G

∂t ∂F

∂s = 0,(24b)

∂G

∂s

∂2G

∂s2+∂F

∂s

∂2F

∂s2= 0,(24c)

where sis the arc length coordinate along the interface. The contact

angle θcis imposed at the wall ( s= 1) through

∂F

∂s tan π

2−θc+∂G

∂s = 0, and F =R. (24d)

The magnetic boundary conditions derive from Eq. 11 after not-

ing that the magnetic liquid and magnet are magnetized, while the

coil has null magnetization. This results in

2−5 : Hr= 0, Br= 0,(25)

9b, 9r, 11 : Ht,1=Ht,2, Bn,1=Bn,2,(26)

14 −17 (coil) : H1=H2,B1=B2,(27)

14 −17 (magnet) : Ht,1=Ht,2, Bn,1=Bn,2,(28)

10,12,13 : H1=H2,B1=B2.(29)

The potentials Ψand Φare truncated at the external contour

(1,6-8) by considering the magnetic dipole term of the coil (c), and

magnetized (m) domains, which are characterized by the moments

mα=mαuzlocated at rα=zαuz(with α=c, m). If the

external contour is sufﬁciently separated from the magnetic sources,

the potential vector of the system becomes similar to that of the

dipole associated with the coil, and then

Ψ(r, z)|1,6−8≈mc

4π

r2

[r2+ (z−zc)2]3/2.(30)

The scalar potential Φis generated by the magnetized media. As-

suming that its contribution to the Hﬁeld at the external contour

is approximately equal to the contribution of the dipole terms of the

magnet and magnetized liquid, the condition

(−∇Φ)1,6−8≈Hdip

m(31)

is satisﬁed, and

−∂Φ

∂r 1,6−8

≈X

i

3mm,i

4π

r(z−zm,i)

[r2+ (z−zm,i)2]5/2

1,6−8

,

(32a)

−∂Φ

∂z 1,6−8

≈X

i

mm,i

4π

2(z−zm,i)2−r2

[r2+ (z−zm,i)2]5/2

1,6−8

.

(32b)

Although the dipoles associated with the coil and magnet can be

calculated beforehand, the dipole produced by the magnetized liq-

uid needs to be approximated iteratively by integrating Min the

domain A.

It should be noted that, if the system only includes magnets and

magnetized liquids, the boundary condition given by Eq. (30) be-

comes unnecessary, since Ψ(r, z) = 0. Similarly, it is possible to

impose Φ1,6−8≈0when a weakly magnetized liquid and a coil

are considered. Although less rigorous, the magnetic isolation con-

dition Φ1,6−8= Ψ1,6−8= 0 can also be imposed for very large

simulation domains.

3.4 Solution procedure

The numerical procedure used in this study is a variation of that

developed in Herrada & Montanero for interfacial ﬂows [32]. As

shown in Fig. 2, the simulation domain is divided into ﬁve blocks

that implement different discretization methods.

Domains A, B and C are is mapped onto a rectangular domain

by means of non singular mappings

rA=F(s, t), zA=G(s, t)ηA,(33a)

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rB=F(s, t), zB=G(s, t) + [h−G(s, t)]ηB(33b)

rC=F(s, t), zC=H+ [htop −h]ηC,(33c)

where htop is the height of the domain, 0≤s≤1,0≤ηA≤1,

0≤ηB≤1, and 0≤ηC≤1.

The derivatives appearing in the governing equations are ex-

pressed in terms of s,η, and t. Then, the resulting equations

are discretized in the sdirection using second-order ﬁnite differ-

ences with nsequally spaced points. In the ηdirection, second-

order ﬁnite differences are also employed with nηA,nηA, and nηC

equally spaced points. This discretization strategy gives rise to

meshes that automatically adapt to any variation of the free liquid

interface (rin =F(s, t), zin =G(s, t)). The ﬁxed domain E is

discretized using second-order ﬁnite differences with nrE(nzE)

equally spaced points in the radial (axial) direction. The results

presented in this work are obtained using ns= 81,nηA= 81,

nηB= 81,nηC= 81,nrE= 51, and nzE= 51.

In order to gain ﬂexibility in the geometrical conﬁguration of

the external magnetic sources, a meshless method is followed to dis-

cretize the domain D. The boundary nodes in contact with domains

A, B, C, and E are forced to have the same coordinates as those in

the connecting domains. This implies that all points in contact with

A and B move in accordance with the ﬂuid. On the contrary, lines

1, 2, 7 and 8 have a ﬁxed discretization. A 2D Matlab-based Delau-

nay mesh generator5is used to obtain the nodes of the grid with a

maximum element size of 3 cm and a maximum growth rate of 0.1

[40]. The spatial derivatives in the D domain are computed from a

set of 7 neighbors surrounding each node, which have the minimum

virtual distance

dv=q(fkr∆r)2+ (fkz∆z)2,(34)

where f≈0.25 is a weighting factor, kr(kz) is a binary index that

is set to 1 if the node belongs to a horizontal (vertical) boundary, be-

coming 0 otherwise, and {∆r, ∆z}are the relative cylindrical coor-

dinates of the neighbors referred to the central node. This algorithm

results in a good conditioning of the boundary nodes by promoting

the selection a set of neighbors in the interior of the domain. Once

the neighbors are deﬁned, a second order Moving Least Squares

algorithm [41] with six polynomial terms and a quartic weight func-

tion with a scale length of 10 cm is employed to compute the col-

location matrices. Since the points in contact with A and B change

for each time step, only a limited number of neighbors and collo-

cation matrices is updated, resulting in an enhanced computational

efﬁciency. The result of the discretization is represented in Fig. 2.

To compute the meniscus (steady-state solution), all the equa-

tions of the system are solved together (monolithic scheme) by em-

ploying a Newton–Raphson technique. In this work, second order

backward differences are used to compute the time derivatives, and

since the method is fully implicit, the time step is chosen to be sufﬁ-

ciently large to ensure that a steady state is reached in a single step.

One of the main characteristics of this procedure is that the elements

of the Jacobian matrix are obtained by combining analytical func-

tions and the collocation matrices of all subdomains. This allows

taking advantage of the sparsity of the resulting matrix to reduce the

computational time on each Newton-Raphson iteration.

5https://github.com/dengwirda/mesh2d. Consulted on:

05/07/2020

4 Veriﬁcation and validation

The magnetohydrodynamic model here introduced is an extension

of the capillary model presented in Ref. 32, which has already

been validated with experimental measurements at the International

Space Station. After verifying that the ﬂuid-dynamic results in the

absence of magnetic ﬁelds are correct, the veriﬁcation and validation

process focuses on the magnetic modules. In this section, funda-

mental results are compared with analogous magnetostatic models

and analytical predictions. The geometrical and physical conﬁg-

urations of the ESA Drop Your Thesis! 2017 experiment [29] is

implemented.

The magnetic (H) and magnetization (M) ﬁelds are ﬁrst com-

pared in Figs. 3 and 4 with a geometrically equivalent solution

from Comsol Multiphysics. In both cases, the ﬁelds are in excel-

lent agreement with the veriﬁcation case, reﬂecting the appropriate

implementation of Eqs. 10 and 11. Similar levels of agreements

are observed when the coil is replaced by a vertically magnetized

magnet.

Figure 5 compares the magnetic force density for I= 20 A

with Comsol Multiphysics. The distributions are practically iden-

tical both in module and direction, which veriﬁes the correct im-

plementation of the magnetic force formulations. Since the force

depends on the spatial derivatives of the magnetic ﬁeld, as reﬂected

by Eq. (5c), it is highly sensitive to irregularities in Hand M. In

the plot, slight instabilities are manifested as knots at the interface

between domains A and D. However, they are much less signiﬁcant

than their Comsol Multiphysics counterpart, reﬂecting a robust im-

plementation of the numerical method.

The implementation of the magnetic force term in the momen-

tum balance is veriﬁed in Fig. 6. According to Eq. 9b, the steady

state pressure lines must be coincident with the constant Hlines for

a linearly magnetized liquid. After implementing a constant mag-

netic susceptibility χvol = 0.1, the comparison between both plots

reﬂects the desired feature.

The shape of the meniscus is ﬁnally represented in Fig. 7 for

the non-magnetic (I= 0 A) and magnetic (I= 11 A) cases and

the physical properties reported in Ref. 29. The solution from a

previous quasi-analytical magnetic sloshing model [33] is repre-

sented for comparison. Although the shapes are identical for the

non-magnetic case, the magnetic result is slightly different. This

may reﬂect fundamental differences between the physical models

(e.g. level of ﬂuid-magnetic coupling), the accumulation of numeri-

cal errors, or an error in the implementation. In order to understand

this divergence and verify the computation of the meniscus, larger

free surface deformations must be induced by simulating stronger

magnets (see for instance Ref. [33]). However, strong magnetic

ﬁelds require an excellent conditioning of the collocation matrices

in region D, which is discretized by following a meshfree methodol-

ogy. Small numerical errors give rise to the instabilities discussed in

Fig. 5. Current efforts are consequently focused on the development

of a robust and well-conditioned discretization for region D.

5 Conclusions

A fully coupled magnetohydrodynamic model has been developed

to analyze the axisymmetric oscillations of magnetic liquids in mi-

crogravity. The model differs from previous works in the coupled

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(a) Model (b) Comsol Multiphysics

Fig. 3: Magnetic ﬁeld comparison for I= 20 A.

(a) Model (b) Comsol Multiphysics

Fig. 4: Magnetization ﬁeld comparison for I= 20 A.

(a) Model (b) Comsol Multiphysics

Fig. 5: Force density comparison for I= 20 A.

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(a) Model (b) Comsol Multiphysics

Fig. 6: Comparison of presssure and magnetic ﬁeld lines for a paramagnetic ﬂuid with χvol = 0.1.

solution of the ﬂuid-magnetic problem and the adoption of a highly

efﬁcient, monolithic, and implicit numerical approach. These key

characteristics enable the analysis of highly susceptible magnetic

liquids, such as ferroﬂuids.

The magnetic solution from the steady-state Maxwell equa-

tions has been veriﬁed with equivalent simulations in Comsol Mul-

tiphysics. The magnetic momentum balance has also been veriﬁed

by comparing the steady-state magnetic ﬁeld and pressure contours

for a linearly magnetized liquid. The comparison of the meniscus

proﬁle with a previous quasi-analytic magnetic sloshing model has

shown a virtually perfect agreement in the non-magnetic case. How-

ever, stronger magnetic ﬁelds (i.e. optimum conditioning of the

meshless region D) are required to perform an accurate veriﬁcation

for the magnetic case.

Although the ESA Drop Your Thesis! 2017 conﬁguration has

been implemented to ease the veriﬁcation of results, the numerical

model here introduced ﬁnds application in many other scenarios of

scientiﬁc and technical interest. Some of them involve the study

of axisymmetric tank geometries, viscous liquids, magnetically-

induced viscosity, free surface stability, or time-dependent simula-

tions. Although signiﬁcant efforts need to be carried out to extend

this work to the 3D case (lateral sloshing) and improve the robust-

ness of numerical computations, this already represents a promising

step towards the development of future applications.

Fig. 7: Comparison of meniscus interfaces with a previous quasi-

analytical model [33] for the non-magnetic and magnetic

(I= 11 A) cases.

6 Acknowledgments

The project leading to these results has received funding

from la Caixa Foundation (ID 100010434), under agreement

LCF/BQ/AA18/11680099. M.A.H. acknowledges the support of

the the Ministerio de Econom´

ıa y Competitividad through Grant

PID2019-108278RB-C31.

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