Advanced numerical simulation of magnetic liquid sloshing in microgravity

Abstract

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.

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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+µ0Mr
∂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+µ0Mr
∂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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