Non-linear stress response of non-gap-spanning magnetic chains suspended in a
Newtonian fluid under oscillatory shear test: A direct numerical simulation
M. R. Hashemi, M. T. Manzari, and R. Fatehi
Citation: Physics of Fluids 29, 107106 (2017);
View online: https://doi.org/10.1063/1.5009360
View Table of Contents: http://aip.scitation.org/toc/phf/29/10
Published by the American Institute of Physics
PHYSICS OF FLUIDS 29, 107106 (2017)
Non-linear stress response of non-gap-spanning magnetic chains
suspended in a Newtonian fluid under oscillatory shear test:
A direct numerical simulation
M. R. Hashemi,
1
M. T. Manzari,
1,2,a)
and R. Fatehi
3,4
1
Center of Excellence in Energy Conversion, School of Mechanical Engineering, Sharif University of Technology,
Tehran, Iran
2
School of Geosciences, University of Aberdeen, Aberdeen, United Kingdom
3
Department of Mechanical Engineering, Persian Gulf University, Bushehr 75168, Iran
4
Oil and Gas Research Center, Persian Gulf University, Bushehr 75169, Iran
(Received 25 May 2017; accepted 15 October 2017; published online 31 October 2017)
A direct numerical simulation approach is used to investigate the effective non-linear viscoelastic stress
response of non-gap-spanning magnetic chains suspended in a Newtonian fluid. The suspension is
confined in a channel and the suspended clusters are formed under the influence of a constant external
magnetic field. Large amplitude oscillatory shear (LAOS) tests are conducted to study the non-linear
rheology of the system. The effect of inertia on the intensity of non-linearities is discussed for both
magnetic and non-magnetic cases. By conducting magnetic sweep tests, the intensity and quality of the
non-linear stress response are studied as a function of the strength of the external magnetic field. The
Chebyshev expansion of the stress response is used to quantify the non-linear intra-cycle behaviour
of the suspension. It is demonstrated that the system shows a strain-softening behaviour while the
variation of the dynamic viscosity is highly sensitive to the external magnetic field. In a series of strain
sweep tests, the overall non-linear viscoelastic behaviour of the system is also investigated for both a
constant frequency and a constant strain-rate amplitude. It is shown that the intra-cycle behaviour of
the system is different from its inter-cycle behaviour under LAOS tests. Published by AIP Publishing.
https://doi.org/10.1063/1.5009360
I. INTRODUCTION
The bulk rheology of electrorheological (ER) and magne-
torheological (MR) fluids can be readily adjusted by applying
an external electric and magnetic field, respectively.
1,2
This
makes these fluids suitable choices for active control mech-
anisms, e.g., dampers and actuators.
3–5
Under the influence
of an external field, a micro-structure is formed by particle
aggregates aligned with the direction of the field. This micro-
structure can lead to either a significant viscosity enhance-
ment or a solid-like behaviour depending on the strength of
the induced bonds and the concentration of the solid parti-
cles.
6
Generally, magnetic bonds are stronger in a conventional
magnetorheological (MR) fluid than electric bonds in an elec-
trorheological (ER) fluid;
7
therefore, MR fluids have become
more attractive in recent years.
Under a steady shear test, as long as the static yield
stress
8
of an MR fluid is not exceeded, there will not be any
flow. Above this static yield stress threshold, the static fric-
tional force exerted by ending particles in the micro-structure
is overcome
9
and an infinite strain is possible.
10
By further
increasing the shear strain, a strain-softening behaviour is
observed due to breakdown of the magnetic clusters. When a
field-induced (chain-like) structure is strained to a rather large
extent, it becomes unstable and eventually breaks apart.
11
At
this point, the MR fluid flows with a finite strain-rate and the
a)
E-mail: mtmanzari@sharif.edu
associated stress is the so-called dynamic (or Bingham
10
) yield
stress.
12
Both the static and dynamic yield stresses are func-
tions of intensity of the magnetic field, particle concentration,
and particle size distribution.
6,13,14
In the post-yield state, the
behaviour of MR fluids is generally shear-thinning.
15,16
Under an oscillatory shear test, MR fluids exhibit a vis-
coelastic behaviour with moduli that primarily depend on
their micro-structure.
17
The linear viscoelastic behaviour of
MR fluids has been thoroughly investigated especially for
the pre-yield state.
18,19
Nevertheless, MR fluids exhibit a lin-
ear behaviour only in a very narrow range of strain ampli-
tude.
8,20
Large Amplitude Oscillatory Shear (LAOS) tests can
be utilized to investigate the nonlinear rheological behaviour
of MR fluids. The LAOS test reveals that MR fluids can
be classified as type III (complex fluids) which exhibits a
strain-softening/shear-thinning behaviour with a slight over-
shoot in the loss modulus.
8,21
The nonlinear behaviour of field
responsive (ER and MR) fluids is commonly attributed to the
breakdown
20,22
and rearrangement
23
of the particle clusters.
As discussed in Ref. 24, an MR fluid behaves as an
elasto-visco-plastic material
25
whose micro-structure has the
principal role in determining its bulk rheology. This role can
be explored using a particle-level numerical simulation.
26,27
In
the literature, numerical simulation has been widely employed
to investigate various aspects of the MR fluids, e.g., time
scales associated with magnetic chain formation,
28
parti-
cle aggregation in a poly-disperse magnetic suspension,
12
micro-structural evolution in a Poiseuille flow,
29
and magnetic
1070-6631/2017/29(10)/107106/18/$30.00 29, 107106-1 Published by AIP Publishing.
107106-2 Hashemi, Manzari, and Fatehi Phys. Fluids 29, 107106 (2017)
clusters exposed to an oscillatory shear test.
19
Theoretical
models
7,30,31
are also useful for evaluating the storage modu-
lus in the linear region
17,19
and estimating the dynamic yield
stress.
16
In the majority of previous particle-level simulations and
theoretical models addressing the effective rheology of a field-
responsive fluid, the field-induced chains were considered to
be gap-spanning with ending particles stuck to the channel
walls.
19,32
For a rather large strain amplitude, these gap-
spanning clusters undergo progressive rearrangements
33
and
eventually break up into smaller non-gap-spanning chains by
further increase in the strain amplitude. Since a magnetic chain
would most probably break from its tip,
34
the blockage ratio
associated with the shortened clusters is still large enough to
significantly affect rheology of the system. However, the indi-
vidual contribution of these clusters to the bulk rheology has
been rarely addressed in the literature. A successful model-
ing of these broken non-gap-spanning magnetic chains needs
a two-way coupling between the suspending fluid flow and the
suspended solid particles, which necessitates utilization of a
direct numerical simulation (DNS) approach.
35
Recently, using the DNS approach, it has been shown that
non-gap-spanning chains can also contribute to the storage of
energy
36
as well as enhance the effective viscosity.
37,38
In that
work,
36
a confined periodic array of non-gap-spanning mag-
netic chains was suspended in a Newtonian fluid exposed to a
small amplitude oscillatory shear (SAOS). It was shown that
the system behaves as a viscoelastic fluid. It was also discussed
how inertia could hinder elasticity, an effect which can be con-
trolled by adjusting the intensity of the external magnetic field.
The main goal of the present work is to qualitatively investi-
gate the non-linear stress response of the non-gap-spanning
magnetic clusters. To this end, an array of suspended mag-
netic clusters similar to the systems presented in the previous
studies
36,37
is simulated under LAOS and the effective stress
response is studied following the methodology introduced in
the literature.
39–41
In the following, first the physical model
and the governing equations are briefly described. Then, the
results of the LAOS tests are presented and the non-linearities
in the intra-cycle and overall rheology of the system are dis-
cussed. The methodologies used to characterize the results of
the LAOS tests are also briefly surveyed during discussions.
II. PHYSICAL MODELING
A direct numerical simulation (DNS) approach is used
to investigate the behaviour of a suspension of paramagnetic
solid particles forming non-gap-spanning chain-like clusters.
Here, the physical system consists of a Newtonian fluid and
a number of (para-)magnetic solid particles confined between
two parallel walls. The system is subjected to a large ampli-
tude oscillatory shear (LAOS) test as schematically shown in
Fig. 1(a). In order to avoid a prohibitive computational cost,
the study is performed on a two-dimensional periodic domain
as shown in Fig. 1(b). The computational domain contains N
neutrally buoyant circular cylinders initially arranged in a ver-
tical row with the middle one being placed at the center. These
solid particles are magnetized under the influence of an exter-
nal magnetic field with a flux density of B
0
. For the current
setup, in order to study the shear rheology of the system, the
spatially averaged stress response is measured as
¯σ
xy
=
1
L
y=0
σ
xy
(x)dx, (1)
where σ
xy
is the local value of the shear stress. In the following,
the over-bar sign is omitted for brevity.
In the present work, the smoothed particle hydrodynamics
(SPH) method
38
is used to solve the governing equations for
both the fluid flow and the magnetostatics. For fluid flows, in
the Lagrangian framework of the weakly compressible SPH
method,
42
the governing equations are the conservation of
momentum
ρ
dv
dt
= −∇p + η
0
∇
2
v (2)
and the continuity
d ρ
dt
= −ρ∇ · v, (3)
where a simple equation of state, p − p
0
= c
2
0
(ρ − ρ
0
), relates
density and pressure. Here, the velocity vector v is subject
to the no-slip boundary condition at a solid surface. In these
equations, ρ is the density, p is the pressure, and η
0
denotes
the dynamic viscosity of the suspending fluid. Also, c
0
is the
artificial speed of sound and subscript 0 denotes the initial state
in the fluid domain.
For a two-dimensional magnetic field, in the absence of
a free current, the Maxwell equations
43
can be combined into
the Poisson equation for the magnetic potential, φ, as
∇ · (µ∇φ) = 0. (4)
In this way, the magnetic field intensity is calculated as
H = ∇φ. Far below the magnetic saturation limit,
44
constant
magnetic permeabilities are considered for the solid bodies
(µ
s
) and the fluid domain (µ
0
). The magnetic flux density is
calculated as
B = µH. (5)
FIG. 1. Schematic of (a) the suspension
of magnetic solid particles shearing in
a channel with oscillating solid walls
and (b) the initial configuration of the
particles in the computational domain.
The computational domain is marked by
dashed lines in (a).
107106-3 Hashemi, Manzari, and Fatehi Phys. Fluids 29, 107106 (2017)
The magnetic field is subject to the conservation of B at the
fluid-solid interface. It must be noted that the external mag-
netic field is imposed by setting φ at the solid walls so that
B
0
= µ
0
∇φ as explained in the literature.
35,38,45
Solid bodies are moved using Newton’s law of motion as
M
s
dv
s
dt
= F
m
s
+ F
h
s
+ F
r
s
(6)
and
I
s
dΩ
s
dt
= M
m
s
+ M
h
s
, (7)
where M
s
and I
s
are the total mass and moment of inertia
of solid body s, respectively. Also, v
s
and Ω
s
are the lin-
ear and angular velocities of s, respectively. The terms on
the right-hand side of Eq. (6) are the magnetic force, F
m
,
the hydrodynamic force, F
h
, and the repulsive force due to
solid-solid collisions, F
r
. In a similar way, the terms on the
right-hand side of Eq. (7) correspond to the magnetic and
hydrodynamic effects. The full description of the numerical
method and boundary conditions was presented in an earlier
article.
38
A. Simulation details
In all test cases solved in this paper, circular cylinders
are of the same radius (a); the number of solid particles ini-
tially arranged in a chain, N = 9, the non-dimensionalized
channel height, H/a = 20, and the non-dimensionalized peri-
odicity length, L/a = 8, are kept constant. This gives a solid
volume fraction of N πa
2
/LH ≈ 0.177. In order to facilitate
the numerical simulation, solid bodies are initially arranged
with a vertical spacing equal to the discretization length,
δ
p
. Here, using a rather small ratio for the magnetic perme-
abilities (µ
s
/µ
0
= 1.1), converged solutions are obtained for
a/δ
p
= 18.75. As discussed previously, when exposed to an
external magnetic field, this system exhibits a viscoelastic
behaviour. The shear rheology of such a system can be investi-
gated using an oscillatory shear test. This study aims to extend
the results obtained in the previous work
36
to LAOS.
In this work, inertia is quantified at the particle scale by
defining the particle Reynolds number as Re
p
= ρ ˙γ
0
a
2
/η
0
.
Also, as discussed in the literature,
15,46
for a steady shear
flow, the viscous force can be non-dimensionalized against
the magnetic force using the Mason number defined as
Mn =
˙γ
0
η
0
µ
0
β
2
H
2
0
, (8)
where ˙γ
0
= 2U
0
/H, β = χ/(3 + χ) is the effective polarization,
and χ = (µ
s
µ
0
)/µ
0
is the magnetic susceptivity. On the other
hand, for an oscillatory shear test, since the time scale can be
properly determined by the frequency of oscillations (ω), a
modified non-dimensional group is defined and used in the
present work,
Mn
∗
= Mn
ω
˙γ
0
. (9)
It must be noted that in Sec. III B, since H
0
is the variable
and ω is constant, (Mn
∗
)
1
is used as a measure of the external
magnetic field. Also, in Sec. III C, where ω is the variable and
˙γ
0
is constant, Mn as defined in Eq. (8) is used as a measure
of the external magnetic field.
B. LAOS theory
The stress response of a viscoelastic material to an oscil-
latory shear strain, γ(t) = γ
0
sin(ωt), is harmonic with the
same frequency, ω, only for a rather small strain amplitude. In
a more general representation which is also valid for a LAOS
test, the stress response can be described using the Fourier
series as
47,48
σ
xy
= γ
0
X
n:odd
|G
∗
n
|(ω, γ
0
) sin(nωt + Ψ
n
). (10)
Here, G
∗
n
and Ψ
n
are the complex modulus and phase angle
corresponding to the nth harmonic, respectively. For a SAOS
test, only the first harmonic is important, while for a LAOS test,
higher harmonics are also significant. In this work, using the
subroutines provided in the MITlaos
40,49
program, the stress
response is calculated using only the first, third, and fifth har-
monics. Normally, amplitudes of the higher harmonics are
either negligible or an order of magnitude smaller than the
third harmonic. It should be noted that for an odd-symmetric
stress response, even harmonics are all negligible.
50
The stress response can be decomposed using its sym-
metry properties,
40
and considering the fact that elasticity and
viscosity are related to the storage and loss of energy, respec-
tively, the elastic stress, σ
0
, and the viscous stress, σ
00
, are
obtained as
50
σ
0
xy
=
σ
xy
(γ, ˙γ) − σ
xy
(−γ, ˙γ)
2
(11)
and
σ
00
xy
=
σ
xy
(γ, ˙γ) − σ
xy
(γ, − ˙γ)
2
. (12)
In this way, dσ
0
/dγ and dσ
00
/d ˙γ are measures of the
local (tangent) elastic modulus and dynamic viscosity, res-
pectively.
In order to quantify the non-linear properties of the (intra-
cycle) rheology of the system, it is more appropriate to express
the elastic (and viscous) stress as a polynomial series
40,50
rather than using the above mentioned Fourier Transform (FT)
rheology.
39
To this end, a framework has been introduced by
Ewoldt et al.
48
that facilitates the physical interpretation of the
non-linear rheology of a material under LAOS.
51
The idea is
to expand σ
0
xy
and σ
00
xy
in series of the Chebyshev polynomials
of the first kind, T
n
, as
48
σ
0
xy
(γ) = γ
0
X
n:odd
e
n
(ω, γ
0
)T
n
(
γ
γ
0
) (13)
and
σ
00
xy
( ˙γ) = ˙γ
0
X
n:odd
v
n
(ω, γ
0
)T
n
(
˙γ
˙γ
0
), (14)
where e
n
= G
0
n
(−1)
(n−1)/2
and v
n
= η
0
n
. Considering only the
first and third harmonics in the stress response, the resulting
polynomials are
σ
0
xy
(γ) ≈
(
e
1
− 3e
3
)
γ + 4e
3
γ
3
γ
2
0
(15)
and
σ
00
xy
( ˙γ) ≈
(
v
1
− 3v
3
)
˙γ + 4v
3
˙γ
3
˙γ
2
0
. (16)
107106-4 Hashemi, Manzari, and Fatehi Phys. Fluids 29, 107106 (2017)
According to Eqs. (15) and (16), e
3
and 3
3
determine the vari-
ation of the tangent elastic modulus (dσ
0
/dγ) and the tangent
dynamic viscosity (dσ
00
/d ˙γ) in a strain-cycle; a positive e
3
leads to an intra-cycle strain-stiffening behaviour and an intra-
cycle shear-thickening behaviour is associated with a posi-
tive 3
3
. Negative e
3
and 3
3
also correspond to the intra-cycle
strain-softening and shear-thinning behaviours, respectively.
III. RESULTS
In the following, first, the effect of inertia on the non-linear
rheology of the system is investigated for both non-magnetic
and paramagnetic solid particles. In Secs. III B and III C,
the results of magnetic sweep tests (changing the intensity of
the external magnetic field while keeping all other parameters
constant) and strain sweep tests are presented, respectively.
A. Non-linearity and the effects of inertia
It is easy to show that for a linear stress response, the
Lissajous-Bowditch curve,
52
a plot of stress versus strain
(-rate), is of an elliptical shape. By increasing the strain ampli-
tude in a LAOS test, higher harmonics become more significant
and non-elliptical Lissajous-Bowditch curves are obtained.
These curves are helpful for a qualitative interpretation of the
results by investigating the variation of the stress response in a
complete strain(-rate) cycle.
50,52
In Fig. 2, Lissajous-Bowditch
curves are shown for γ
0
= 1.2/π with ω = 2π rad/s and different
particle Reynolds numbers. Here, 0.0015 ≤ Re
p
≤ 0.0188 is
changed by altering density while all other parameters are kept
constant. Results are presented for both the non-magnetic case
and the magnetic case with Mn
∗
= 0.419.
Previously,
36
it was discussed that for a purely viscous
system (or a viscoelastic system with weak elasticity) with
finite inertia, an obtuse phase angle, i.e., Ψ
1
> π/2, is obtained.
The larger the Reynolds number, the larger the phase angle
is. The effect of inertia on the orientation of the Lissajous-
Bowditch curve is schematically shown in Fig. 3(a). For the
present test cases in the absence of an external magnetic field,
the stress response is almost purely viscous and a similar
behaviour is observed in Fig. 2(a). Also for a viscoelastic
system, by increasing inertia, the phase angle increases and
consequently the effective elasticity (G
0
) is decreased
36
and
the Lissajous-Bowditch curve rotates in the clockwise direc-
tion as schematically shown in Fig. 3(b). For the present test
cases with Mn
∗
= 0.419, the stress response is viscoelastic and
a similar behaviour is observed in Fig. 2(c). It is also worth
FIG. 2. Lissajous-Bowditch curves obtained for ˙γ
0
= 2.4 s
−1
, ω = 2π rad/s, and different Reynolds numbers, with B
0
= 0 for (a) and (b) and Mn
∗
= 0.419
for (c) and (d).
