Computers and Fluids 126 (2016) 170–180
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Computers and Fluids
journal homepage: www.elsevier.com/locate/compfluid
Numerical modeling of non-Newtonian biomagnetic fluid flow
K. Tzirakis a, L. Botti b, V. Vavourakis c, Y. Papaharilaou a,∗
a Institute of Applied and Computational Mathematics (IACM), Foundation for Research and Technology-Hellas (FORTH), Heraklion Crete, Greeceb Universtità degli Studi di Bergamo, Dipartimento di ingegneria e scienze applicate, Dalmine (BG) 24044, Italyc Centre for Medical Image Computing, University College London, London, WC1E 6BT, United Kingdom
a r t i c l e i n f o
Article history:
Received 16 December 2014
Revised 24 July 2015
Accepted 28 November 2015
Available online 4 December 2015
Keywords:
Biofluid
Magnetization force
Continuous/discontinuous Galerkin
Symmetric Weighted Interior Penalty (SWIP)
Herschel–Bulkley fluid
a b s t r a c t
Blood flow dynamics have an integral role in the formation and evolution of cardiovascular diseases. Simu-
lation of blood flow has been widely used in recent decades for better understanding the symptomatic spec-
trum of various diseases, in order to improve already existing or develop new therapeutic techniques. The
mathematical model describing blood rheology is an important component of computational hemodynam-
ics. Blood as a multiphase system can yield significant non-Newtonian effects thus the Newtonian assump-
tion, usually adopted in the literature, is not always valid. To this end, we extend and validate the pressure
correction scheme with discontinuous velocity and continuous pressure, recently introduced by Botti and Di
Pietro for Newtonian fluids, to non-Newtonian incompressible flows. This numerical scheme has been shown
to be both accurate and efficient and is thus well suited for blood flow simulations in various computational
domains. In order to account for varying viscosity, the symmetric weighted interior penalty (SWIP) formu-
lation is employed for the discretization of the viscous stress tensor. We disregard the dependency of the
viscosity on spatial derivatives of the velocity in the Jacobian computation. Even though this strategy yields
an approximated Jacobian, the convergence rate of the Newton iteration is not significantly affected, thus
computational efficiency is preserved. Numerical accuracy is assessed through analytical test cases, and the
method is applied to demonstrate the effects of magnetic fields on biomagnetic fluid flow. Magnetoviscous
effects are taken into account through the generated additive viscosity of the fluid and are found to be im-
portant. The steady and transient flow behavior of blood modeled as a Herschel-Bulkley fluid in the presence
of an external magnetic field, is compared to its Newtonian counterpart in a straight rigid tube with a 60%
axisymmetric stenosis. A break in flow symmetry and marked alterations in WSS distribution are noted.
© 2015 Elsevier Ltd. All rights reserved.
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1. Introduction
In recent years significant research work has been directed to-
wards studying the effects of magnetic fields on biomagnetic fluid
flow, with ample applications in bioengineering and the medical sci-
ences [1,2]. The most common biofluid is blood which behaves as a
magnetic fluid because of the hemoglobin molecule that is present in
red blood cells. To this end, Haik et al. [3] developed a Biomagnetic
Fluid Dynamics (BFD) model by considering the Langevin equation
for the magnetization of classical fluids. Haik’s model does not take
into account the electric properties of biofluids. As a result, magnetic
effects appear solely in terms of the field gradients generating a corre-
sponding magnetization force. Experiments on cow and sheep blood
though have shown appreciable dielectric properties for blood [4],
∗ Corresponding author. Tel.: +30 2810 391783; fax: +30 2810 391728.
E-mail address: [email protected] (Y. Papaharilaou).
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http://dx.doi.org/10.1016/j.compfluid.2015.11.016
0045-7930/© 2015 Elsevier Ltd. All rights reserved.
hich produces a Lorentz force even in the case of constant magnetic
elds [5]. The idea of considering the magnetic and electric proper-
ies of blood under a unified mathematical model was introduced by
zirtzilakis [6]. To this end, both forces were included in the Navier–
tokes equations, where blood was assumed to behave as a Newto-
ian fluid [7–10].
The assumption of Newtonian behavior for blood has been widely
sed in the literature. Even though it may be valid for flows that are
haracterized by shear rates higher than 100 s−1 where deviations
rom the Newtonian behavior may be small [11], the Newtonian hy-
othesis becomes problematic for lower shear rates. In addition, dur-
ng the end-deceleration phase of pulsatile cycles, shear rates can de-
line to values lower than the 100 s−1 limit, generating potentially
mportant non-Newtonian effects. This deviation becomes even more
rofound in small arteries and veins or in numerous diseased condi-
ions [12,13].
The transition from the Newtonian to a non-Newtonian con-
ideration for biofluids such as blood is accompanied by a choice
K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180 171
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Table 1
Flow models for various shear rate dependent viscosities.
μ = κ Newtonian
μ(γ̇ ) = κγ̇ n−1 Power-law
μ(γ̇ ) = (τ0/γ̇ )[1 − exp(−mγ̇ )] + κγ̇ n−1 Herschel–Bulkley
2
2
l
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ρ
∇
u
σ
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σ
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or the mathematical model describing it. There are many mod-
ls for the characterization of non-Newtonian blood behavior. Cas-
on [14,15] and Herschel–Bulkley [16] are the models amongst
thers (Power-law, Carreau, Bingham) that appear most often in
iterature. The Herschel–Bulkley compared to the Casson fluid model
hough has two distinctive advantages. For arterioles with di-
meters less than 0.065 mm, the Casson model does not cap-
ure velocity profiles accurately [17]. In addition, the constitu-
ive equation of the Herschel–Bulkley model has two degrees of
reedom (instead of one for the Casson model) yielding a bet-
er description of blood flow under a wider range of realistic
onditions.
The combined goal of accurately describing blood behavior and
f altering its flow using externally applied magnetic fields can be
pplied to targeted drug deposition, yielding higher drug concentra-
ion at specific sites and reducing total required dosage. This can be
chieved by injecting magnetic nanoparticles into blood flow which
nteract with magnetic fields around a desired target location [18]. In
ddition, external magnetic fields will alter viscosity due to generated
agnetoviscous effects. Specifically, it has been shown by Haik et al.
19] that a static magnetic field of 4T can increase the apparent vis-
osity of human blood by approximately 11.5%. It is thus important to
ccount for magnetoviscous effects on blood flow exposed to external
agnetic fields.
The available literature on non-Newtonian flows is expanding.
tarting with Shukla et al. [20] many researchers have studied
on-Newtonian flows in arterial stenosis [12,13,21,22]. More re-
ently, Kröner et al. [23] presented a fully implicit Local Discontin-
ous Galerkin (LDG) discretization for non-Newtonian incompress-
ble flows. Due to the computational expense of the scheme though
hey considered 2D computations only. Recently, Kwack and Masud
24] presented a stabilized mixed FEM to non-Newtonian shear-rate
ependent flows, where viscosity is considered a nonlinear func-
ion of shear-rate. Along these lines, they developed a stabilized
umerical scheme using the Variational Multiscale framework to
he underlying generalized Navier–Stokes equations. Here we em-
loy the pressure correction formulation proposed by Botti and Di
ietro [25], which has demonstrated to be effective for high-Reynolds
emodynamic simulations in real patient geometries [26]. Specifi-
ally, pressure gradients in hemodialysis patients were simulated and
ompared with experiments for various steady conditions and wide
ange of Reynolds numbers (100–2000). The results closely followed
he experimental data, yielding an accurate solver for convection-
ominated incompressible flows.
In this study, we consider the Symmetric Weighted Interior
enalty (SWIP) formulation for the stress tensor, ignoring the
ependence of viscosity on the velocity solution in the Jacobian
omputation. The resulting approximated Jacobian does not alter
he convergence rate of the Newton iteration significantly, retaining
he computational efficiency unchanged. The novelty of the pro-
osed scheme is a fast and accurate algorithm for realistic blood
ow simulations where in some cases the computational domain
onsists of hundreds of thousands if not millions of elements. To the
uthors knowledge, this is the first numerical simulation study of
on-Newtonian biomagnetic fluid flow using the SWIP formalism,
nd is a generalization of a previous work on Newtonian biofluid
ow exposed to external magnetic fields [27], to biofuids that are
haracterized by non-constant viscosity.
The paper is organized as follows: Section 2 analyzes the mathe-
atical framework for the laminar flow of a non-Newtonian fluid in
he presence of external magnetic fields. Section 3 describes the nu-
erical implementation used for the solution of the Navier–Stokes
quations. Section 4 examines the numerical validation of the pro-
osed method and Section 5 contains the results and comparison
ith exact solutions when these are available. Finally, Section 6
resents the summary and conclusions.
. General setting
.1. Flow model
Let � ⊂ Rd, d = 2, 3, denote a bounded, connected open set, and
et tF > 0 denote the final simulation time. We consider the un-
teady incompressible Navier-Stokes equations with Dirichlet and
raction-free outflow boundary conditions to be imposed on the do-
ain boundaries ∂�D and ∂�N respectively,
∂u
∂t+ ρu · ∇u − ∇ · σ = ρf in � × (0, tF ), (1a)
· u = 0 in � × (0, tF ), (1b)
= gD on ∂�D × (0, tF ), (1c)
· n = h on ∂�N × (0, tF ), (1d)
(·, t = 0) = u0 in �, (1e)
here u0 is the initial condition, gD is the Dirichlet velocity bound-
ry condition, h is the prescribed boundary traction vector, ρ and f
re the density, and body force respectively. The homogeneous ex-
ression of (1d) corresponds to the traction-free boundary condition
hich is widely imposed as artificial outflow boundary. Additionally,
is the stress tensor given by the following expression,
(u, p) = −pI + τ(u). (2)
he shear-stress tensor, τ(u), is written in terms of the deformation
ate tensor, D(u) ≡ 12
[∇u + (∇u)T], and its three invariants as,
(u) = 2μ(D)D(u) = 2μ(ID(u), IID(u), IIID(u))D(u). (3)
ince the flow is incompressible (ID(u) = 0) and assuming the third
nvariant is negligible for shear flows we find that,
(u) = 2μ(IID(u))D(u). (4)
efining finally shear rate γ̇ ≡ 2√
IID(u), the magnitude of the shear-
tress tensor takes the form,
= μ(γ̇ ) γ̇ , (5)
here in Cartesian coordinates,
˙ 2 = 2
(∂ux
∂x
)2
+ 2
(∂uy
∂y
)2
+ 2
(∂uz
∂z
)2
+(
∂ux
∂y+ ∂uy
∂x
)2
+(
∂ux
∂z+ ∂uw
∂x
)2
+(
∂uy
∂z+ ∂uw
∂y
)2
. (6)
he different expressions of the shear rate dependent viscosity mod-
ls that are considered in this paper are presented in Table 1. The
implest non-Newtonian constitutive equation yields the so-called
ower-law model which is characterized by two parameters. The con-
istency index, κ , and power index, n. The Newtonian case is then
imply obtained by setting μ = κ and n = 1. For blood flow simula-
ions, the Herschel–Bulkley model is considered. Due to discontinuity
hough at yield stress, τ 0, the generalization proposed by Papanas-
asiou [28] as expressed by the exponential term and regularization
arameter, m, is also included. As such, the above models are chosen
n the basis of being effective in providing good fits to blood viscosity
172 K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180
Table 2
Magnetization force components for the magnetic field given by Eq. (10).
Coordinate (m) Magnetization force component (N/kg)
x fMx = − 4αC2
ρ(x−xi )
3
[(x−xi )2+(y−yi )
2+(z−zi )2]
6
y fMy = − 2αC2
ρ (y − yi)3(x−xi )
2+(y−yi )2+(z−zi )
2
[(x−xi )2+(y−yi )
2+(z−zi )2]
6
z fMz = − 2αC2
ρ (z − zi)3(x−xi )
2+(y−yi )2+(z−zi )
2
[(x−xi )2+(y−yi )
2+(z−zi )2]
6
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measurements (for different shear-rate regimes) as well as capturing
shear thinning effects.
The explicit form of the external body forces, f, in Eq. (1a) depends
on the coupling of the electromagnetic and flow fields. In the most
general case, ionized flows as described by the Navier–Stokes equa-
tions will be affected due to presence of electric and magnetic fields,
yielding a coupled system of equations [29]. Assuming though a time
independent electric field, and applying Ohm’s law,
J = σ (E + v × B), (7)
in order to drop the electric field altogether, it is possible to obtain
the induction equation for magnetic fields which is coupled to the
Navier–Stokes as in the MHD approximation [30,31]. As a result, a
constant in time magnetic field interacts with an electric and mag-
netic biofluid generating Lorentz and magnetization forces given by,
fL = J × B, (8a)
fM = μ0(M · ∇)H, (8b)
where H, B, and M are the field intensity, flux density, and magnetiza-
tion respectively. In this work, the current density is given by Eq. (7),
and the magnitude of M by the following linear relation valid for
isothermal flows,
M = χH, (9)
where χ is the magnetic susceptibility (see Tzirakis et al. [27] and
references therein for a detailed description of the magnetization and
its dependence on flow parameters).
2.2. Geometric model, magnetic field, and magnetoviscosity
Three geometrical models are considered for the needs of this
study. The flow of a power-law fluid between two infinitely long par-
allel plates and through a pipe are simulated for different values of
exponent n. A comparison of results with corresponding analytical
solutions is performed, in order to establish validity of the method.
Lastly, flow of Newtonian and Herschel–Bulkley fluids through a
straight rigid tube with a 60% axisymmetric stenosis, in the presence
of an external magnetic field are also considered. The applied mag-
netic field resembles that generated by an ideal dipole, and can be
used in real patient treatment. Its components along x, y, and z are,
Bx = −C
(2(x − xi)
2 − r2
r6
), (10a)
By = −C
(2(x − xi)(y − yi)
r6
), (10b)
Bz = −C
(2(x − xi)(z − zi)
r6
), (10c)
respectively, where r =√
(x − xi)2 + (y − yi)
2 + (z − zi)2. The mag-
nitude then takes the form,
|B(x, y)| = C
r4, (11)
expressed in terms of parameter C, and distance, r, from some arbi-
trary point (xi, yi, zi). The generated magnetization force can then be
found using,
H = 1
μB = 1
μ0(1 + χ)B, (12a)
M = χH = 1
μ0
(χ
1 + χ
)B, (12b)
ielding the components presented in Table 2, in terms of the param-
ter α given by,
= 1
μ0
χ
(1 + χ)2. (13)
In addition to the magnetization force, the presence of the exter-
al magnetic field can affect the flow by altering the viscosity. This
appens because red blood cells orient at a specific angle with re-
pect to the lines of the magnetic field creating an additive viscosity
19]. The generated magnetoviscous effect is proportional to the mag-
itude of the field, and disappears when the field is switched off. For
eld magnitudes up to 4 T and following [19], we parametrize the ra-
io of the viscosity when the field is on, μ∗, to the viscosity when the
eld is off, μ, as follows,
μ∗
μ= α + βH + (1 − α)eH, (14)
here α = 0.9986 and β = 0.01425. It is clear therefore that the ex-
ernal magnetic field affects viscosity quite substantially yielding an
dditive effect up to 11.5%. In the present analysis, the effect is taken
nto account by readjusting the fluid viscosity in terms of the mag-
etic field magnitude throughout the computational domain.
. Numerical method
We consider the extension of the incompressible Navier–Stokes
olver strategy devised by Botti and Di Pietro [25] to non-Newtonian
uids. The finite element formulation is implemented in the open-
ource hemodynamics solver Gnuid and is based on the pressure-
orrection scheme proposed by Guermond and Quartapelle [32]. It
ombines a discontinuous Galerkin (dG) approximation for the ve-
ocity and a continuous Galerkin (cG) approximation for the pressure.
his space couple is LBB stable for equal order velocity-pressure for-
ulations allowing to exploit first polynomial degree discretization
or both velocity and pressure. The Temam device [33] is adopted in
rder to construct a skew-symmetric version of the convective term
nd ensure kinetic energy conservation.
For time discretization we partition the domain, (0, tf), into equal
ntervals, �t, yielding at step n, tn ≡ n�t. We define the velocity-
ressure pair (un+1, pn+1) iteratively by solving the following prob-
ems,
ρβ0
�tun+1 − 2μ∇·D + ρun+1 · ∇un+1 + 1
2ρ(∇·un+1)un+1
= −ρβ1
�tun − ρβ2
�tun−1 − ∇p∗ + ρf in �,
(15a)
n+1 = gD on ∂�D, (15b)
μD · n − p∗n = 0 on ∂�N, (15c)
nd,
∂2(pn+1 − pn) = −ρβ0 ∇·un+1 in �, (16a)
�tK. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180 173
∂
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(
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Pf
u
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P
i
i
r
n(pn+1 − pn) = 0 on ∂�D, (16b)
pn+1 = 0 on ∂�N. (16c)
A homogeneous boundary condition is enforced on the pres-
ure correction on ∂�N in order to take into account the out-
ow conditions, as described by Guermond, Minev and Shen
34]. Coefficients β0 = 3/2, β1 = −2, β2 = 1/2 are provided by BDF2,
hile p∗ is a pressure extrapolation computed according to the BDF
rder.
For space discretization we consider a division, T , of � into ele-
ents T. All interior elements will then share faces among each other.
f Ff is a common face of two different elements, say T+ and T−, we
an define for any function φ : � → R the quantities,
[φ]] ≡ φT+ − φT− , and {φ} ≡ 1
2(φT+ + φT− ), (17)
s the jump and average of φ respectively. By definition, if the face
f belongs to the boundary, [[φ]] = {φ} = φT . Moreover when φ is
ector-valued or tensor-valued the average and jump operators act
omponentwise on φ. The velocity and pressure then will stem from
he spaces Uh and Ph,
h ≡ [dG(k)]d, and Ph ≡ cG(k)/R,
where k ≥ 1 and,
G(k) ≡ {vh ∈ L2(�)|∀T ∈ T , vh ∈ Pkd(T )},
cG(k) ≡ {qh ∈ C0(�̄)|∀T ∈ T , qh ∈ Pkd(T )},
with Pkd(T ) defining all polynomials of order less or equal to k for d
ariables.
As opposed to the discretization scheme proposed in [25] we con-
ider the discretization of the full viscous stress tensor instead of the
implified Laplacian formulation, as required for heterogeneous dif-
usion problems. To this end, two real and non-negative numbers,
T+ and ωT− , are assigned for the common face Ff, such that,
T+ + ωT− = 1. (18)
he generalization then of the average for scalar-valued functions φweighted average) takes the form,
φ}ω ≡ ωT+φT+ + ωT−φT− . (19)
As before, in the special case where Ff belongs to the bound-
ry, {φ}ω = φT . For heterogeneous diffusion problems, the standard
rithmetic definition for the average (17) is insufficient, and a more
eneral expression must be adopted. Following Ern et al. [35] the fol-
owing diffusion dependent weights, ωT ±, are defined as,
T− ≡ μT+
μT+ + μT−, and ωT+ ≡ μT−
μT+ + μT−. (20)
It is clear that for the case of homogeneous diffusion, definition
20) reduces to the arithmetic average (17). The modification of the
Table 3
Power-law fluid flow between two parallel plates. L2
for the velocity and its gradients for a dG(1)-cG(1) sc
Mesh Velocity
L2 error (×10−3) Convergence ra
32 × 32 3.411
64 × 64 0.947 1.85
128 × 128 0.253 1.9
256 × 256 0.0656 1.95
512 × 512 0.0166 1.98
iffusive term, ah
(un+1
h, vh
), using weighted averages was first in-
roduced by Dryja [36] and is usually referred to as the Symmetric
eighted Interior Penalty (SWIP) scheme. It takes the form,
h,SWIP(uh, vh) ≡∫�
2μDh(uh) : Dh(vh)
+∑F∈Fh
ηk2γμ
hF
∫F
1
2
([[uh ⊗ nF ]] : [[vh ⊗ nF ]]
+ [[uh ⊗ nF ]] : [[vh ⊗ nF ]]T)
−∑F∈Fh
∫F
({2μDh(uh)}ω : [[vh ⊗ nF ]]
+ [[uh ⊗ nF ]] : {2μDh(vh)}ω), (21)
here the symbol : denotes the Frobenius inner product, η is a posi-
ive penalty parameter independent of the mesh size h and the poly-
omial degree k, and hF ≡ minT∈TF
|T |d|∂T |d−1with TF denoting the set of
lements belonging to boundary F. Finally, the diffusion dependent
enalty parameter, γ μ, is defined for all internal faces as [37],
μ ≡ 2μT+μT−
μT+ + μT−, (22)
hich corresponds to the harmonic mean of the two diffusion coeffi-
ients on both sides of the interface. The generalization of the familiar
ymmetric Interior Penalty (SIP) scheme [38], to the heterogeneous
iffusion case is thus achieved through definitions (20) and (22). Ob-
iously, whenever viscosity is constant in �, the SWIP bilinear form
21) reduces to the SIP formulation.
It is interesting to remark that if first degree velocity discretization
s employed, as in all numerical computations here performed, vis-
osity models yield a piecewise constant viscosity over each T ∈ T .
or higher than first order discretization the viscosity is piecewise
mooth. This is in agreement with the SWIP mesh compatibility con-
ition which requires the mesh to be compatible with singularities of
he diffusion coefficient [39].
. Numerical validation
The open source framework libMesh [40] is used for the solver
mplementation. The polynomial space Pkd
considered is monomi-
ls for the discontinuous spaces and Lagrange polynomials for the
ontinuous spaces. Parallelization is performed using MPICH, and the
ETSc toolkit [41] is chosen for data structures and routines needed
or the numerical solution of the linear systems involved. They both
se the MPI communication protocol. Finally, mesh partitioning is
xecuting using the METIS library [42], or its parallel counterpart
arMETIS.
It should be noted that the temporal accuracy of the scheme (21)
s not examined since it has been confirmed for the Newtonian case
n [25] using the Taylor vortex and Couzy decoupling error tempo-
al tests. It was shown that the convergence rates for velocity and
errors and corresponding convergence rates
heme.
Velocity gradients
te L2 error (×10−1) Convergence rate
2.735
1.342 1.03
0.663 1.02
0.33 1.01
0.164 1.00
174 K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180
Table 4
Power-law fluid flow between two parallel plates. L2 errors and corre-
sponding convergence rates for pressure for a dG(1)-cG(1) scheme.
Pressure
Mesh L2 error (×10−3) Convergence rate
32 × 32 3.852
64 × 64 1.032 1.9
128 × 128 0.267 1.95
256 × 256 0.068 1.97
512 × 512 0.0174 1.98
Table 5
Non-dimensional velocity for different values of n at x/L = 50 as given by
Eq. (23), and percentage RMS error with respect to umean .
n u/umean % RMS error w.r.t. umean
3/6 1.3 5.8 · 10−3
4/6 1.4 5.6 · 10−3
5/6 1.45 4.8 · 10−3
x/Lu
/um
ean,y
/L=
0
0 10 20 30 40 50
1.35
1.4
1.45
1.5
Numer.: n=3/6Theory: n=3/6Numer.: n=4/6Theory: n=4/6Numer.: n=5/6Theory: n=5/6
Fig. 1. Non-dimensional velocity for Couette flow of a power-law fluid for different
values of exponent n at y/L = 0. All simulations converge to corresponding analyti-
cal solutions yielding a percentage relative error no more than a few hundredths of a
percent.
y/H
u/u m
ean,x
/L=
50
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Numer.: n=3/6Theory: n=3/6Numer.: n=4/6Theory: n=4/6Numer.: n=5/6Theory: n=5/6
Fig. 2. Non-dimensional velocity profiles for Couette flow of a power-law fluid for dif-
ferent values of exponent n at x/L = 50. The numerical results deviate no more than a
few hundredths of a percent with respect to the analytical solutions.
5
5
i
a
a
k
1
fi
T
pressure in L2 and L∞ norms at the final simulation time step agree
with the results of E and Liu [43]. In order to numerically assess
the spatial convergence rates, the flow of a power-law fluid between
two parallel plates is considered in the two-dimensional domain
(0, 10) × (−1, 1). The discretization scheme is an equal order dG(1)-
cG(1). The simulation is performed at Re = 80, and weakly Dirichlet
boundary is enforced to the exact solution. The initial conditions on
the computational domain are set to zero velocity and pressure. In
order to obtain a steady solution, a time integration is performed as-
suming a fixed step �t = 0.1 s. L2 errors for both velocity and its gra-
dient, as well as the resulting convergence rates are shown in Table 3.
Similarly, L2 errors for pressure and the resulting convergence rates
are shown in Table 4.
Theoretical convergence rates of hk+1 for L2 error on velocity, and
of hk for L2 error on velocity gradients are confirmed for the dG(1)-
cG(1) discretization. Convergence rates for pressure are higher than
expected due to the linear nature of the pressure exact solution,
while first order convergence is expected in general for a dG(1)-cG(1)
discretization.
5. Results
All computational domains are discretized with linear hexahedral
elements. Hexahedral meshes are preferred to tetrahedral or pris-
matic since they require less number of elements for a fixed level
of accuracy. Specifically, De Santis et al. [44] showed that same ac-
curacy can be achieved with six times fewer hexahedral elements
compared to tetrahedral or prismatic meshes. In addition, simula-
tions converge much faster requiring 14 times less CPU hours. The
computational grid for parallel plates and pipe was constructed us-
ing Gmsh [45], a 3D finite element mesh generator with build-in CAD
tools and post-processor, with approximately 105 elements for the
former and 8 · 105 for the latter. The grid of the stenosis was con-
structed using ANSA v15 (Beta CAE Systems, Greece) using approxi-
mately 1.2 · 106 elements. In all cases, elements are clustered close to
the wall in order to successfully resolve high velocity gradients, and
the fully developed profile of Newtonian flow is prescribed at the in-
let of computational domains. The domains are also extended to the
appropriate length to allow for sufficient flow development. In addi-
tion, for the stenosis case both inlet and outlet are located fifteen and
twenty diameters away from the location of maximum constriction
respectively, in order to ensure that generated gradient forces due to
the magnetic field are negligible at the two boundaries. In all cases,
the outlet is modeled as a free surface with constant pressure. The
biomagnetic fluid considered duplicates the rheological properties
of blood with ρ = 1050 kg/m3 [46], and behaves as deoxygenated
blood with χ = 3.5 · 10−6. Finally, blood is assumed to behave as a
Herschel–Bulkley fluid, with the following set of parameters: τ0 =0.0035 Pa, n = 0.8375, κ = 0.008 Pa · s0.8375, and m = 1000 s. These
parameters were extracted from stress strain measurements of hu-
man blood samples using non-linear regression, under healthy phys-
iological conditions[47].
.1. Validation cases
.1.1. Case I - Couette flow
The Couette flow of a power-law model is considered as a val-
dation test case. The channel’s width and length are L = 0.01 m
nd l = 0.5 m respectively. Three different values of the exponent
re considered (n = 3/6, 4/6, 5/6), whereas the consistency index is
ept constant and equal to 0.0035 Pa · sn. Simulations run at Re =00, �t = 2 · 10−3 s, and fully developed Newtonian velocity pro-
le is prescribed at the inlet, yielding u/umean = 1.5 at the center.
he numerical result is then compared to the analytical solution for
K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180 175
Table 6
Non-dimensional velocity for different values of n at x/D = 25 as given by
Eq. (24), and percentage RMS error with respect to umean .
n u/umean % RMS error w.r.t. umean
3/6 1.6 1.4 · 10−2
4/6 1.8 1.3 · 10−2
5/6 1.90 1.4 · 10−2
x/D
u/u
mea
n,y
/D=
0
0 5 10 15 20 251.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2
2.05 Numer.: n=3/6Theory: n=3/6Numer.: n=4/6Theory: n=4/6Numer.: n=5/6Theory: n=5/6
Fig. 3. Non-dimensional velocity for the Poiseuille flow of a power-law fluid for differ-
ent values of exponent n at y/D = 0. All simulations converge to corresponding analyt-
ical solutions yielding a percentage relative error no more than a few hundredths of a
percent.
y/R
u/u m
ean,x
/D=
25
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10
0.5
1
1.5
2
Numer.: n=3/6Theory: n=3/6Numer.: n=4/6Theory: n=4/6Numer.: n=5/6Theory: n=5/6
Fig. 4. Non-dimensional velocity profiles for Poiseuille flow of a power-law fluid for
different values of exponent n at x/D = 25. Numerical results deviate no more than a
few hundredths of a percent with respect to the analytical ones.
C
w
l
c
s
t
f
r
n
r
p
n
p
n
5
a
D
e
s
p
t
fl
w
s
w
d
l
m
d
n
5
fl
T
d
F
y
z
Fig. 5. Contours of magnetic field (10) with C = 1.72 · 10−10 Tm4, yielding |B(x, y, z)|max =intensity is plotted on a natural logarithmic scale.
ouette flow of a power-law fluid as given by,
u
umean= 2n + 1
n + 1
[1 −
(y
H
)1/n+1], (23)
here y denotes distance in the transverse direction from the center-
ine, and H = L/2. Using the assumed values for the exponent we can
onstruct Table 5 at x/L = 50, in order to examine the accuracy of the
imulation, as expressed by the percentage RMS error with respect to
he mean velocity, umean.
Fig. 1 presents the non-dimensional velocity along the centerline
or the three cases considered. All simulations converge to the cor-
esponding analytical solutions yielding a percentage relative error
o more than a few hundredths of a percent. Additionally, the length
equired for convergence is inversely related to the value of the ex-
onent, thus requiring more channel lengths for smaller values of
. Fig. 2 presents numerical and analytical non-dimensional velocity
rofiles at x/L = 50. Flattening of the profiles with decreasing expo-
ent, a property of shear-thinning fluids, is clearly shown.
.1.2. Case II - Poiseuille flow
The Poiseuille flow of a power-law model is considered as an
dditional validation test case. The pipe’s diameter and length are
= 0.01 m and l = 0.25 m respectively. All power-law flow param-
ters are kept the same as with the Couette flow, but for this set of
imulations Re = 50, and �t = 5 · 10−3 s. A fully developed parabolic
rofile is again prescribed at the inlet, yielding u/umean = 2 at the cen-
er, and results are compared with analytical solution for Poiseuille
ow of a power-law fluid,
u
umean= 3n + 1
n + 1
[1 −
(r
R
)1/n+1], (24)
here R is the pipe radius. Table 6 presents the RMS error with re-
pect to umean between numerical and analytical solutions, where as
ith the Couette flow the error is small. Figs. 3 and 4 present non-
imensional axial velocities along the centerlines at y/D = 0 and ve-
ocity profiles at x/D = 25 respectively. Similar conclusions can be
ade as with the Couette flow, such as flattening of the profile with
ecreasing exponent, and the inverse relation between the length
ecessary for flow convergence and exponent value.
.2. Stenosis flow
Steady and pulsatile flow of Newtonian and Herschel–Bulkley
uids through an axisymmetric stenosis are presented in this section.
he stenotic geometry is generated assuming a hyperbolic secant
ependence on the axial coordinate, x, [27,48] defining its shape as,
(x) = D/2 − Asech[B(x − x0)], (25a)
= F(x) cos θ, (25b)
= F(x) sin θ, (25c)
4 T at (x0/D, y0/D, z0/D) = (±0.171,−0.31, 0). For visualization purposes the field
176 K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180
x/D
u/u m
ean,y
/D=
0
-15 -10 -5 0 5 10 15 20
2
3
4
5
6
7
8
NO MF: NewtonianNO MF: Herschel-BulkleyMF: NewtonianMF: Herschel-Bulkley
Fig. 6. Non-dimensional centerline velocity in the absence of external magnetic fields for Newtonian (black solid) and Herschel–Bulkley (green dashed) fluids, and when the field
of Eq. (10) is turned on for Newtonian (red dashed-dotted) and Herschel–Bulkley (blue dotted) fluids. In all cases, a parabolic profile is prescribed at the inlet.
Fig. 7. TOP: Contour plot of axial velocity difference between Newtonian and Herschel–Bulkley fluids along the z = 0 plane as expressed by the dimensionless quantity (uNew −uHB)/umean . BOTTOM: WSS of the two fluids. Due to flow symmetry, WSS that corresponds to positive values of y is only shown.
a
n
i
m
c
fl
l
s
d
t
c
t
o
a
i
d
m
c
t
where x0 and D are the position of maximum constriction and diam-
eter of the non-stenosed pipe respectively. Parameters A and B de-
termine the degree of constriction and extension of the stenosis. In
this work, the stenosis is parametrized using x0 = 0, and D = A/0.3 =6/B = 0.01 m. In addition, following the notation of Eq. (10) the
magnetic field is placed at (xi/D, yi/D, zi/D) = (0, −0.5, 0) yielding
|B(x, y, z)|max = 4 T for C = 1.72 · 10−10 Tm4 at (x0/D, y0/D, z0/D) =(±0.171,−0.31, 0) as shown in Fig. 5.
The simulations run at Re = 100, �t = 5 · 10−3 s for the steady
case, and Remean = 100 (Repeak = 150), �t = 10−3 s for the pulsatile
case. As before, the fully developed parabolic profile is prescribed at
the inlet. For the assumed magnetic field magnitudes, the generated
Lorentz force affects the flow minimally as pointed out in [8,27], and
it is not taken into account as an external body force in Eq. (1a). As a
result, only the magnetization force is acting upon the fluid yielding
the following results for the steady and pulsatile cases.
5.2.1. Steady flow:
Fig. 6 presents the centerline axial velocity of the four possible
combinations between the two fluids and intensity of the externally
pplied magnetic field of Eq. (10). The shear-thinning effect of the
on-Newtonian fluid yields a flattened profile and thus a lower max-
mum velocity. This result does not depend on the presence of the
agnetic field since, as can be seen, its effect is weak at maximum
onstriction. The addition of the magnetic field though pushes the
ow in both Newtonian and non-Newtonian fluid cases towards the
ower wall, reducing the streamwise component along the axis of
ymmetry. In both cases, this effect diminishes approximately nine
iameters downstream of the stenotic region. Fig. 7 compares the
wo fluids in the absence of magnetic fields. At the top of Fig. 7 a
ontour plot of the dimensionless variable (uNew − uHB)/umean at
he z = 0 plane is shown. It is again clear that the flattened profile
f the Herschel–Bulkley fluid results in lower velocity along the
xis of symmetry and a steeper velocity gradient with respect to
ts Newtonian counterpart, for mass to be conserved. As a result,
(uNew − uHB)/umean accepts positive values in an area symmetrically
istributed around the axis. As expected, these values decrease when
oving away from the axis of symmetry yielding eventually a sign
hange and negative values near the wall. An additional manifesta-
ion of the non-Newtonian viscosity model for moderate Reynolds
K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180 177
Fig. 8. TOP: Contours of apparent viscosity for the Herschel–Bulkley fluid in the absence of magnetic field inducing shear viscous effects. BOTTOM: percentage viscosity difference
when the field is turned on due to the generated magnetoviscous effects. Since the differences are mainly localized in the vicinity of the stenosis the result is shown on a natural
logarithmic scale. The field is placed at (xi/D, yi/D, zi/D) = (0,−0.5, 0).
x/D
WS
S(P
a)
-1 -0.5 0 0.5 1 1.5 20
1
2
3
4
5
6
MF, y>0: Herschel-BulkleyMF, y<0: Herschel-Bulkley
x/D
WS
S(P
a)
-0.1 -0.05 0 0.05
4
5
Fig. 9. TOP: Axial velocity contours for a Herschel–Bulkley fluid when the magnetic field of Eq. (10) is switched on. BOTTOM: the presence of the external magnetic field breaks
the flow symmetry yielding different values of WSS along upper and lower wall (positive and negative y values respectively).
n
t
a
y
N
r
a
v
c
T
t
m
p
t
(
i
s
I
p
w
umbers is the relocation of the reattachment point, xr. It is found
hat for the Herschel–Bulkley fluid the reattachment point is located
lmost one diameter upstream towards the maximum constriction,
ielding xr,HB/D = 4 as opposed to xr,New/D = 5 for the corresponding
ewtonian case. This is to be expected since the characteristic shear
ate for the Herschel–Bulkley fluid γc = 8umean/3R = 17.7̄ s−1. As
result, μc = 0.0052 Pa · s and Rec = 67.2 for the characteristic
iscosity and Reynolds number respectively, satisfying the positive
orrelation between Reynolds number and reattachment length.
he lower part of Fig. 7 presents wall shear stress magnitudes in
he vicinity of the stenosis, where differences appear small and
ainly located near maximum constriction. The top part of Fig. 8
resents contours of viscosity for the Herschel–Bulkley fluid when
he field is switched off. Areas characterized by low shear rates
such as the axis of symmetry) are associated with higher viscos-
ty (red). As the fluid is forced to flow through the stenosis, the
hear rate increases generating lower values for viscosity (blue).
t is interesting also to note the two symmetrical features in the
ost-stenotic region and close to the wall. These are associated
ith the recirculations regions and are formed by the minimization
178 K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180
0.5
0.75
1
1.25
1.5
0 0.25 0.5 0.75 1
Q(t
)/Q
mea
n
t/T
A
B
C
D
Fig. 10. Imposed flow rate waveform for the pulsatile simulation. Letters indicate time
moments in the cycle where results are obtained.
a
t
e
m
fl
z
s
u
5
v
i
F
v
e
F
s
u
w
k
of the dominant components of the velocity derivatives in the shear
rate. The lower part of Fig. 8 shows the percentage change of the
viscosity when the magnetic field is switched on. For visualization
purposes the result is plotted on a natural logarithmic scale. As ex-
pected, a marked increase of apparent viscosity with a maximum of
x/
u/u
*
-15 -10 -5 00
1
2
3
4
5
6A
x/
u/u *
-15 -10 -5 00
1
2
3
4
5
6B
x/
u/u
*
-15 -10 -5 00
1
2
3
4
5
6C
x/
u/u
*
-15 -10 -5 00
1
2
3
4
5
6D
Fig. 11. Centerline velocity divided by mean (cycle-averaged) centerline inlet velocity, u∗ , for
the magnetic field defined by Eq. (10). A: early systole (t/T = 0), B: peak systole (t/T = 0.25)
cases, a parabolic profile is prescribed at the inlet. (For interpretation of the references to col
pproximately 11.5% can be seen in a small restricted region around
he lower part of maximum constriction. Obviously, magnetoviscous
ffects diminish very rapidly following the steep decline of the
agnetic field. Finally, Fig. 9 examines the flow of a Herschel–Bulkley
uid with the magnetic field switched on. The generated magneti-
ation force breaks the flow symmetry (top) resulting in higher wall
hear stress on the lower part of the post-stenotic region, while the
pper part is minimally affected (bottom).
.2.2. Pulsatile flow
The effect of the external magnetic field given by Eq. (10) on a time
arying flow of Newtonian and Herschel–Bulkley fluids is presented
n this section. In both cases, the sinusoidal flow rate waveform of
ig. 10 is considered. The inverse Womersley method computes the
elocity profile from a prescribed flow rate, as opposed to the Wom-
rsley method where the pressure gradient is the needed quantity.
or any volumetric flow rate, Q(t), it is possible to calculate the corre-
ponding velocity profile, u(r/R, t), as follows [49],
(r
R, t
)= Q(t)
πR2
[αi3/2J0(αi3/2) − αi3/2J0(αi3/2r/R)
αi3/2J0(αi3/2) − 2J1(αi3/2)
], (26)
here J0 and J1 are the modified Bessel functions of zero and first
ind respectively, i = √−1, and α is the dimensionless Womersley
D 5 10 15 20
Newt: t/T=0HB: t/T=0
D 5 10 15 20
Newt: t/T=0.25HB: t/T=0.25
D 5 10 15 20
Newt: t/T=0.5HB: t/T=0.5
D 5 10 15 20
Newt: t/T=0.75HB: t/T=0.75
Newtonian (black solid) and Herschel–Bulkley (green dashed) fluid in the presence of
, C: mid-deceleration phase (t/T = 0.5), D: end-deceleration phase (t/T = 0.75). In all
or in this figure legend, the reader is referred to the web version of this article.)
K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180 179
p
α
w
s
αe
i
s
i
a
t
i
u
c
a
t
t
6
N
d
d
p
a
m
i
J
i
e
r
m
fi
g
R
d
a
o
c
s
d
A
f
G
n
E
R
[
[
[
[
[
[
[
[
[
[
arameter,
= R
√ωρ
μ, (27)
hich measures the unsteadiness of the flow. For the parameters as-
umed in this section and a period of pulsation T = 1 s, we find that
= 6.86. Setting the initial conditions to zero velocity and pressure,
ight cycles are computed with the time periodic solution of Eq. (26)
n order to ensure that all transient effects are washed out before re-
ults are collected.
Fig. 11 presents the centerline velocity when the magnetic field
s on, for both fluids at early systole, peak systole, mid-deceleration,
nd end-deceleration phases. Even though flow characteristics of the
wo fluids are similar, differences in flow patterns are clearly vis-
ble. Due to shear-thinning, the Herschel–Bulkley fluid recovers its
nperturbed state earlier compared to the Newtonian along the
enterline, as is clearly illustrated in Fig. 11D. Both fluids though
re affected by the magnetic field creating an oscillatory flow in
he post-stenotic region that diminishes while receding from the
hroat.
. Conclusions
We present a pressure-correction scheme for the flow of non-
ewtonian and incompressible fluids. It consists of a combined
iscontinuous Galerkin approximation for velocity, and a stan-
ard continuous Galerkin approximation for pressure. Use of the
rojection method in order to decouple the momentum equation
nd the incompressibility constraint ensures the efficiency of the
ethod. The stress-tensor is not discretized separately but rather
s computed explicitly thus disregarding its non-linearity in the
acobian computation. The convergence rate, however, of the Newton
teration was not significantly affected preserving the computational
fficiency of the method. The ability of the method to accurately
esolve 2D and 3D benchmark problems was demonstrated. The
ethod is subsequently utilized to assess the effects of magnetic
elds on biomagnetic fluid flow. To this end, the magnetization force
enerated by an externally applied magnetic field is added in the
HS of the momentum equations, resulting in considerable flow
eviation, even for moderate field intensity. Magnetoviscous effects
re also taken into account through the generated additive viscosity
f the fluid and were found to be important. Applications of interest
an be foreseen by exploiting magnetic fields for blood flow control,
uch as reduction of blood loss during surgery and targeted drug
elivery.
cknowledgments
The authors thank BETA CAE Systems, Greece, customer service
or support on mesh generation using ANSA v15, and Dr. Domenico
iordano at ESA-ESTEC for useful discussions on coupling of biomag-
etic fluids with electromagnetic fields. This work was supported by
SA TRP Contract 4200022319/09/NL/CBI.
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