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: yannisp@iacm.forth.gr (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

s

t

m

ρ

∇

u

σ

u

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p

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σ

σ

T

r

τ

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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

y

e

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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)

�t

K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180 173

∂

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ω

T

(

{

a

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g

l

ω

(

d

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W

a

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γ

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d

v

(

i

c

F

s

d

t

4

i

a

c

Pf

u

e

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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