This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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.
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).
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.
eferences
[1] Ruunge EK, Rusetski AN. Magnetic fluid as drug carriers: targeted transport of
drugs by a magnetic field. J Magn Magn Mater 1993;122:335–9.[2] Plavins J, Lauva M. Study of colloidal magnetite binding erythrocytes: prospects
for cell separation. J Magn Magn Mater 1993;122:349–53.[3] Haik Y, Chen J, Pai V. Development of bio-magnetic fluid dynamics. In: Proceed-
ings of the IX International Symposium on Transport Properties in Thermal FluidsEngineering, Singapore, Pacific Center of Thermal Fluid Engineering, SH Winoto,
YT Chew, NE Wijeysundera (ed) Hawaii, USA; 1996. p. 121–6.
[4] Jaspard F, Nadi M, Rouane A. Dielectric properties of blood: an investigation ofhaematocrit dependence. Physiol Measurement 2003;24(1):137–47. doi:10.1088/
0967-3334/24/1/310.[5] Davidson PA. An introduction to magnetohydrodynamics. Cambridge University
Press; 2001.[6] Tzirtzilakis EE. A mathematical model for blood flow in magnetic field. Phys Fluids
2005;17(7):077103–15. doi:10.1063/1.1978807.[7] Raptis A, Xenos M, Tzirtzilakis E, Matsagkas M. Finite element analysis of mag-
netohydrodynamic effects on blood flow in an aneurysmal geometry. Phys Fluids
2014;26(10):101901–14. doi:10.1063/1.4895893.[8] Kenjereš S. Numerical analysis of blood flow in realistic arteries subjected to
strong non-uniform magnetic fields. Int J Heat and Fluid Flow 2008;29(3):752–64. doi:10.1016/j.ijheatfluidflow.2008.02.014.
[9] Türk Ö, Tezer-Sezgin M, Bozkaya C. Finite element study of biomagnetic fluidflow in a symmetrically stenosed channel. J Comput Appl Math 2014;259:760–
70. doi:10.1016/j.cam.2013.06.037.
[10] Türk Ö, Bozkaya C, Tezer-Sezgin M. A fem approach to biomagnetic fluid flowin multiple stenosed channels. Comput Fluids 2014;97:40–51. doi:10.1016/j.
compfluid.2014.03.021.[11] Pedley TJ. The fluid mechanics of large blood vessels. Cambridge University Press;
1980.[12] Mishra JC, Patra M, Mishra S. A non-newtonian fluid model for blood flow through
arteries under stenotic conditions. J Biomech 1993;26(9):1129–41. doi:10.1016/
S0021-9290(05)80011-9.[13] Tu C, Deville M. Pulsatile flow of non-newtonian fluids through arterial stenoses.
J Biomech 1996;29(7):899–908. doi:10.1016/0021-9290(95)00151-4.[14] Chaturani P, Samy RP. Pulsatile flow of Cassons fluid through stenosed arteries
with applications to blood flow. Biorheology 1986;23(5):499–511.[15] Siddiqui SU, Verma NK, Mishra S, Gupta R. Mathematical modelling of pulsatile
flow of Cassons fluid in arterial stenosis. Appl Math Comput 2009;210(1):1–10.
doi:10.1016/j.amc.2007.05.070.[16] Sankara DS, Lee U. Mathematical modeling of pulsatile flow of non-newtonian
fluid in stenosed arteries. Commun Nonlinear Sci Numer Simul 2009;14(7):2971–81. doi:10.1016/j.cnsns.2008.10.015.
[17] Lida N. Influence of plasma layer on steady blood flow in microvessels. Jpn J ApplPhys 1978;17(1). doi:10.1143/JJAP.17.203.203–
[18] Goya GF, Grazu V, Ibarra M. Magnetic nanoparticles for cancer therapy. Curr
Nanosci 2008;4(1):1–16. doi:10.2174/157341308783591861.[19] Haik Y, Pai V, Chen CJ. Apparent viscosity of human blood in a high static magnetic
field. J Magn Magn Mater 2001;225(1-2):180–6. doi:10.1016/S0304-8853(00)01249-X.
20] Shukla JB, Parihar RS, Rao BRP. Effects of stenosis on non-newtonian flowof the blood in an artery. Bull Math Biol 1980;42(3):283–94. doi:10.1016/
S0092-8240(80)80051-6.
[21] Jr JRB, Kleinstreuer C, Comer JK. Rheological effects on pulsatile hemody-namics in a stenosed tube. Comput Fluids 2000;29(6):695–724. doi:10.1016/
S0045-7930(99)00019-5.22] Razavi MRM, Seyedein SH, Shahabi PB. Numerical study of hemodynamic wall
parameters on pulsatile flow through arterial stenosis. Int J Ind Eng Prod Res2006;17(3):37–46.
23] Kröner D, Ruzicka M, Toulopoulos I. Local discontinuous galerkin numerical solu-tions of non-newtonian incompressible flows modeled by p-Navier–Stokes equa-
[24] Kwack J, Masud A. A stabilized mixed finite element method for shear-rate de-pendent non-newtonian fluids: 3d benchmark problems and applications to
blood flow in bifurcating arteries. Comput Mech 2014;53(4):751–76. doi:10.1007/s00466-013-0928-6.
25] Botti L, Di Pietro DA. A pressure-correction scheme for convection-dominated in-compressible flows with discontinuous velocity and continuous pressure. J Com-
put Phys 2011;230(3):572–85. doi:10.1016/j.jcp.2010.10.004.
26] Botti L, Koen VC, Kaminsky R, Claessens T, Planken RN, Verdonck P, et al.Numerical evaluation and experimental validation of pressure drops across a
patient-specific model of vascular access for hemodialysis. Cardiovasc Eng Tech-nol 2013;4(4):485–99. doi:10.1007/s13239-013-0162-6.
[27] Tzirakis K, Papaharilaou Y, Giordano D, Ekaterinaris J. Numerical investigationof biomagnetic fluids in circular ducts. Int J Numer Methods in Biomed Eng
2014;30(3):297–317. doi:10.1002/cnm.2603.
28] Papanastasiou TC. Flows of materials with yield. J Rheol 1987;31(5):385–404.doi:10.1122/1.549926.
29] MacCormack RW. Numerical simulation of aerodynamic flow including inducedmagnetic and electric fields. In: Proceedings of the 39th AIAA Plasmadynamics
and Lasers Conference; 2008. http://dx.doi.org/10.2514/6.2008-4010.30] Shercliff JA. A textbook of magnetohydrodynamics. Pergamon Press; 1965.
[31] Pai SI. Magnetohydrodynamics and magnetogasdynamics. PN 1955.
32] Guermond JL, Quartapelle L. On the approximation of the unsteady Navier–Stokes equations by finite element projection methods. Numerische Mathematik
la mthode des pas fractionnaires. Arch Ration Mech Anal 1969;33(5):377–85.doi:10.1007/BF00247696.
34] Guermond JL, Minev P, Shen J. An overview of projection methods for incom-
pressible flows. Comput Methods in Appl Mech Eng 2006;195(44–47):6011–45.doi:10.1016/j.cma.2005.10.010.
[35] Ern A, Stephansen AF, Zunino P. A discontinuous Galerkin method with weightedaverages for advection-diffusion equations with locally small and anisotropic dif-
fusivity. IMA J Numer Anal 2008;29(2):235–56. doi:10.1093/imanum/drm050.
180 K. Tzirakis et al. / Computers and Fluids 126 (2016) 170–180
[
[36] Dryja M. On discontinuous Galerkin methods for elliptic problems with discon-tinuous coefficients. Comput Methods in Appl Math 2003;3(1):76–85. doi:10.
2478/cmam-2003-0007.[37] Di Pietro DA, Ern A, Guermond JL. Discontinuous Galerkin methods for anisotropic
semi-definite diffusion with advection. SIAM J Numer Anal 2008;46(2):805–31.doi:10.1137/060676106.
[38] Arnold DN. An interior penalty finite element method with discontinuous ele-ments. SIAM J Numer Anal 1982;19(4):742–60. doi:10.1137/0719052.
[39] Di Pietro DA, Ern A. Mathematical aspects of discontinuous Galerkin methods.
Springer Berlin Heidelberg; 2012.[40] Kirk BS, Peterson JW, Stogner RH, Carey GF. libmesh: A c++ library for paral-
lel adaptive mesh refinement/coarsening simulations. Eng Comput 2006;22(3–4):237–54. doi:10.1007/s00366-006-0049-3.
[41] Balay S, Abhyankar S, Adams M, Brown J, Brune P, Buschelman K, et al. Petscusers manual. Argonne National Laboratory, ANL-95/11 - Revision 35 2014. http:
//wwwmcsanlgov/petsc
[42] Karypis G, Kumar V. A fast and highly quality multilevel scheme for parti-tioning irregular graphs. SIAM J Sci Comput 1998;20(1):359–92. doi:10.1137/
S1064827595287997.
[43] E W, Liu JG. Projection method i: convergence and numerical boundary layers.SIAM J Numer Anal 1995;32(4):1017–57. doi:10.1137/0732047.
44] De Santis G, Mortier P, De Beule M, Segers P, Verdonck P, Verhegghe B.Patient-specific computational fluid dynamics: structured mesh generation from
coronary angiography. Med Biol Eng Comput 2010;48(4):371–80. doi:10.1007/s11517-010-0583-4.
[45] Geuzaine C, Remacle JF. Gmsh: A 3-d finite element mesh generator withbuilt-in pre- and post-processing facilities. Int J Numer Methods in Eng
2009;79(11):1309–31. doi:10.1002/nme.2579.
[46] Pedley TJ. The fluid mechanics of large blood vessels. Cambridge University Press;1980.
[47] Valant AZ, Žiberna L, Papaharilaou Y, Anayiotos A, Georgiou G. The influence oftemperature on rheological properties of blood mixtures with different volume