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60 Chapter 4
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where l and m are fluid (viscosity) parameters and p is the pressure. We assume no
dependence of material parameters, density, and pressure on the temperature. The
assumption is valid for the fluids of constant temperature (isothermal), which is relevant
for many blood flows. For the incompressible fluid, the constitutive equation reduces to
s ¼p þ 2mDðvÞ: (4.17)
Summarizing Eqs. (4.13), (4.15)e(4.17), we arrive at the following set of equations for the
basic fluid model used in this book, the system of the NaviereStokes equations for the
incompressible viscous Newtonian isothermal fluid,
8
vv
>
<
>
:
þðv$VÞ v 2mdiv DðvÞþVp ¼ rf
r
vt
div v ¼ 0
(4.18)
with constant density and viscosity coefficients r > 0 and m > 0 and given body forces f,
e.g., gravity force. Eq. (4.18) is written in the Eulerian coordinates. Later, we discuss other
useful formulations of the NaviereStokes equations and introduce initial and boundary
conditions necessary to close the system.
In Newtonian fluids, the viscosity coefficient m in Eq. (4.17) does not depend on the rate
of strain. For flows of blood plasma and blood flows in arteries with diameters larger than
w1 mm, it is plausible to assume that m is constant, and experimental measurements show
m ˛ ½0:003; 0:004 Pa$s [43]. In smaller vessels, in particular vessels with diameters below
0.1 mm, the blood may exhibit non-Newtonian fluid behavior, and more accurate models
should include a dependence of m on the invariants of the rate of strain tensor [193, 43].
For hemorheology in the microcirculation, we refer to the review paper [42].
4.2.2 Energy balance
Conservation of energy is a fundamental law of nature, which was not explicitly used so
far to deduce the governing Eq. (4.18). It is, however, easy to show that any (smooth)
solutions to Eq. (4.18) satisfy suitable energy balance. To see this, let us consider any
material volume VðtÞ consisting of fluid particles. We further take an inner product of the
momentum equation (4.18) with v and integrate over V ðtÞ. Using the notion of the
material derivative, we get
Z
ðr_v $ v 2mdiv DðvÞv þv $ VpÞ dx ¼
VðtÞ
Z
rf$v dx:
VðtÞ
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Now we apply the Reynolds transport theorem (4.7) to f ¼jv
2
, the incompressibility
j
condition (4.16), and integrate by parts the viscous and pressure terms. Recalling the
relation for the stress tensor (Eq. 4.17), we get the following energy balance:
Z
12d
dt
Vðt
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
rate of change of kinetic energy
Þ
rjv
2
dx
j
¼2 m
Z
2
DðvÞj
j
Þ
Vðt
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
viscous energy dissipation
dx
Z
þ
ðsnÞ$vds
Þ
vVðt
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
work of surface forces
Z
þ
rf$vdx:
Þ
Vðt
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
work of body forces
(4.19)
Thus, all smooth solutions to the NaviereStokes Eq. (4.18) obey the fundamental energy
balance (Eq. 4.19). The identity (4.19) can take slightly different forms, depending on
boundary conditions and the choice of V ðtÞ. For numerical methods, it is important to
ensure that computed (approximate, discrete) solutions satisfy an analog of the energy
balance (Eq. 4.19).
4.2.3 Boundary conditions
For the system of fluid equations to be closed, we need to supply it with initial and
boundary conditions. For the initial condition, it is common to assume that the velocity of
fluid is known everywhere in the flow domain, vð0; xÞ¼v
Although in many cases knowing the initial state v
is not feasible, the influence of the
0
error in the initial guess on the computed solution is not significant for larger t.For
obvious reasons, practically important blood flows happen and being computed in bounded
volumes. We introduce the notation U for the domain in Eulerian coordinate system where
the fluid equations are posed (U may depend on t). The choice of suitable boundary
conditions on vU is important.
ðxÞ with some given v0.
0
We shall distinguish between the no-slip, no-penetration, inflow, outflow, and interface
conditions. Interface conditions are imposed on interfaces between fluid and other
continuum medium (e.g., a different fluid or elastic body), which motion is unknown and
finding it is a part of the problem. Interface conditions relevant for cardiovascular
applications will be discussed in Section 4.4.1. To introduce other conditions, recall that
the mapping xðtÞ:bU/U from the reference domainbU to the physical domain U defines
the material trajectories for allbx ˛bU. The condition that fluid does not penetrate through
the part of the boundary, vU
ns
3vU, then reads
n $ v ¼ n$x
t
+ x
1
on vU
ns
; (4.20)
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i.e., the normal fluid velocity equals the normal velocity of the boundary. The
no-penetration condition (4.20) is not enough and should be supplemented with a
tangential condition. For viscous fluids, the most common is the no-slip condition,
n v ¼ n x
t
+ x
1
on vU
ns
; (4.21)
which states that the tangential component of fluid velocity coincides with the tangential
velocity of the boundary (fluid particles attach to the boundary). Conditions (4.20) and
(4.21) together, of course, mean that the velocity of fluid along vU is the same as the
velocity of material points on the boundary,
t
+x
1
on vUns: (4.22)
One can recognize v ¼ 0 on vU
v ¼x
ns
for the case of stationary fluid domain.
The Dirichlet condition,
v ¼v
is often imposed on inflow parts of the boundary vU
on vUD; (4.23)
D
D
, where the flow profile vDis
assumed to be known. The Neumann condition,
sn ¼g on vU
N
; (4.24)
is useful on artificial outflow boundaries, where one lacks any suitable information on flow
profile. It is common to set g equal to zero or an approximate average value of normal
stress exerted by an exterior medium on the fluid. If vU
consistence condition
R
n xt+x1ds þ
ns
vU
R
n vDds ¼ 0.
D
vU
N
¼ B, then one should ensure the
No-penetration, no-slip, inflow (4.23), and outflow (4.24) conditions are the most often
used boundary conditions in the modeling of blood flows. It is not rare, however, that
other type of boundary conditions is better suited for modeling and simulations. On
fluidesolid interfaces, the no-slip condition (4.21) models the adherence of fluid to the
solid. This assumption is sometimes argued as not sufficiently justified, especially on the
macroscopic level (see [201] and references therein). In certain situations, it makes sense
to assume that the fluid may slip along a part vU
slip
of the solid boundary possibly
generating friction forces. To model this phenomenon, one applies the Navier boundary
condition also known as slip with friction condition [178]:
where a > 0 is a coefficient and a
of the fluid to slip along the boundary. We see that slip with friction condition assumes a
linear resistance of fluid to slip. In even more general situations, nonlinear relations
between normal stress and slip velocity can be used (see, e.g., [201]).
ðsn þavÞn ¼ 0onvU
1
is known as slip length, which measures the ability
slip
; (4.25)
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Next boundary condition looks more ad hoc but is proved to be useful in numerical
simulations of many cardiovascular blood flow problems with artificial boundaries.
For numerical purposes, it is common to cut a fluid domain along an imaginary
surface vU
aux
(artificial boundary). It is a well-known problem to prescribe suitable
boundary conditions on such a boundary, since little a priori information is available about
a flow there. One popular option is setting the normal stresses to zero, i.e., Eq. (4.24) with
g ¼ 0. It appears that, for problems with incoming (reverse) flows through artificial
boundaries setting, Eq. (4.24) leads to energetically unstable models, since incoming flow
may lead to a blowup of energy or nonphysical oscillations for the computed solution. To
compensate for this inflow of excessive energy, one adds extra term to Eq. (4.24), which
models boundary forces that diffuse the energy in the case of incoming flow [118, 114]:
sn þ
with a parameter a ˛½0; 1andðx
a
ð
Þ
v ¼ 0 on vU
v$n
2
Þ
1
ðj
¼
xjxÞ. Condition (4.26) is known in the
2
aux
; (4.26)
literature as directional do-nothing boundary condition. For other options of stabilizing
boundary conditions for physiological flows on artificial boundaries, see, e.g., [110] and
references therein.
4.2.4 Fluid equations in Lagrangian and quasi-Lagrangian coordinates
One can rewrite fluid equations in the reference domain, using Lagrangian velocity and
pressure variables. In Lagrangian coordinates system, the inertia terms in Eq. (4.18)
simplify, but the viscous terms and incompressibility constrain get more complicated. In
this section, all derivatives are assumed with respect to the reference coordinatesbx. Recall
the notations F ¼ Vx for deformation gradient tensor and J ¼ detðFÞ for its determinant.
Changing variables and applying the product rule for computing derivatives in Eq. (4.18),
one obtains the set of the NaviereStokes equations in Lagrangian coordinates:
(
b
rbv
J1divJðs+xÞF
t
1
b
divJF
v¼ 0
with body forcesbf ¼ f +x. Note that we have_v+x ¼bv
T
¼brbf
. The constitutive relation (4.17) in
t
(4.27)
the reference domain reads
s + x ¼bpI þ mVbvF
1þFT
ðVbvÞ
T
: (4.28)
In practice, it may happen that the motion of U is given by a non-Lagrangian mapping
from the reference domain to U, i.e., xðbx; t Þ does not define a material trajectory for any
b
x ˛bU. One example of non-Lagrangian mapping x is considered in Section 5.4.4 where it
is recovered from a sequence of medical images of the motion of heart walls. Let such
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mapping x be level preserving, i.e., x
b
U; t¼ UðtÞ for all t ˛ ½0; T, and J > 0. If one uses
such a non-Lagrangian mapping to formulate the fluid system in the reference coordinates,
then the identity_v+x ¼bv
is no longer true, and using Eq. (4.1) and the product rule, we
t
get
1
b
_
v+x ¼ðv
þðVxvÞvÞ+x ¼bvt xt$ðVxvÞ+x þðV
t
þðV
¼bv
t
b
vÞF
^
x
1
ðbv xtÞ:
^
x
vÞF
b
v
We see that the inertia term does not simplify as in Eq. (4.27), and the momentum
equation reads
b
b
r
v
þðVbvÞF1ðbv xtÞJ1divJðs + xÞF
t
T
¼brbf: (4.29)
4.2.5 Arbitrary LagrangianeEulerian formulation
Eq. (4.27) shows that inertia terms in Lagrangian formulation look simpler than in the
Eulerian form, but viscous terms become more complicated. From computational point of
view, it is sometimes convenient to handle the time derivative in the reference domain, but
to treat all other terms in the physical domain. For doing this, it is sufficient to know any
smooth mapping xðtÞ :bU/UðtÞ (not necessary Lagrangian) as in Eq. (4.29). Then
identifyingbvðt;bxÞ¼vðt; xðt;bxÞÞ, one computes v
formulation of Eq. (4.18) known as Arbitrary LagrangianeEulerian (ALE) formulation
8
vbv
>
<
>
:
r
þððv x
vt
Þ$VÞv 2mdiv DðvÞþVp ¼ rf ;
t
div v ¼ 0:
¼bvt xt$Vv. This leads to the
t
(4.30)
Vector field x
is called ALE velocity.
t
4.2.6 Flow regimes
The quantities in the flow problem (4.18) have the following physical units: kgm3for
density r, kg=ðs $mÞ for viscosity coefficient m, and m=s for velocity v. Assuming for a
moment, that the domain is stationary and L is a characteristic linear dimension for this
problem (for example, L ¼ diamðUÞ), one can define the dimensionless quantity,
Re ¼
r VL
m
;
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where V is a characteristic velocity for this problem (for example, the maximum of the
inflow velocity v
). In fluid dynamics, this important dimensionless quantity is known as
D
Reynolds number. It helps to predict flow patterns and dynamic similarity in different fluid
flow situations.
To see its role, one may rescale unknowns and variables in Eq. (4.18) to make them
nondimensional, x ¼ Lx
0
, v ¼ Vv0, t ¼ LV1t0, p ¼ rV2p0. We arrive at the
nondimensional form of the NaviereStokes equations,
8
>
<
>
:
vv
vt
þðv$VÞ v
2
div DðvÞþVp ¼ f
Re
div v ¼ 0:
(4.31)
From Eq. (4.31), we see that the Reynolds number is the ratio of inertial forces to viscous
forces in the momentum equation.
When the viscous forces dominate (Re number is small), then one can neglect the
nonlinear terms in Eq. (4.31) and arrive at the Stokes system. After further rescaling of
pressure, right-hand side, and time variable, the Stokes problem takes the form:
8
>
vv
<
div DðvÞþVp ¼ f
>
:
vt
div v ¼ 0:
(4.32)
The Stokes system (Eq. 4.32) is used to model slow flows such as blood flow in capillary
vessels.
In more general, for lower Reynolds numbers, fluid flows are characterized by low
velocities, regular flow patterns, and dynamics happening over a narrow range of spatial
and temporal scales. Such flows are called laminar. When the Reynolds number increases,
the flow eventually becomes turbulent. The turbulent flow regime is characterized by
complex vortical flow patterns, chaotic fluctuations in flow velocity and pressure, and the
presence of flow dynamics over a wide range of length and time scales. One can also
identify transitional flow regimes between very regular laminar and chaotic turbulent flows
(see, e.g., [167]).
Most of hemodynamics is characterized by laminar blood flows (Fig. 4.1). Flows in
aneurysms, aorta, and stenosed arteries are often classified as transitional [203, 197].
Turbulent blood flow regimes may be observed in the presence of some pathologies such
as severe stenoses [197].
Direct numerical simulation of turbulent flows can be prohibitively expensive in terms of
computational resources since one, in general, needs too many degrees of freedom to
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(A)
(C) (D)
Figure 4.1
Blood flow: (A) laminar flow in vessel, (B) turbulent flow behind stenosis, (C) turbulent flow
near aneurysm, (D) flow pattern in heart cavities may be complex.
(B)
resolve all spatial and temporal scales in the solution. Most often, one solves for averaged
flow statistics such as averaged velocity and pressure. Deducing a complete set of
equations for such averaged quantities is a well-known challenge, and available models
of turbulence are based on certain empirical modeling assumptions. We do not
elaborate this topic further in the book and instead refer the interested reader to the
literature (see, e.g., [168]).
4.3 Equations for deformable bodies
In mathematical biomechanics, soft biological tissues are treated as deformable elastic
bodies. Equations governing the motion of elastic medium are derived from the
fundamental principles already discussed in Section 4.1. In fact, all equations we deduced
in that section remain valid. There are, however, two important differences between how
one proceeds from Eqs. (4.12) and (4.13) in the case of fluids and solids. First, the
equation of motion for deformable elastic bodies is conventionally written and treated
numerically in Lagrangian coordinates, i.e., in the reference domainbU. Thus, we consider
the conservation of momentum equation in the Lagrangian form (Eq. 4.27):
Hereafter, we shall use the short notation S ¼ s+x for the stress tensor in Lagrange
coordinates and refer to JSF
b
rbv
J1divJSF
t
T
as the PiolaeKirchhoff tensor.
T
¼brbf: (4.33)
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Second, the constitutive equation for solids relates stress and deformation, rather than
stress and rate of deformation as happens for fluids. Finding a constitutive law, which
would be the best to describe a material at hand, is an important modeling question. In the
following, we discuss a general framework and several models that have been proved to be
useful in cardiovascular simulations.
4.3.1 General framework for hyperelastic materials
We further consider general constitutive laws for deformable bodies that relate the Cauchy
stress S and the deformation gradient F. The dependence of S on material properties and
F is very diverse depending on a material. To present many useful models within a single
framework, we take advantage of the notion of hyperelasticity. Recall that a material is
elastic if its constitutive law is defined entirely by the current deformation: S ¼ SðF;bxÞ.
A material is said to be hyperelastic if the mechanical work spent for its deformation
depends on the initial and final states only, i.e., the work is independent of the loading
path and, therefore, is spent to change an energy of the material. Although, for many
solids, hyperelasticity is only an idealized model, it was experimentally found to describe
material properties of most biological tissues rather accurately [122].
Hyperelastic materials are characterized by an elastic potential (also known as a strain energy
density function or a stored energy function) JðFÞ, which defines the Cauchy stress tensor as
where matrix
vJ
has entries
vF
vJ
vF
1JvJ
S ¼
vJ
¼
. Indeed, the mechanical work for deformation
vF
ij
ij
T
; (4.34)
F
vF
of the unit volume of a nondeformed body is [113, Section 5.2]
t
Z
from which we have J
vJ
J
¼
t
vF
, and therefore, we obtain Eq. (4.34).
: F
t
JðFÞ¼
¼ðJSFTÞ : Ft. On the other hand, the chain rule gives us
t
JSF
t
0
T
dt (4.35)
: F
t
Material is said to be isotropic at pointbx if its constitutive law is independent of any
rotation Q of the coordinate frame with origin atbx, SðF;bxÞ¼SðFQ;bxÞ, i.e., its elastic
properties are the same in all directions. Material is isotropic if it is isotropic at any point
b
x. The elastic potential of an isotropic material should be independent of any rotation
transformation as well, JðFÞ¼JðQFÞ: It is shown [124] that in this case J is a function
of the right CauchyeGreen tensor C ¼ F
T
F, which defines the change of distance
between two material points during the deformation. Therefore, we further assume that J
is given as a function of C. One can use the chain rule to compute
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vJ
vF
3
X
vJ
vC
¼
ij
k;m¼1
X
¼
m¼1
vC
3
vJ
vC
km
vF
Fimþ
jm
km
ij
¼
X
k;m¼1
3
k¼1
3
X
vJ
vC
vJ
vC
km
Fik¼ 2F
kj
k
d
þ Fikd
F
im
j
vJ
vC
ij
m
j
:
Hence, Eq. (4.34) can be rewritten in the equivalent form:
2
vJ
T
S ¼
F
J
: (4.36)
F
vC
The 3 3 right CauchyeGreen tensor C has three principal invariants under a rigid
coordinate transformation:
I
¼tr C; I2¼
1
1
2
ðtr CÞ
2
tr C
2
; I
¼ detC ¼ J2: (4.37)
3
The principal invariants can be written in the componentwise notations:
3
X
I
where ε
¼
1
is 1 for even permutation of i; j; k, 1 for odd permutation of i; j; k, and 0 for
ijk
Cii; I2¼
i¼1
i ¼ j or j ¼ k or i ¼ k. One also defines the left CauchyeGreen tensor B ¼ FF
has the same invariants I
, I2, I3. We recall that J ¼ detF shows the relative change of
1
3
X
1
ðCiiCjjCijCijÞ; I3¼
2
i; j¼1
3
X
i;j;k¼1
ε
Ci1Cj2Ck3;
ijk
T
, which
volume under the deformation. Materials featuring J ¼ 1 are called incompressible.
It is shown [124] that the elastic potential of an isotropic and homogeneous hyperelastic
material is a function of the principal invariants of the right CauchyeGreen tensor C
(Eq. 4.37), J ¼ JðI
where
vI
vC
The last identity follows from CayleyeHamilton theorem. Using these identities, one
derives the general constitutive law for isotropic and homogeneous hyperelastic material:
2
S ¼
J
or in terms of B:
; I2; I3Þ. Therefore, the constitutive relation becomes
1
F
1
¼ I;
vJ
vI
S ¼
S ¼
vI
2
¼ I
vC
vJ
þI
1
vI
1
2
vJ
J
vI
I C;
1
þI
2
þI
1
3
X
2
F
J
i¼1
vI
vJ
3
vC
vJ
I
2
vI
3
vJ
B
1
vI
2
!
vI
i
T
F
þI
2
B2þ 2J
2
1
;
vJ
vI
C þ
3
vJ
vI
vI
vC
i
I I1C þ C2hI3C1:
¼ I
2
vJ
vI
2JvJ
vI
I: (4.39)
3
vJ
vI
2
T
C
3
; (4.38)
F
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A practically important group is incompressible elastic materials, i.e., the ones preserving
the volume under the deformation. In laboratory tests, incompressible materials are more
easy-to-handle. Therefore, it is a common practice that a basic elastic potential J
inc
for
incompressible material is known. Then, a generalization of the potential to the
compressible case is built as a “perturbation” of the incompressible potential:
J
cmp
¼J
inc
þ J
ðJÞ
; (4.40)
vol
where the last term is added to capture the pressureevolume response. The
incompressibility constrain J ¼ 1 requires an extra term pðJ 1Þ to be added to the
incompressible elastic potential J
term vanishes for J ¼ 1, its derivative with respect to I
, where p is the Lagrange multiplier. Although this
inc
¼ J2in Eq. (4.39) generates an
3
extra “pressure” type term pI in the constitutive law.
We briefly mention heterogeneous materials. If the body material is heterogeneous, then it
can be decomposed into homogeneous patches described by their own constitutive laws.
The final problem formulations should take into account matching conditions on interfaces
between different materials. The physically motivated interface conditions are the
continuity of displacements and stresses.
Many biological tissues are considered to be composed of an isotropic matrix and elastic
fibers, which introduce material anisotropy. In this case, the elastic potential can be
represented in the form [122,152]:
J ¼J
iso
þ J
; (4.41)
aniso
where the first term describes the isotropic part of the material and the second term
involves additional rigidity of particular directions.
Let an anisotropic incompressible material have one family of fibers whose direction is
given by a unit vector a in the undeformed reference configuration. If the material
demonstrates rotational symmetry about the direction of a, then it is called locally
transversally anisotropic. Elastic potential J of locally transversally anisotropic material is
a function of the right CauchyeGreen tensor C and the structure tensor a5a.
Incompressibility of the material (I
of three) invariants I
, I2, whereas dependence on a5a leads to dependence on two
1
invariants of the structure tensor, I
¼ J2¼ 1) yields dependence of J on two (instead
3
¼ a$ðCaÞ, I5¼ a$C2a:
4
According to Eq. (4.39), the Cauchy stress is given by
S ¼pI þ 2J
JðFÞ¼JðI
B þ 2J
1
2
; I2; I4; I5Þ¼J
1
I
B B
1
2
þ 2J
ðI1; I2ÞþJ
iso
4aF5aF
ðI4; I5Þ:
aniso
þ 2J5ðaF5 BaFþBaF5 aFÞ;
(4.42)
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