Unsteady flow of a generalized Oldroyd-B fluid using a
director theory approach
FERNANDO CARAPAU
Universidade de Évora
Dept. Matemática and CIMA/UE
R. Romão Ramalho, 59, 7001-651, Évora
PORTUGAL
[email protected]
ADÉLIA SEQUEIRA
Instituto Superior Técnico
Dept. Matemática and CEMAT/IST
Av. Rovisco Pais, 1049-001, Lisboa
PORTUGAL
[email protected]
Abstract: Using a 1D model based on the Cosserat theory which reduces the exact three-dimensional
equations to a system of partial differential equations depending on time and on a single spatial variable,
we analyze the axisymmetric unsteady flow of an incompressible viscoelastic generalized Oldroyd-B fluid
with shear-dependent viscosity. In this study we consider an uniform rectilinear pipe with circular crosssection and give the relationship between average pressure gradient and volume flow rate over a finite
section of the pipe and the corresponding equation of the wall shear stress. An application to blood flow
is also given in the steady case.
Key–Words: Cosserat theory, generalized Oldroyd-B fluid, steady solution, shear-thinning flow, volume
flow rate, average pressure gradient, blood flow.
1
Introduction
Blood is a very complex fluid and the mathematical modeling of blood flow is a difficult and
challenging problem. Blood is a suspension of
particles (red blood cells, white blood cells and
platelets) in a fluid called plasma and the vessels
can be regarded as elastic (or mildly viscoelastic) tubes with different scales. Blood flow interacts with the vessel wall and can be modeled using a fluid-structure interaction problem (see e.g.
[20]). In large arteries blood may be considered
as an homogeneous fluid, with the standard Newtonian behavior. However, blood exhibits nonNewtonian charateristics mainly due to shearthinning viscosity (i.e. the viscosity decreases
with increasing shear rate) and viscoelasticity related to stress relaxation and normal stress difference effects. In particular, in the case of small arteries, arterioles and venules, the microstructure
and rheological behavior of blood should not be
neglected since the dimension of the blood particles are of the same order of that of the vessels.
Phenomena like aggregation and deformability of
red blood cells have great influence on the rheological behavior of blood, specially on its viscosity
at low shear rates, and blood can be modelled as
an homogeneous shear-thinning and viscoelastic
fluid (see e.g. [3], [22], [23]).
In this paper we introduce a one-dimensional
model for viscoelastic non-Newtonian generalized
Oldrody-B flows in an axisymmetric pipe with
circular cross-section, based on the director approach (Cosserat theory) with nine directors developed by Caulk and Naghdi [9]. The theoretical basis of this approach (see Cosserat [10], [11],
based on the work of Duhen [12]) is to consider an
additional structure of deformable vectors (called
directors) assigned to each point on a space curve
(the Cosserat curve). With this approach and integrating the axial component of linear momentum for the flow field over the pipe cross-section,
the 3D system of equations is replaced by a system of partial differential equations which, apart
from the dependence on time, depends only on a
single spatial variable. Using this one-dimensional
Cosserat theory we can predict some of the main
properties of the three-dimensional problem. For
additional background information, the Cosserat
theory has been used in studies of rods, plates
and shells, see e.g. Ericksen and Truesdell [13],
Truesdell and Toupin [24], Green et al. [17], [16]
and Naghdi [19]. Later, this theory has been
developed by Caulk and Naghdi [9], Green and
Naghdi [18], and Green et al. [15] in studies of
unsteady and steady flows, related to fluid dynamics. Recently, the nine-director approach has
been applied to blood flow in the arterial system
by Robertson and Sequeira [21] and also by Cara-
pau and Sequeira [6], [7], [8], considering Newtonian and non-Newtonian flows.
In this work we study the initial boundary
value problem of an incompressible homogeneous
generalized Oldroyd-B fluid model in a straight
circular rigid and impermeable small vessel with
constant radius (see e.g. [4], [5] for further discussion) where the fluid velocity field given by the
director theory can be approximated by the following finite series1:
v∗ = v +
k
X
xα1 . . . xαN W α1 ...αN ,
(1)
N =1
Figure 1: Fluid domain Ω with the components of the
surface traction vector τ1 , τ2 and pe . Γw is the lateral wall
of the pipe with equation φ(z, t), and Γ1 , Γ2 are the upstream part and downstream districts of the pipe, respectively.
with
v = vi (z, t)ei , W α1 ...αN = Wαi 1 ...αN (z, t)ei . (2)
Here, v represents the velocity along the axis of
symmetry z at time t, xα1 . . . xαN are the polynomial weighting functions with order k (the number k identifies the order of hierarchical theory
and is related to the number of directors), the
vectors W α1 ...αN are the director velocities which
are completely symmetric with respect to their
indices and ei are the associated unit basis vectors. From this velocity field approach, we obtain
the unsteady relationship between average pressure gradient and volume flow rate, and the correspondent equation for the wall shear stress.
Our goal is to develop a nine-director theory
(k = 3 in (1)) for the steady flow of a generalized
Oldroyd-B fluid in a straight pipe with constant
radius, to compare the average pressure gradient
for different values of both Reynolds and Weissenberg numbers.
at z and time t. The components of the threedimensional equations governing the motion of
a generalized Oldroyd type fluid (with sheardependent viscosity) are given in Ω × (0, T ) by2
 ∗
∂v

ρ

+ v,i∗ v∗i = ti,i ,


∂t






∗

 vi,i = 0,


ti = −p∗ ei + σij ej , t = ϑ∗i ti ,








λ2 O

 σeij + λ1 σ
Dij ,
=
2
µ(|
γ̇|)
−
eij
λ1
with the initial condition
v ∗(x, 0) = v 0 (x) in Ω,
Formulation
Problem
of
the
Model
Let us consider a homogeneous fluid inside a circular straight and impermeable pipe, the domain
Ω ⊂ R3 , with boundary ∂Ω composed by the
proximal cross-section Γ1 , the distal cross-section
Γ2 and the lateral wall Γw , see Fig.1.
Let xi (i = 1, 2, 3) be the rectangular Cartesian coordinates and for convenience set x3 = z.
Consider the axisymmetric motion of an incompressible fluid without body forces, inside a surface of revolution, about the z axis and let φ(z, t)
denote the instantaneous radius of that surface
1
Latin indices subscript take the values 1, 2, 3, Greek
indices subscript 1, 2. Summation convention is employed
over a repeated index.
(4)
and the boundary condition
v∗ (x, t) = 0 on Γw × (0, T ),
2
(3)
(5)
where v ∗ = vi∗ei is the velocity field, ρ is thecon∗ + v∗
stant fluid density and Dij = 12 vi,j
j,i are
the components of the rate of deformation tensor
D. Equation (3)1 represents the balance of linear
momentum and (3)2 is the incompressibility condition. In equation (3)3, p∗ is the pressure and
σij are the components of the (symmetric) extra
stress tensor given by
∗
∗
σij = µn vi,j
+ σeij ,
+ vj,i
where σeij are the components of its viscoelastic part. Here t denotes the stress vector on the
surface whose outward unit normal is ϑ∗ = ϑ∗i ei ,
2
∗
∗ ∗
We use the notation vi,j
= ∂vi∗ /∂xj and v,i
vi =
vi∗ ∂v ∗ /∂xi adopted in Naghdi et al. [9], [14].
and ti are the components of t. In equation (3)4
O
the symbol σeij represents the objective Oldroyd
derivative of the tensor σeij given by (see e.g. [25])
O
σeij
∂σeij
∂σeij
∗
∗
+ vk∗
+ σeik vk,j
− vj,k
∂t
∂xk
h
∗
∗
∗
∗
− vi,k
σekj − a vi,k
σekj
− vk,i
+ vk,i
i
∗
∗
,
(6)
+ σeik vk,j
+ vj,k
=
where a ∈ [−1, 1] is a real given parameter,
µ(|γ̇|) : R+ → R+
is the shear-dependent viscosity function
and γ̇
√
is the shear rate such that |γ̇| := 2D : D. The
initial velocity field v 0 is assumed to be known.
Finally, µn = (µλ2)/λ1 is the Newtonian viscosity and µ = µn + µe denotes the viscosity coefficient with µe = µ 1 − λ2 /λ1 = µλ the elastic
viscosity, and λ1 and λ2 (with 0 6 λ2 < λ1) are
the relaxation and the retardation times, respectively. Models with λ2 = 0 are called of ”Maxwell
type” and those with λ2 > 0 of ”Jeffreys type”.
Oldroyd-B fluids correspond to Jeffreys type fluids with a = 1 (in equation (6)) and Oldroyd-A
fluids correspond to a = −1.
The lateral surface Γw of the axisymmetric
domain is defined by
φ2 = xα xα ,
(7)
and the components of the outward unit normal
to this surface are
ϑ∗α =
xα
φ 1 + φ2z
∗
1/2 , ϑ3 = −
φz
1 + φ2z
1/2 ,
(8)
where a subscript variable denotes partial differentiation. Since equation (7) defines a material
surface, the velocity field must satisfy the condition3
d 2
φ − xα xα = 0.
dt
As a consequence
φφt + φφz v3∗ − xα vα∗ = 0
(9)
at the boundary (7).
Let us consider S(z, t) as a generic axial section of the domain at time t defined by the spatial
variable z and bounded by the circle defined in (7)
3
d
(·) is the material time derivative.
dt
and let A(z, t) be the area of this section S(z, t).
The volume flow rate Q is defined by
Z
Q(z, t) =
v3∗(x1 , x2, z, t)da,
(10)
S(z,t)
and the average pressure p̄ is defined by
Z
1
p̄(z, t) =
p∗(x1 , x2, z, t)da.
A(z, t) S(z,t)
(11)
In what follows, this general framework will be
applied to the specific case of the nine-director
theory in a rigid pipe, i.e. φ = φ(z). Using condition (1), with k = 3, it follows from Caulk and
Naghdi [9] that the approximation for the threedimensional velocity field v∗ is given by
h
x2 + x2 2φz Q i
v ∗ = x1 1 − 1 2 2
e1
φ
πφ3
h
x2 + x2 2φz Q i
e2
+ x2 1 − 1 2 2
φ
πφ3
h 2Q
x21 + x22 i
1
−
e3
+
(12)
πφ2
φ2
where the volume flow rate Q(t) is
π
Q(t) = φ2 (z)v3(z, t).
(13)
2
We remark that the initial condition (4) is satisfied when Q(0) = ct. The stress vector on the
lateral surface Γw in terms of the outward unit
normal and tangential components (see Fig.1) is
given by
tw = τ1 λ − pe ϑ∗ + τ2eθ ,
(14)
where λ, eθ are unit tangent vectors defined by
xα
eθ =
(15)
eαβ eβ , λ = ϑ∗ × eθ ,
φ
so that λ, ϑ∗, eθ form a right-handed triad, with
eαβ the permutation symbol defined by
e11 = e22 = 0, e12 = −e21 = 1.
Now, with the help of equations (8) and (15), the
expression for the stress vector (14) can be rewritten as
h
1
tw =
τ 1 x1 φz − p e x1
φ(1 + φ2z )1/2
i
− τ2 x2 (1 + φ2z )1/2 e1
h
1
τ 1 x2 φz − p e x2
+
φ(1 + φ2z )1/2
i
+ τ2 x1 (1 + φ2z )1/2 e2
i
h
1
τ
e3. (16)
+
+
p
φ
1
e
z
(1 + φ2z )1/2
where τ1 represents the wall shear stress in the axial direction of the flow. Instead of satisfying the
momentum equation (3)1 pointwise in the fluid,
we impose the following integral conditions
Z
h
i
∂v∗
ti,i − ρ
(17)
+ v ∗,i vi∗ da = 0,
∂t
S(z,t)
Z
h
∂v∗
ti,i − ρ
∂t
S(z,t)
+ v∗,i vi∗
i
xα1 . . . xαN da = 0,
(18)
where N = 1, 2, 3.
Using the divergence theorem and integration
by parts, equations (17) − (18) for nine directors,
can be reduced to the four vector equations:
∂n
+ f = a,
∂z
(19)
∂mα1 ...αN
+ lα1 ...αN = kα1 ...αN + bα1 ...αN , (20)
∂z
where n, k α1 ...αN , mα1 ...αN are resultant forces
defined by
Z
Z
α
n=
t3 da, k =
tα da,
(21)
S
tα xβ + tβ xα da,
kαβγ =
(22)
S
Z tα xβ xγ + tβ xα xγ + tγ xαxβ da, (23)
S
mα1 ...αN =
Z
t3 xα1 . . . xαN da.
(24)
S
The quantities a and bα1 ...αN are inertia terms
defined by
Z ∗
∂v
a=
(25)
ρ
+ v ∗,i vi∗ da,
∂t
S
Z ∗
∂v
α1 ...αN
b
=
ρ
+ v∗,i vi∗ xα1 . . . xαN da, (26)
∂t
S
and f , lα1 ...αN , which arise due to surface traction
on the lateral boundary, are defined by
Z 1/2
f=
1 + φ2z
tw ds,
(27)
∂S
l
α1 ...αN
=
Z
Results and Discussion
In this section we apply the director theory approach with nine directors to both Oldroyd-B and
generalized Oldroyd-B models, to capture the effects of the shear-dependent viscosity in a straight
circular rigid and impermeable walled pipe with
constant radius, i.e. φ = ct. As will be seen later
equation (3)4 introduces some difficulties in handling the general case φ = φ(z).
3.1
Oldroyd-B fluids
Using a constant viscosity µ0 in equation (3)4,
we obtain the Oldroyd type fluid model. We Replace the solutions of equations (21) − (28) into
equations (19) − (20) to obtain the relationship
between average pressure and volume flow rate in
a rigid axisymmetric straight pipe with constant
radius φ, given by
1 + φ2z
1/2
tw xα1 . . . xαN ds. (28)
∂S
The equation relating the average pressure gradient with the volume flow rate will be obtained
using these quantities (21) − (28).
8µn
4ρ
Q̇(t)
Q(t) −
πφ4
3πφ2
2
4
(ψ33)z − 4 ($33)z
φ2
φ
1
4
(ψ11)z + 4 ($11)z ,
(29)
φ2
φ
p̄z (z, t) = −
+
−
S
Z kαβ =
3
where the functions ψij and $ij are defined by4
Z
Z
β
ψij =
σeij da, $ij δα =
σeij xα xβ da,
S
S
the viscoelastic part of the stress tensor (due to
compatibity conditions) takes the particular form


0
0
σe11
σe =  0
(30)
σe11
0 ,
0
0
σe33
and, the corresponding wall shear stress τ1 is given
by
τ1 =
+
−
4µn
ρ
1
Q(t) +
Q̇(t) −
(ψ33)z
3
πφ
6πφ
2πφ
2
1
($33)z +
(ψ11)z
3
πφ
2πφ
2
($11)z .
(31)
πφ3
We consider now the following dimensionless variables5
x
x̂ = , φ̂ = 1, t̂ = ω0 t,
φ
4 β
δα
5
is the two-dimensional Kronecker symbol.
In cases where a steady flow rate is specified, the nondimensional flow rate Q̂ is identical to the classical Reynolds
number used for flow in pipes, see [21].
Q̂ =
2ρ
φ2 ρ
φ2 ρ
Q, p̄ˆ = 2 p̄, σˆe = 2 σe ,
πφµ
µ
µ
where ω0 is a characteristic frequency for unsteay
flow. Substituting these dimensionless variables
into equations (29) and (3)4, we obtain, respectively
2
˙
= −4 1 − λ Q̂(t̂) − W02 Q̂(t̂) + 2 ψ̂33 ẑ
3
− 4 $̂33 ẑ − ψ̂11 ẑ + 4 $̂11 ẑ ,
(32)
p̄ˆẑ
and
O
σ̂eij
+ We σ̂ eij = 2λD̂ij
(33)
p
where W0 = φ2 (ρω0)/µ is the Womersley number, which reflects the unsteady flow phenomena,
We = λ1ω0 is the Weissenberg number, related
with the flow viscoelasticity and
Dij =
µ
D̂ij .
φ2 ρ
Substituting the given dimensionless variables
into equation (31), we obtain
τ̂1
fluid model. From, (30) and (33) (dimensionless
forms), we obtain
σˆe11 = σˆe22 = 0,
φ2 ρ
τ̂1 = 2 τ1.
µ
Integrating condition (32) over the interval [ẑ1, ẑ],
with ẑ1 fixed, we obtain the following relationship between average pressure gradient and volume flow rate
ˆ
pp(ẑ,
t) = p̄ˆ(ẑ, t) − p̄ˆ(ẑ1, t)
(35)
= 4 1 − λ Â1 (ẑ) Q̂(t̂)
2 2
˙
+
W0 Â2 (ẑ) Q̂(t̂)
3
+ 2 ψ̂33(ẑ, t) − ψ̂33(ẑ1, t) B̂4 (ẑ)
+ 4 $̂33(ẑ, t) − $̂33(ẑ1, t) B̂5 (ẑ)
+ ψ̂11(ẑ, t) − ψ̂11(ẑ1, t) B̂6 (ẑ)
+ 4 $̂11(ẑ, t) − $̂11(ẑ1, t) B̂7 (ẑ),
where Â1 (ẑ) = Â2 (ẑ) = B̂5 (ẑ) = B̂6 (ẑ) = ẑ1 − ẑ
and B̂4 (ẑ) = B̂7 (ẑ) = ẑ − ẑ1 . Now, considering
equations (34) and (35) in the steady case and
fixing a = 1 in (6) we deal with the Oldrody-B
(36)
and
1 ˙
1
= 2 1 − λ Q̂(t̂) + W02 Q̂(t̂) −
(ψ̂33)ẑ
12
2π
2
1
2
+
($̂33)ẑ +
(ψ̂11)ẑ − ($̂11)ẑ , (34)
π
2π
π
where
Figure 2: Nondimensional average pressure gradient (35)
in the steady case of an Oldroyd-B fluid for different values
of the Reynolds number (Q̂s = (0.001, 0.5, 1, 5, 10, 20)) and
Weissenberg number (We = (0.01, 0.1, 1, 50, 100)).
σˆe33 = exp
. (37)
ẑ
We Q̂s x̂21 + x̂22 − 1
Using (36), (37) and the approximation
exp
ẑ
We Q̂s ζ 2 − 1
' exp −
−
ζ2
We Q̂s
ẑ
We Q̂s
exp −
ẑ
We Q̂s
we get ψ̂11 = $̂11 = 0,
ẑ 1 πẑ
ẑ ψ̂33 = π exp −
−
,
exp −
2 We Q̂s
We Q̂s
We Q̂s
and
$̂33 =
−
1
ẑ π exp −
4
We Q̂s
1 πẑ
ẑ .
exp −
6 We Q̂s
We Q̂s
Again due to compatibility conditions, these results are only valid when λ ' 0, i.e. λ1 ' λ2.
Shown in Fig.2 is the normalized nine-director average pressure gradient steady solution (35) for an
Oldroyd-B fluid for different values of Reynolds
and Weissenberg numbers in [0, ẑ]. We conclude
,
havior of blood over a large range of shear rates
but it has its limitations, since the relaxation
times do not depend on the shear rate, which
does not agree with experimental observations
(see e.g. [1]). More recently, Arada and Sequeira
[2] have studied the existence of strong solutions
for the steady case of a generalized Oldroyd-B
fluid, where the viscosity function satisfies some
regularity and growing conditions, and is given by
the general expression
q
µ(|γ̇|) = µ∞ + µ0 − µ∞ 1 + |γ̇|2 .
Figure 3: Nondimensional wall shear stress (34) of an
Oldroyd-B fluid in the steady case for different values of
the Reynolds number (Q̂s = (5, 10, 20)) and Weissenberg
number (We = (0.0001, 0.001, 0.01, 0.1)).
that the behavior of the steady solution with fixed
Reynolds number does not change when we increase the Weissenberg number. However, with
fixed Weissenberg number we can observe a slight
change of the steady solution behavior, with increasing Reynolds number. Also, we compare the
corresponding wall shear stress (34) for different
values of the Reynolds and Weissenberg numbers,
see Fig.3, and conclude that it undergoes a small
perturbation for ẑ close to zero and We 0.001.
However, for higher values of the Weissenberg
number the wall shear stress becomes constant.
3.2
Generalized Oldroyd-B fluids
The steady generalized Oldroyd-B fluid model
(a = 1 in the objective Oldroyd derivative (6))
has been studied by some authors, using analytical and numerical approaches for different viscosity functions, in particular in view of applications
to haemodynamics. For instance, Yeleswarapu el
al. [26], used the shear-thinning viscosity function
given by
µ(|γ̇|) = µ∞ + µ0 − µ∞
h 1 + ln(1 + k|γ̇|) i
,
1 + k|γ̇|
where k is a positive material constant, and µ0 ,
µ∞ (with µ∞ < µ0 ) are the asymptotic apparent viscosities as |γ̇| → 0 and |γ̇| → ∞, respectively. This viscosity function has been obtained
by fitting experimental data in uni-directional
flows and generalizing such curve fits to threedimensions. It captures the shear-thinning be-
If q = 0, the Oldroyd-B fluid with constant viscosity is recovered. Moreover, if q < 0 the viscosity has shear-thinning behavior and is shearthickening (viscosity increases with shear rate) if
q > 0. This general model can also be considered
as a good approach of blood flow in small and
medium sized arteries where the shear-thinning
and viscoelastic behavior of blood should not be
omitted, see e.g. [22].
In our work, we consider this general viscosity
function µ(|γ̇|) with asymptotic apparent viscosities µ0 and µ∞ as |γ̇| → 0 and |γ̇| → ∞, respectively. Using equation (3)4, i.e.
λ2 ∗
O
∗
σeij + λ1 σ eij = µ(|γ̇|) −
vi,j + vj,i
λ1
we can relate the viscoelastic components σeij
with the velocity field (12), and with λ1 and λ2,
for different values of a ∈ [−1, 1]. Now, with the
nondimensional variables already defined in last
section we get the same solution (32) for the pressure gradient, the same solution (34) for the wall
shear stress, and
1
+ We σ̂ eij = 2
µ |γ̇| − 1 − λ D̂ij (38)
µ0
O
σ̂eij
where the viscoelastic part of the stress tensor
(due to compatibity conditions) takes the particular form


0
0
σ̂e11
σ̂eij =  0
(39)
σ̂e11
0 .
0
0 σ̂e33
From results (32), (38) and (39) in the steady
case with fixed a = 1 in (6) we obtain the same solution (35) of the average pressure gradient in the
steady flow, with the same values for the functions
σˆe11 , σˆe33 and consequently for ψ̂11, ψ̂33, $̂11, $̂33.
Again due to compatibility conditions between
Figure 4: Nondimensional average pressure gradient
(35) in the steady case of a generalized Oldroyd-B fluid
for different values of the Reynolds number (Q̂s =
(0.001, 0.5, 1, 5, 10, 20)) and Weissenberg number (We =
(0.01, 0.1, 1, 50, 100)).
Figure 5: Nondimensional wall shear stress (34) of a generalized Oldroyd-B fluid in the steady case for different
values of the Reynolds number (Q̂s = (5, 10, 20)) and Weissenberg number (We = (0.0001, 0.001, 0.01, 0.1)).
(38) and (39), these results for steady flow in a
uniform rectilinear pipe are only valid when
On the other hand for generalized Oldroyd-B fluids with shear-dependent viscosity, the directors
approach is only possible with
1
µ |γ̇| − 1 − λ ' 0.
µ0
1
µ |γ̇| − 1 − λ ' 0.
µ0
In the limiting viscosity condition |γ̇| → 0,
i.e. λ ' 0, we recover the Oldroyd-B fluid results.
Here we present the results when |γ̇| → ∞, i.e.
In the limit viscosity case |γ̇| → 0 (i.e. λ ' 0)
we recover the Oldroyd-B fluid results. However,
when |γ̇| → ∞, i.e.
λ ' 1−
µ∞
µ0
with the relevant parameters related to blood flow
µ0 = 0.56 poise and µ∞ = 0.0345 poise (see
e.g. [26] and cited references). Fig.4 shows that
the nondimensional nine-director average pressure
gradient steady generalized Oldroyd-B solution
(35) for different values of Reynolds and Weissenberg numbers in [0, ẑ] has the same qualitative
behavior as the Oldroyd-B solution, when |γ̇| →
∞, Fig.2. Analyzing the results of Fig.5 and Fig.3
for the steady nondimensional wall shear stress,
we also conclude the same qualitative behavior.
4
Conclusion
Unlike Newtonian, generalized Newtonian and
second order fluids (see e.g. [9], [21], [6], [7], [8],
respectively) where the 1D director approach has
been applied without restrictions to rectilinear
flows, in the case of Oldroyd-B fluids, the 1D theory is only possible when the relaxation and retardation times are close to each other, i.e. λ1 ' λ2.
λ' 1−
µ∞
,
µ0
the results for generalized Oldroyd-B and
Oldroyd-B fluids are qualitatively similar. The
effects of the shear-thinning viscosity are mainly
visible in terms of quantitative results. One of
the possible extensions of this work is the application of the 1D director approach to blood flow
in compliant straight and curved vessels.
Acknowledgements:
This
work
has
been
partially
supported
by
projects
POCTI/MAT/41898/2001,
HPRN-CT-200200270 of the European Union and by the research
centers CEMAT/IST and CIMA/UE, through
FCT´s funding program.
References:
[1] M. Anand and K. R. Rajagopal, A shearthinning viscoelastic fluid model for describing the flow of blood, Int. J. Cardiovascular
Medicine and Science, 4, 2004, pp. 59-68.
[2] N. Arada, and A. Sequeira, Strong Steady
Solutions for a generalized Oldroyd-B
Model with Shear-Dependent Viscosity in a
Bounded Domain, Mathematical Models &
Mehods in Applided Sciences (M3AS), 13,
no.9, 2003, pp. 1303-1323.
[3] O.K. Baskurt, and H.J. Meiselman, Blood
rheology and hemodynamics, Seminars in
Trombosis and Hemostasis, 29, 2003, pp.
435-450.
[4] F. Carapau, Development of 1D Fluid Models
Using the Cosserat Theory. Numerical Simulations and Applications to Haemodynamics,
PhD Thesis, IST, 2005.
[5] F. Carapau, and A. Sequeira, Unsteady flow
of Oldroyd-B fluids in an uniform rectilinear pipe using 1D models, Proceedings of the
WSEAS Int. Conf. on Continuum Mechanics
CM’06, 2006, in press.
[6] F. Carapau, and A. Sequeira, Axisymmetric flow of a generalized Newtonian fluid in
a straight pipe using a director theory approach, Proceedings of the 8th WSEAS Int.
Conf. on Applied Mathematics, 2005, pp.
303-308.
[7] F. Carapau, and A. Sequeira, 1D Models for
blood flow in small vessels using the Cosserat
theory, WSEAS Transactions on Mathematics, Issue 1, 5, 2006, pp. 54-62.
[8] F. Carapau, and A. Sequeira, Axisymmetric
motion of a second order viscous fluid in a circular straight tube under pressure gradients
varying exponentially with time, Proceedings
of the Sixth Wessex Int. Conf. on Advances
in Fluid Mechanics, May 2006, in press.
[9] D.A. Caulk, and P.M. Naghdi, Axisymmetric motion of a viscous fluid inside a slender
surface of revolution, Journal of Applied Mechanics, 54, 1987, pp. 190-196.
[10] E. Cosserat, and F. Cosserat, Sur la théorie
des corps minces, Compt. Rend., 146, 1908,
pp. 169-172.
[11] E. Cosserat, and F. Cosserat, Théorie des
corps déformables, Chwolson´s Traité de
Physique, 2nd ed. Paris, 1909, pp. 953-1173.
[12] P. Duhem, Le potentiel thermodynamique et
la pression hydrostatique, Ann. École Norm,
10, 1893, pp. 187-230.
[13] J.L. Ericksen, and C. Truesdell, Exact theory
of stress and strain in rods and shells, Arch.
Rational Mech. Anal., 1, 1958, pp. 295-323.
[14] A.E. Green, and P.M. Naghdi, A direct theory of viscous fluid flow in pipes: I Basic
general developments, Phil. Trans. R. Soc.
Lond. A, 342, 1993, pp. 525-542.
[15] A.E. Green, and P.M. Naghdi, A direct theory of viscous fluid flow in pipes: II Flow of
incompressible viscous fluid in curved pipes,
Phil. Trans. R. Soc. Lond. A, 342, 1993, pp.
543-572.
[16] A.E. Green, N. Laws, and P.M. Naghdi,
Rods, plates and shells, Proc. Camb. Phil.
Soc., 64, 1968, pp. 895-913.
[17] A.E. Green, P.M. Naghdi, and M.L. Wenner,
On the theory of rods II. Developments by
direct approach, Proc. R. Soc. Lond. A, 337,
1974, pp. 485-507.
[18] A.E. Green, and P.M. Naghdi, A direct theory of viscous fluid flow in channels, Arch.
Ration. Mech. Analysis, 86, 1984, pp. 39-63.
[19] P.M. Naghdi, The Theory of Shells and
Plates, Flügg´s Handbuch der Physik,
Vol. VIa/2, Berlin, Heidelberg, New York:
Springer-Verlag, pp. 425-640, 1972.
[20] A. Quarteroni, and L. Formaggia, Modelling
of living systems, in: Mathematical Modelling and Numerical Simulation of the Cardiovascular System, Handbook of Numerical Analysis Series, ed. N. Ayache, Elsevier,
2003, pp. 83-88.
[21] A.M. Robertson, and A. Sequeira, A Director
Theory Approach for Modeling Blood Flow
in the Arterial System: An Alternative to
Classical 1D Models, Mathematical Models &
Methods in Applied Sciences, 15, nr.6, 2005,
pp. 871-906.
[22] A. Sequeira and J. Janela, An overview of
some mathematical models of blood rheology, in A portrait of research at the Technical University of Lisbon (M.S. Pereira Ed.),
Springer, 2006, in press.
[23] G.B. Thurston, Viscoelasticity of human
blood, Biophys. J., 12, 1972, pp. 1205-1217.
[24] C. Truesdell, and R. Toupin, The Classical
Field Theories of Mechanics, Handbuch der
Physik, Vol. III, pp. 226-793, 1960.
[25] J.G. Oldroyd, On the Formulation of Rheological Equations of State, Proc. Roy. Soc.
London, A200, 1950, pp. 523-541.
[26] K.K. Yeleswarapu, Evaluation of Continuum
Models for Characterizing the Constitutive
Behavior of Blood, Phd thesis, University of
Pittsburgh, 1996.