C. R. Acad. Sci. Paris, t. 332, Série I, p. 1–??, 2001
Rubrique/Heading
(Sous-rubrique/Sub-Heading)
- PXMA????.TEX -
Effective equations describing the flow of a viscous
incompressible fluid through a long elastic tube
Sunčica Čanić a , Andro Mikelić b
a
Department of Mathematics, University of Houston, Houston TX 77204-3476, UNITED STATES
E-mail:canic@math.uh.edu
b
UFR Mathématiques, Université Claude Bernard Lyon 1, Bât. 101, 43, bd. du 11 novembre 1918, 69622
Villeurbanne Cedex, FRANCE
E-mail:andro@lan.univ-lyon1.fr
(Reçu le jour mois année, accepté après révision le jour mois année)
Abstract.
We study the flow of a viscous incompressible fluid through a long and narrow elastic tube
whose walls are modeled by the Navier equations for a curved, linearly elastic membrane.
The flow is governed by a given small time dependent pressure drop between the inlet and
the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which
correspond to a Biot type viscoelastic equation for the effective pressure and the effective
displacement. The approximation is rigorously justified by obtaining the error estimates for
the velocity, pressure and displacement. Applications of the model problem include blood
flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. c 2001 Académie
des sciences/Éditions scientifiques et médicales Elsevier SAS
Equations efficaces décrivant l’écoulement d’un fluide visqueux incompressible
dans un tuyau long et élastique
Résumé.
Nous considérons l’écoulement d’un fluide incompressible visqueux à travers un tuyau long
et de faible épaisseur, ayant la paroi latérale obéissant aux équations de Navier pour une
membrane courbe élastique linéaire.L’écoulement est régi par une petite chute de pression entre l’entrée et la sortie du tuyau et on a un écoulement lent décrit par les équations
de Stokes. En utilisant la méthode des développements asymptotiques, avec la réduction
dimensionnelle dans la partie mince , nous obtenons les équations limites. Elles correspondent aux équations viscoélastiques de Biot pour la pression efficace et les déplacements
efficaces. L’approximation est justifiée rigoureusement en obtenant une estimation d’erreur pour la vitesse , la pression et les déplacements ” dilatés ”. Des applications incluent
l’écoulement sanguin dans des arterioles. Nous retrouvons la bien connue Loi de Laplace
et donnons un nouveau modèle amélioré lorsque le module de cisaillement de la paroi n’est
pas négligeable. c 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Note présentée par First name NAME
S0764-4442(00)0????-?/FLA
c 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
1
Elastic wall dynamics
Version française abrégée
Nous considérons l’écoulement quasi-statique axisymétrique d’un fluide Newtonien incompressible à
travers un cylindre droit, à petite épaisseur,
"!#%$'&()+*-,/.+102*3$4*357698
(1)
La vitesse du fluide : et la pression ; satisfont les équations de Stokes dans des coordonnées cylindriques
D
D
D
D
:
:
;
:
:
<#=?>A@CB @CB $
@ L0 dans NMOP?
@ < (2)
@ BFE @ BGEIH @ BJKE @ <#=?> @CB : Q @CB $ : Q @ : Q
@ ; $ L0 et @ : D @ : $ Q : D L0 dans RNMPS8
(3)
@ BFE @ BGE H @ J-E @
@ E @ E
VUWXG,/.2YCMZ[09%5&
Nous supposons que la paroi latérale du cylindre T
est élastique et que sa déformation
est décrite par les équations de Navier
,
h
&
i
g
^
,
%
&
d
j
^
,
&
^
,
&
\ D ] ^,&_`^,& > ,/a . @dc $
< ]
@CB $
] @B , .
(4)
<ba
@ e B3Elk'm
@ E B e BfJ @on e B
B ^,&
^
,
&
`
_
a @
\ Q < H]
>
B
@
c
] ,& @ B c 8
$
p
,
. $
(5)
<ba
@ B E
@on B
B
@ e J Elkm
H ^,& radial et c le déplacement longitudinale dans_qlesrcoordonnées
Dans (4)-(5),
est le déplacement
de
_`,&
]
]
Lagrange 0`
(voir
Figure
1),
est
l’épaisseur
de
la
membrane,
sa
densité
,
le
module
* ea *
grsgi^,&
k9m cisaillement et jOVjd^,& le facteur
d’Young,
le coefficient de Poisson,
le module de
\
\
de correction de Timoshenko (voir [7]). D et Q sont les composantes radiale et longitudinale de la force
H
exercée par le fluide sur la paroi et données par
\ Du t D \ Q9u t Q Iv ; xw <byp={z : &| u t D }f~ T NMPS
(6)
&
E
où z :
est le tenseur des vitesses de déformation. La vitesse du fluide : est liée avec la vitesse de la
paroi par
@Cc sur T MO P 8
:D @
et : Q
(7)
@oe n
@n
Nous ajoutons au système (1)-(7) les conditions aux limites (13)-(16).
On va obtenir les équations asymptotiques qui décrivent l’écoulement à l’échelle du temps dominante
€ et

aussi les oscillations
de
la
membrane
induites
par
le
réponse
du
matériau
élastique.
Nous
posons
n
n
€ ,B
avec
d’énergie (2.1) , qui nous permet d’obtenir les estimations a
= . Ce choix conduit à l’égalité
^
,
h
&
_Z‚7%ƒ„"…† ‚ ] , _`,& gX‚7%ƒ"„…† ‚ gi,& ] ^,&jd,&‡,
priori pour la solution. Soient
et
. Nous utilisons l’approche
^
,
&ˆ$9%/ &+
[$oh,// &
[$o[2])ou
:
de^,Ciarlet
n
n et
&ˆ$9%/ et&
al. (cf
h,// & onremplace les inconnues par leurs ” dilatées
‰Š7‹” :
;
;
.
devient
le
domaine
”
non-mince
”
.
La
vitesse
et
la
pression
”
n
n
dilatées ” satisfont les estimations a priori (18)-(20). Ensuite nous cherchons les équations efficaces en
utilisant les développements asymptotiques suivantes
^,&ˆ$9%/ A& , B ‚ , Œ
n = Œ T :
[$o &A,
n
e
Œ [$o/ & ,p&x[$o/ &A ‚ [ $o &
ΠΠ[$o/ &
;
;
Œ[T ‹Ž ;
n
n
n
n
ΠΠ[$o & [$o &A
ΠΠ[E o$ &(
Œ[T ‚oŽ
n
c
n ŒT ‚oŽ c
n
e
‚
qui nous donnent le problème parabolique suivant pour la pression efficace ;
;
:
(8)
(9)
2
Elastic wall dynamics
PETIT
*),+ +-) ),.
*/&+ 10 p(z,0) =0
NON-NÉGLIGEABLE
" !# ' (
&
$
%$
* ),+ )2. */#+ 10 (
!
3 *),+ )2. "!& */#+ 54 4 0 ! 7 6 98 ! ;:=< ?> ?> ED
>
"
#
!
"
#
!
(10)
A@C B +
3 GF &
!
H +
5@ B ;:
(11)
N
K
O
K
K
6
I J +-K+ 3 ,P
L + : 3 + + I M K :
LK
(12)
g ‚ L0
Notons que
, on obtient la loi (21), liant la pression efficace ; et le déplacement radial efficace
pour
y (21) est exactement la Loi de Laplace (voir [4]).
. Pour a
US#TVU XWYTVU Y
Nous justifions le problème efficace par une estimation d’erreur. Soit
la solution pour les
e
HRQ
équation de la couche limite (22)-(23). Alors on a
,p&A ,p& <
W , [ ,& , = : ^,& < : Q‚ u t Q < , : D‚ u t D S , ^,& < ‚
Z
;
;
T HEOREM
0.1.
–
Soient
c
c
c
,&A < , ‚
E
E
B
et
. Alors on a
\ ^,& \]^_ ‚` acb dfe \ ^,& \
]^_hgji%_ ‚k` ale(enmpo , 7q e
e
e
[
Z
B
^
,
&
E
‚ksl}t"sr avuxw g ‚ \ @ , $ & \
]^_ ‚k` ] e w _ ‚ \ ^, ,& & \
]^_ ‚k` ] e \ @Cc $ ,p& & \]^_ ‚` ] eRy=mzo , 7q 8
B
n
n
n
@
e@
e
E
E
et les expressions suivants pour les autres inconnus
1.
Statement of the Problem
We study the quasi-static axisymmetric flow of a Newtonian incompressible fluid in a thin right cylinder
, given by (1), whose radius is small with respect to its length. Fluid velocity : and pressure ; satisfy
Stokes equations in Eulerian cylindrical coordinates (2)-(3). We assume that the lateral wall of the cylinder
ηε
sε
εR
z
Ω
ε
Figure 1: Wall displacement
U l ,p.2Y G
M [09%5 &
is elastic and that it satisfies the Navier equations (4)-(5). We denote
by
^,&
the radial and by c the longitudinal displacement from
state, see Figure 1,0 u * ] * is the
_ the_`reference
,p&
e
a
membrane thickness,
is the Young modulus,
the
g Lgi^k ,m& the wall volumetric mass,
j`Fjd^,&
Poisson ratio,
is the shear modulus and
is the Timoshenko shear correction factor (see
H
T
3
Elastic wall dynamics
\
&
[7]). \ The radial and the longitudinal component
of the forces exerted by the fluid on the lateral wall, D
and Q , and given by (6), where z :
is the rate of the strain tensor. Fluid velocity : is coupled with
the
Eulerian quantity : and
the velocity of the wall through
between
(7). We note that
the true
coupling
&
& &Z &[) &
@
@dc
the Lagrangian quantities
c is given by :
c n
n . After linearization,
@
n
@
n
E
e
compatible with linear elasticity
and creeping flow, we eapproximate the interface by its initial position and
e
the interface condition for the velocity by (7).
Initially, the cylinder is ‚ filled with fluid at velocity zero, and the entire structure is in an equilibrium at
some reference pressure ,
@ C@ c 0
c
@e n
@ n
on
M UW0fY 8
T (13)
e
The following boundary conditions
are motivated by the fact that only the pressure field is correctly observed at the outlet boundary:
0
G0
R UW$ L0Y/&MOPS
D
and
(14)
:
;
G0
& on @ R UW$ L5#Y/&MOPS
and ;
on
(15)
:D
G0
n L0 @ $ 5
8
$
L
f
0
P
@Cc $
for
for
and n
(16)
c
@
&
e
The time-dependent pressure
drop n between thee inlet and the outlet boundary
‚ drives the flow. We are
assuming that the pressure drop is small compared to the reference pressure
so that the acceleration of
the fluid is negligible compared to the effects of the fluid viscosity. Blood flow in small arteries satisfies
these assumptions [6].
2.
Main Results
'&
Wexderive
the asymptotic equations that describe:
the flow occurring at the leading order time scale
&
and
the oscillations of the membrane caused
by
a
response
of the elastic material. Since they occur at
€ we introduce the scaling n  3€ n . Energy estimate (2.1) below implies that € G, B = 8
different time scales
This choice of
balances the term involving the time derivative
2 & of the velocity and the inertia term.Q It
needs to be compatible with the oscillations of the pressure drop n , which is related to the heart pulsation,
blood viscosity and the size of the vessel via the so called Womersley number [7]. The results that follow
are all given in terms of the rescaled time n  where the wiggle sign is dropped for notational simplicity.
Energy Estimate. We first state the energy equality
]
]
]
‚ vt t | $ gi,&jd,p& ‚ $ ‹ _ e ^ > a ‚ v! < " # |
€ ] ^ ,&‡,p.
t u ^€ &
Q
B
B
B
Q B
B
B
km
E
E
E
]
<ba & ‚ v ! | y y/= g _ t*e z && $4 < ‚ ! & Q / 5 &) : n B
Q B
B
n : n
E H
E
J E
G€ @ and : Q 3€ @dc on T Mb[09!$7& , which provides the following a priori estimate
with : D
@e n
@n : D : Q & satisfies the following a priori estimate.
P ROPOSITION 2.1. – Solution
c
]
€ ] ,p&_`^,&
t
& $ e , . \ @dc & \
&a &)( ‚
'
$
% ,p.+
B B
n B
n B]^+* = ( ‚ ( g z : & $
@
E
t
B
E
] e & $ m=o [.+%5& ,-,
E 2/.9& 2.9& y .f8 E
H
(17)
€ gi^,&%jd^,&‡,p. ( @ $ n B
B 1@ 0
B
= (‚ u
‚
@
E
e
4
;
Elastic wall dynamics
A priori solution estimates follow from here. We use the approach^,introduced
&ˆ$9%/ &O by Ciarlet
ˆ$9, / et& al. (see e.g.
:
[2] and the references therein) and consider the scaled velocity :
^,&_`pressure
^,&
n
n and
^,&ˆ$9%/ &` [ $oh,// &
Z‹
_ ‚ ƒ"„…† ‚ ] ,
;
defined on the scaled, fixed domain
. Let
and
n
n
g ‚ %ƒ„"…† ‚ gi^,& ] ,p%& jd^,& ,
. Then (17) leads to the following a priori estimates for the scaled unknowns.
^,&( ,p& &
C OROLLARY 2.1. – Scaled solution :
a priori estimates
;
c satisfies the
^
,
&
p
,
&
^
,
&
,
&
\ : D \
] ^ _hg i_ ‚` a%e*e \ @ : Q , @ : D \
] ^ _hg e i_ ‚k` ale*e \ @ : D \] ^ _hgji%_ ‚k` a%e*eAm=o , B \ \ $
(18)
=
@
@
@
E
E
E
^
,
&
= \ @ : Q \
]^_hgji%_ ‚k` ale(e \ ,& \
]^
_hg i_ ‚` a%e*e
\
\
,
$
, @ ; ^,& B]^
_ ‚k` ab _hg e*e mpo \ \ (19)
; @ \ \
@
& ] ^ _ ‚k` ] e E \ @dc $ & \ ] ^ _ ‚` ] e g_#E ‚‚ H, \ @ $ n & \ ] ^ _ ‚k` ] e mzo _ ‚ \ \ ,
(20)
B
B
B
n B
@ n
@
B
E
E
B
e
H
H
a
\ \ ‚ e / .9& 2/ .9& 6 .
where
.
B
B @0
B
E
These a priori estimates are insufficient
to control the velocity field correctly. Further calculations imply
\ @ : ^,& Q \
] ^ _hgji%_ ‚k` ale(e , \ @ : ,& D \] ^ _hgji%_ ‚k` a%e*eAm , B > \ \ , \ @dc \
] ^ _(_ ‚` ] e i%_ ‚k` a%e*e 8
$
=
@
@
@n
E
E
J
We observe
that
the
above
estimates
also
hold
for
the
time
derivatives
of
the
unknowns,
but
with
replaced
#t ˆ0 &AL0
by @
, assuming the compatibility condition
.
Asymptotic Expansions. The above a priori estimates provide the multi-scale asymptotic expansions
listed in (8)-(9).
The Reduced Problem. By plugging the multi-scale expansions into the scaled Stokes equations and by
taking into account the boundary conditions which are the Navier equations
‚ for the membrane, we obtain
the following initial-boundary value problem for the effective pressure ;
;
SMALL
*),+ +-) ),. */&+ 1
0 p(z,0)=0
NON-NEGLIGIBLE
" !# ' (
&
$
%
$
*),+ )2. */#+ 10 !
3 *),+ )2. "!& */#+ 54 4 0 and formulas (10)-(12) for the other effective unknowns. Denote by the initial value problem for ; coupled
with (10)-(12). In the case when the shear stress terms are negligible we find that pressure is directly related
to the radial displacement via
[0&
_ ‚ h& . > %ƒ"„…† ‚ ] ,&h_`,p& ,p& . <
. &%&(
(21)
;
< a y
<ba y
e B
J H
E e
y (incompressible materials) our calcuwhere denotes the
area of the vessel.
H
Q
H
Q For a
0 cross-sectional
lation implies c
and equation (21) reduces to the Law of Laplace [4] typically used in literature [1, 6, 7]
HRQ
to model vessel walls via the independent ring model. We note that if the inertia terms have been taken
into account to model the flow by the Navier-Stokes equations, the reduced equation for the pressure (or
effective displacement) would have been hyperbolic. Creeping flow leads to a parabolic problem for the
effective pressure, shown above in .
Convergence Result. As in the study of the Stokes and the Navier-Stokes equations in thin, rigid,
cylinders (see [3] and subsequent works by numerous authors), the a priori estimates
, 0 and uniqueness for
problem allow us to justify the effective model by obtaining the weak limit as
.
5
Elastic wall dynamics
g ‚ 0
UIF5 #& F5 ˆ#&(Y
T HEOREM 2.2. – Suppose for simplicity
and let
, ^,& Q ^,&( , & that
5 [09!$? M
5 B ˆ#&M D ‹ [09%5B & & .
:B
;
Then the sequence =
converges
weakly
in
c B 5 ˆ0f!$ & B to
= ,pB & 0
D
a unique solution of the
. Furthermore, , :
weakly in B
.
e Q
Q initial value problem
Error Estimates. Since weak convergence does not provideB enough information
about
the
accuracy
of the approximation of the true solution by the solution of the reduced problem , we find the order of
approximation by calculating the error estimates of the difference between the two solutions. We found
that
$ the
5 reduced problem gives rise to a boundary layer in the velocity and pressure at the outlet boundary
. We construct the boundary layer
by considering
abstract problem on a
-Mexplicitly
U 2*.2YSthe
M U following
0Y :
semi-infinite rigid-wall cylinder
, where
S D TVU : D‚ / 5 &(
n
< S T(U W T(U 0
and
‚
S Q VT U W TVU E <Zy @ <Zy @ : $ Q n %/%5&
@ E
@
div
S TVU 0
on
L0
in and
S TVU 0
on
@
KM
(22)
8
(23)
The results from [5] give
the existence of a unique smooth velocity and pressure fields, exponentially
S %when
decreasing
and the boundary layer pressure
/%$'&` ,S TVU /. We
& define
W %the
/%$'boundary
& , WYT(U%layer
/ velocity
Q ] now
Q ] & , respectively.
to be
and
These
functions are
n
n
n
B n
exponentially small away from the outlet g boundary.
Assuming
that
the
solution
of
problem
is
sufficiently
‚
_ ‚
smooth and that the convergence toward
and
is
sufficiently
fast,
we
obtain
‚
‚
‚
^,& , = ^,& < Q u t Q < , D u t D S ^,&A
:
:
:
,c
c < c
,p& ,
E
E
B
<
. Then we have
\ ^,& \]^_ ‚` acb dfe \ ^,& \
]^_hgji%_ ‚k` ale(enmpo ,p7q e
e
e
[
Z
B
^
,
&
^
,
E
‚ \ @ , $ & \]^_ ‚k` ] e
‚ \ , & & \]^_ ‚k` ] e \ @dc $ ,& & \
]^
_ ‚` ] e y p
‚ s%}t"s%r a x
%
g
_
m o , 7q 8
u w
w
B
n
n
n
@
e@
E
e
E
T HEOREM
‚ – Let Z
2.3.
^,&A
;
^,& <
;
W
,[
and
(24)
(25)
Therefore, we have shown that the error between the solution of the full set of axisymmetric Stokes equations
coupled with the Navier equations for the elastic tube wall, and the solution of the reduced problem
7q
, modified by the boundary layer velocity and pressure, is of order Ž B , where Ž is the ratio between the
radius and the length of the elastic tube. We point out that in the interior of the domain, away from the
outlet boundary, boundary layer has negligible effect and the accuracy
of the approximation of the solution
to the original, non-reduced problem, by the solution of problem away from the outlet boundary is Ž B .
Acknowledgements. This research was made possible by the support from the Texas Higher Education Board, ARP
Mathematics, grant number 004652-0112-2001 and in part by the National Science Foundation, grant DMS 9970310.
References
[1] S. Čanić. Blood flow through compliant vessels after endovascular repair: wall deformations induced by the
discontinuous wall properties, Comput. Visual. Sci. 4(3) (2002), 147-155.
[2] P.G. Ciarlet, Plates and junctions in elastic multi-structures, R.M.A. 14, Masson, Paris, 1990.
[3] H. Dridi, Comportement asymptotique des équations de Navier-Stokes dans des domaines applatis, Bull. Sc. Math.
106 (1982), 369–385.
[4] Y.C. Fung. Biomechanics: Mechanical Properties of Living Tissues. Springer New York 1993.
[5] W.Jäger, A.Mikelić , On the effective equations for a viscous incompressible fluid flow through a filter of finite
thickness, Communications on Pure and Applied Mathematics, Vol. LI (1998), 1073–1121.
[6] M.S. Olufsen, C.S. Peskin, W.Y. Kim, E.M. Pedersen, A. Nadim, J. Larsen, Numerical Simulation and Experimental Validation of Blood Flow in Arteries with Structured-Tree Outflow Conditions, Annals of Biomedical
Engineering, 28 (2000), 1281-1299.
[7] A. Quarteroni, M. Tuveri, A. Veneziani, Computational vascular fluid dynamics: problems, models and methods.
Survey article , Comput. Visual. Sci. 2 (2000), 163-197.
6