Non-Newtonian Mathematical Model and Numerical
Simulations for the Blood Flow in Capillary Vessels
Balazs ALBERT*,1, Vitalie VACARAS2, Titus PETRILA1,3
*Corresponding author
Babes-Bolyai University
No. 1 Mihail Kogalniceanu, 400084, Cluj-Napoca, Romania
[email protected]*
2
University of Medicine and Pharmacy “Iuliu Hatieganu”,
No. 43 Victor Babes, 400012, Cluj-Napoca, Romania
3
Academy of Romanian Scientists,
No. 54 Splaiul Independentei, 050094 Bucharest, Romania
1
DOI: 10.13111/2066-8201.2016.8.1.1
Received: 19 January 2016 / Accepted: 08 February 2016
Copyright©2016. Published by INCAS. This is an open access article under the CC BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: In this paper, taking into consideration the rheological Cross type non-Newtonian model,
we elaborate an axial-symmetric mathematical model for the blood flow in capillary vessel with
adequate numeric algorithms. We take into account the elastic and porous behavior of the vessel wall
which leads to a more realistic approach of the problem. We also accept that the change of substances
through these vessels complies with the Starling hypothesis. This hypothesis states that the mass debit
through the capillary wall is proportional to the pressure difference between outside and inside the
capillaries. The existence of a slip condition along the permeable surface is also accepted using the
results of Beavers and Joseph. The numerical experiments are made using COMSOL Multiphysics
3.3. Some numerical results with respect to the velocity field, pressure variation and the wall shear
stress are presented.
Key Words: rheological Cross model, blood flow in capillary vessel, elastic and porous walls,
numerical simulations.
1. INTRODUCTION
The capillaries are the smallest blood vessels. There are different types of capillaries as
continuous capillaries, fenestrated capillaries and sinusoidal capillaries. Through their thin
walls oxygen and nutrients pass to the tissue cells, in exchange for carbon dioxide and other
products of cellular activity. Despite the small size and thin walls of the capillaries, the blood
pressures may be quite high, as, for instance, in the legs of a person in a motionless upright
position. In certain disease states there is increased fragility of the capillary wall, with
resultant hemorrhages into the tissues. Vitamin C deficiency and a variety of blood disorders
may be associated with increased capillary fragility.
The capillaries are freely permeable to water and small molecules but ordinarily are not
highly permeable to proteins and other materials. In some pathological situations, such as in
certain allergic states (e.g., hives) or because of local injury, as in burns, there may be local
areas of permeability, with escape of fluid high in protein into the surrounding tissues. If the
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ISSN 2066 – 8201
Balazs ALBERT, Vitalie VACARAS, Titus PETRILA
4
disease affects the entire body, a significant amount of plasma leaks into the nonvascular
spaces, with resultant loss in blood volume [11].
We also remark that concerning the blood-to-brain passage, along all capillaries in the
brain, excepting those present in the circumventricular organs, the so-called blood-brain
barrier (BBB) is present [2].
In the case of hypertension, etc. the disruption of the blood-brain barrier may occur,
which can lead to further difficulties in the change of solutes between the blood (capillaries)
and the nervous system [4].
2. MATHEMATICAL MODEL FOR CAPILLARY VESSELS
In this paper we accept for the blood flow a non-Newtonian model with a non-constant
viscosity of Cross type. More, we will assume, in spite of the presence of different types of
moving blood cells, that the blood behaves as a homogenous fluid, as the flow has a laminar
and incompressible character in the absence of any exterior field forces.
Concerning the vessels’ walls they are considered to be elastic and permeable, the
permeable (porous) character of the capillaries dominating the elasticity of their walls.
We also accept that the change of substances through these vessels, very small in
volume, observes the Starling hypothesis [9]. This hypothesis (checked experimentally by
Mauro [5], Meschia [6]) states that the mass debit through the capillary wall is proportional
to the pressure difference between outside and inside the capillaries . The existence of a slip
condition along the permeable surface which is “covered” by a porous media (a condition
confirmed experimentally) is also accepted using the results of Beavers and Joseph [3].
For the sake of simplicity we accept the axial symmetry of the blood flow, the axis of
the symmetry being Oz.
Using the cylindrical coordinates (r , , z ) , the motion domain will be
(t )  {( r, , z) / r  R  ( z, t ),  [0,2), z  (0, L)}, at every time t, where R and L are the
(initial) radius and the length of the tube (vessel), respectively and  is the elastic
displacement of the wall (t )  {r  R  ( z, t ), z  (0, L)} at the considered moment t.
In the meridian plane  =constant if u z and ur are the components of the blood velocity
in z and r directions, while p is the pressure (evaluated versus a reference pressure pref); then
we get the following working equations (in cylindrical coordinates)
1 
u
(ru r )  z  0 (continuity equation),
r r
z
and
ur
u
u
 2u
p
1 ur  2ur
 ur r  u z r )    (  )( 2r 
 2 )
t
r
z
r
r
r r
z


 ur  ur u z
 L  M [  2
 (

)],
r
r
r r z z
r
(
u z
u
u
 2u
p
1 u z  2u z
 ur z  u z z )    (  )( 2z 
 2 )
t
r
z
z
r
r r
z


 u z  ur u z
 L  M [  2
 (

)].
z
z
z r r z
r
(
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Non-Newtonian Mathematical Model and Numerical Simulations for the Blood Flow in Capillary Vessels
The last (flow) equations are obtained from the general Cauchy motion equations for a
deformable continuum, where we use the following representation for the stress tensor:
T  [ p  (
K
 2
 
K )]I  2(s  RBC )D,

p
where D is the rate of strain tensor while I is the unity tensor, p the physical pressure and
 RBC is given by the Cross model
RBC 
Here   4I 2
1/ 2
*0
1  (k )1n
, I 2 being the second invariant of the rate of strain tensor D, s is the
plasma viscosity,  p and *0 the viscosity coefficient of the blood,  the “relaxation time”,
k is a time constant for the shear thining behavior, n the shear thining index,  the mobility
parameter, while the function
K(  ) 
*0
1
(
  p ), for   0
 1  (k )1n
is the so-called “normal function” of the variable  which measures the variation of
deformation.
For the sake of simplicity we denote
( )  K( )  s   p  s  RBC
L
2 K
 2K
  2  and
 p 

M 
k*0 (1  n)(k )  n
.
[1  (k )1n ]2
In fact we got above only two scalar flow equations (in ur and u z ), u being zero.
To these equations we must add the boundary conditions which express both the
presence of a pressure gradient along the Oz axis and the elastic character of the permeable,
porous wall
u z
 0 and ur  0 for r  0 ,
r
u z


u z and ur  K ( p  ) for r  R  ( z, t ) .
r
K
The first relation expresses the Beavers-Joseph slip condition with the slip parameter 
while K is the specific permeability of the porous media, meantime ur  K ( p  ) is the
consequence of the Starling law,  is built by the interstitial and osmotic pressure supposed
to be fixed. Concerning the pressure we have
INCAS BULLETIN, Volume 8, Issue 1/ 2016
Balazs ALBERT, Vitalie VACARAS, Titus PETRILA
p
cos(t )
 pm for z  0 , where a  0 ,
a
p
cos(t )
 pm for z  L , where a  0 ,
a 1
6
R
where pm 
 f (r )dr  f () , f being a primitivable and derivable function according to r,
0
R
with a maximum at r  0 and a minimum at r  R   ; obviously p z 0  p z L at any time
of the motion (0, T).
Remarks: These boundary conditions on the “edges” z  0 and z  L of the capillary
cos(t )
are in accord with the acceptance of a representation for the pressure of p 
 f (r )
a 1
 
type, namely of a pressure gradient (in the cylindrical reference ez , er ) under the form

cos(t ) 
gradp  
e  f (r )er .
2 z
(a  1)
cos(t )
 gradp r  R( z ,t ) , in
If f (0)  0 and f ( R  ( z, t ))  0 we have gradp Oz  
(a  1) 2
accord with the pressure gradient in the interior of the capillary [8].
On the other hand, accepting for the capillary wall the linear elastic membrane model,
the radial component of the membrane stress is expressed by the radial displacement  as
follows
Tr  m h
 2
hE 1


  pref ,
2
2
2
t
1  R
R
where h is the thickness of the membrane, E the Young modulus,  the Poisson coefficient,
 m is the density of the capillary wall while pref is the reference pressure in
the”unperturbed” state.
It is evident that this stress must coincide with the stress generated by the blood on the
 
 
same radial direction, namely T  er  Tn  er  Tr , which represents a relation for determining
f (the pressure) or ( z, t ) .
At the same time the kinetic condition must be satisfied on the elastic wall, i.e.,


 ur ( R  ( z, t ), z ) but also u z ( R  ( z, t ), z)  0 , what leads to
 K ( p ( z , r , t )  )
t
t
u z
 0 for r  R   , respectively.
and
r
It can be remarked that the last relation, together with u z ( R  ( z, t ), z)  0 , implies
u z
 0 and ur  0 for the
u z  0 in neighborhood of the elastic wall while the conditions
r


axis r  0 show that u  u z ez depends only on z and t so that we have a pulsating flow along
the axis Oz, which “calms down” on the elastic wall ( u z  0 ) where the exterior imposed
pressure will have a minimum.
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Non-Newtonian Mathematical Model and Numerical Simulations for the Blood Flow in Capillary Vessels
At
the
same
time
from

 K ( p ( z , r , t )  )
t
r  R   ( z ,t )
we
obtain
 2
- sin(t)
 K(
)
.
2
t
a 1
r  R   ( z ,t )
 2
allows to make precise the condition on the capillary wall
t 2
 
(linear elastic membrane), namely the expression of the “equilibrium” condition Tn  er  Tr
in cylindrical coordinates. More precisely, if we note
This last evaluation for
P  [ p  (
K
 2
 
K )]
 
p
the equilibrium condition becomes


P
2(  ) ur
u u
z



( r  z)

r
z
r



1  ( )2
1  ( )2
1  ( )2
z
z
z
  m h( 
Ksin(t)
hE 1

)
  pref , [8]
2
2
a 1
1  R
R
which provides an equation to determine the deformation of the capillary wall, namely
( z, t ) , so that the whole set of unknowns of our problem can be determined.
3. NUMERICAL SIMULATIONS
The above described mathematical model has been tested in the case of the blood flow in
capillary vessel, taking into consideration the elasticity and the porosity of the limiting walls.
The calculations were made using COMSOL 3.3.
Let us consider a small vessel “segment” of 0.01mm in radius and 0.1mm in length. The
dynamic viscosity of the blood is given by the Cross rheological model
(  )  s 
0
1  k 1n
*
where s  103 Pa  s , *0  1Pa  s , k  100 , n  0.2 .
The mass density is 1060kg / m3 . The other parameters of the proposed model are
  100 ,   50 . We consider an oscillatory pressure pin  10100Pa  150cos(t ) Pa on the
input boundary (z = 0) and a constant pressure pout  10000Pa on the output boundary.
At the points of the vessel axis of symmetry r = 0 we have imposed the axial symmetry
requirements while on the vessel walls the non-slip condition together with the permeability
condition ur  K ( p  ) . The permeability constant is K  0.5 1010 m2 s/kg and the
osmotic pressure is   9900Pa. The initial conditions are ur = uz = 0 m/s and p = 10150 Pa.
The velocity field at the moment t = 5s is sketched in figure 1 and the variation of the
pressure along the capillary tube is presented in figure 2.
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Figure 1. Velocity field at t = 5s
Figure 2. Pressure variation along the capillary tube at t = 5s
The variation of the pressure through 5 seconds in the middle of the capillary segment
(at the point r = 0, z = 0.05mm) is presented in figure 3 bellow.
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Non-Newtonian Mathematical Model and Numerical Simulations for the Blood Flow in Capillary Vessels
Figure 3. Variation of the pressure in the middle of the capillary vessel
Different researches show that there is significant difference in wall shear stress (WSS)
between the specified conditions which simulate disease state and normal conditions. As a
result it may be possible to use this parameter as for diagnostic purposes. The wall shear
stress changes activate certain mechanism at the endothelium which eventually leads to
vasoconstriction or vasodilatation of the capillaries. Therefore understanding how wall shear
stress changes according to conditions present in the circulation is an important field of study
u u
[10]. In what follows, we calculate the value of the wall shear stress ( WSS  ( r  z ) [1]
z
r
and [7]) on the capillary wall in points P1, P2 and P3 (as can be seen in figure 1). The results
according to the variation of the WSS are presented in figure 4.
Figure 4. Variation of the WSS through 1 second in points P1, P2 and P3
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In figure 5 the dependence of the WSS (in absolute value) versus the shear rate in point
P2 is presented.
Figure 5. WSS (in absolute value) versus the shear rate at point P2
This result seems to be in accordance with the results got by Tan et al. [10]. This
multidisciplinary research has been done by a group formed by physicians, mathematicians
and specialists in continuum mechanics.
Balazs Albert, on the one hand, and Vitalie Vacaras, on the other hand, contributed
equally to the present paper.
The whole research has been performed under the leadership of Professor Titus Petrila,
the coordinator of the research group on blood flow modeling.
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[11] * * * http://www.britannica.com/EBchecked/topic/720793/cardiovascular-disease/33668/Functional-disease.
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