The derivation of transport coefficients of the Fluid Particle Model
for non-linear Casson fluid.
Maciej Paszyński
Robert Schaefer
Institute of Computer Science, Jagiellonian University
Nawojki 11, 30-072 Kraków, POLAND
e-mail: mp@robobat.pl, schaefer@softlab.ii.uj.edu.pl
The problem of three dimensional modeling of pulsatille blood flow in cardiovascular
system in man is of the great importance and motivates itself [13]. Fluid Particle Model
(FPM) invited by Espaňol [3] constitutes advantageous alternative method of flow modeling
in comparison to the classical viariational methods such as Finite Elements, especially in the
case of complicated flows in the moving boundary domains and by the presence of composed,
non-stationary boundary conditions [2], [8], [10]. The three dimensional blood flow in some
part of the artery belongs to the class of problems mentioned above [13], [7], [6].
Fluid Particle Model was originally formulated to solve Newtonian flow (see [3]).
This method postulates to spread the entire fluid area into N separated bodies called particles.
The fluid flow simulation consists in updating the state of each particle in consecutive time
steps, by means of its interactions with neighboring particles distanced up to RCUT threshold
distance.
Inter-particle forces are parameterized by some scalar functions A, B that depend on
the relative distance between interacting particles rij
 
3  rij  A rij   ;
 
3  rij  B rij  
(1)
The derivation presented by Espaňol leads to obtaining the approximation of the stress
tensor in fluid. It can be expressed by some FPM parameters and the macroscopic values of
the Green strain velocity tensor. This expression has the similar form as the Newton
constitutive equation, so that the so called transport coefficients (see [3], [1]) may be
identified with the Newtonian viscosity parameters  ,  . Comparing constitutive equations
of Newtonian fluid
1
pij( a )   divu ij ;
3
pij( d )  2eij( d )
(2)
with the formulae obtained by Espaňol, the following relationships may be derived

 A2  D  2 
KB T

B2  

2D
 2D
 D A0  B0  

   mn 2 

(3)
1
2
KB T 
 A2

 B2  

 2
   A0  B0  
   mn 2 
In the formula above:
A0 
RCUT

0
1
A RdR , B0 
D
RCUT
1
0 B RdR , A2  D
RCUT

0
1
A R R dR , B2 
D D  2
2
RCUT
 B RR dR ,
2
0
and D=3 is the spatial dimension of the flow problem. Moreover  , m , n , T are some
model constants and K B is the Boltzmann constant.
For the classical FPM for Newtonian fluid inter-particles force parameters (1) are
established at the beginning of the simulation by using the formulas (3) in such a way, that
Newtonian viscosity coefficients  ,  are correct. The formulas of interactions remain
unchanged during the entire simulation.
It follows from the medical investigations, that rheological properties of blood can be
described by the non-linear Casson constitutive equations [5]. The paper presents the
modification of the classical FPM, that allows to simulate the flow of the fluid in this nonlinear case. The derivation is dedicated to the Casson fluid, but it may be easily generalized to
the wide class of rheo-stable fluids [4].
Fig. 1 Successive linear approximating of the Casson constitutive equation.
We started from the modified, three dimensional Casson constitutive equation [13],
where the viscosity depends on the norm of the shearing part of Green tensor e ( d ) .
For each pair of interacting particles, inter-particle force parameters (1) are estimated
at the beginning of current time step, and remains valid only during the one time step. We
intent to obtain the local, linear approximation of the Casson formula, in particular the local
values of the viscosity  N ,  N .
d
1

1 p
d

2 e


N 

0
.
5


Y
d

2 e


1
2

   ;

2
3
N  N
(4)
The viscosity coefficients may be locally adjusted by the modification of the interparticle parameters (1) by using the formulas (3). The new relationships for A and B should be
invented in such way, that the dependency on the local value of the norm of Green tensor are
involved.




3    rij , e d  A rij , e d   ;




3    rij , e d  B rij , e d  
(5)
The main result of this paper consists in derivation of transport coefficients from the
modified inter-particle forces, taking into account the non-linear Casson stress. We show that
in this case in which scalar functions (1) depend also on e d (see (5)) it is possible to perform
similar derivation of transport coefficient that has been originally given by Espaňol for the
linear case. We obtained the following interdependence between viscosity coefficients and the
model parameters
 
 e
d
    D  2 B  e  
 D A
2D

 A ed

2 2
  mn
 2D


 
 ed
d
2
  e   B  e 

d
2

d
d
0
   B  e  
 

 A ed
1
2 2
  mn

2
2


KB T
0


,

(6)
KB T
 A  e   B  e 
d
d
0
0




that lead us to non-linear constitutive equations
  13 divu
pij( a )   e d
ij
;
 
p ij( d )  2 e d eij( d )
(7)
Viscosity coefficients are now functions of the norm of the Green tensor. Moreover, it
is possible to define inter-particle force parameters (5) in such way, that the non-linear Casson
viscosity (8) can be expressed.
 

 e d    Y   2 e d

1

1
2

   ;

 
2
3
 
 ed   ed
(8)
The quality of presented modification has been confirmed by the comparison of results
of the modified FPM simulations (for blood understood as a Casson fluid) both to result of the
classical FPM (for blood understood as a Newtonian fluid), and to the medical measurements,
performed on the carotid artery in man (see [12]). The Fig.3 presents comparison of
simulations results obtained by using classical and modified FPM. On the Fig.3 only particles
with non-zero velocity vectors are indicated by drawing they velocity vectors that were scaled
proportionally to they lengths. As opposite to classical FPM algorithm, the modified FPM
algorithm, models in the right way the phenomenon of the peristaltic wave propagation in
artery, comparable with medical measurements data.
Fig.2 Pulse wave propagation along the artery - results after 10 time steps of the classical (left picture) and
modified (right picture) FPM simulations.
The blood flow simulation by Fluid Particle Model seems to be more effective that by
the classical Finite Element Methods [9]. FPM requires only updating of the state of each
particle, taking into account the states of neighboring particles up to some threshold distance.
The complexity of the algorithm is linear with respect to the increasing number of particles,
and depends only on the density of particles within the sphere of the threshold radius.
Keywords: Fluid Particle Model, successive linear approximation, non-linear blood flow
simulation.
References
1. Chapman S., Cowling T.G.. (1970), The mathematical theory of non-uniform gases.
Cambridge Mathematical Library.
2. Dzwinel W., Alda W., Kitowski J., Yuen D.A.. (1999), “Using discrete particles as a
natural solver in simulating multiple-scale phenomena”, UMSI 99/131.
3. Español, P. (1998), “Fluid particle model”, Physical Review E, 57(3), 2930-2948.
4. Ferguson J., Kembłowski Z. (1995), Reologia stosowana płynów, MARCUS, Łódź
5. Fung Y.C. (1984), Biomechanics, Springer-Verlag, New York.
6. Jensen J.A. (1996), Estimation of blood velocities using ultrasound, Cambridge.
7. McDonnald D.A. (1974), Blood Flow in Arteries, Edward Arnold, London.
8. Paszyński M. (2001), “Calculations of blood flow in arteries - comparison of Fluid
Particle Model with Finite Elements Methods”, Proceedings of VII National Conference
on Application of Mathematics in Biology and Medicine, AGH, Poland.
9. Paszyński M., Pelc A. (2001), “Numerical analysis of peristaltic blood flow in arteries”,
CAMES, 8(4).
10. Paszyński M. (2002), “Object-oriented software system that performs FPM simulation in
the area with moving boundary, and its application to the blood flow problem”, accepted
to CAMES.
11. Paszyński M. (2002), “Simulation of a Casson fluid within the Fluid Particle Model”,
invited paper ASEM’02 Conference, Pusan, Korea (to appear).
12. Schaefer R., Paszyński M. (2002), ”Solvability of non-linear Stokes problem for blood
flow in peristaltic deformed vessel”, accepted for Tenth International Congress on
Computational and Applied Mathematics, Leuven, Belgium.
13. Taylor C.A., Hughes T.J.R., Zarris C.K. (1998), “Finite element modeling of blood flow
in arteries”, Computer methods in applied mechanics and engineering, 158.