Biomedical Engineering Reference
In-Depth Information
This condition is clearly met in blood flow regimes, together with the Newtonian
rheological behavior of blood in large arterial systems. Non-Newtonian rheological
models appropriate for simulating blood flow in medium or small-sized arteries, such
as the Casson, Carreau, or Carreau-Yasuda models, can be also incorporated within
the LB approach [ 6 , 20 ].
The LB method is based on the collective dynamics of fictitious particles on
the nodes of a regular lattice where the basic quantity is f p ð
x
;
t
Þ
,representingthe
probability of finding a “fluid particle p” at the mesh location x and at time t and
traveling with discrete speed c p . “Fluid particles” represent the collective motion of a
group of physical particles (often referred to as populations). We employ the common
three-dimensional 19-speed cubic lattice (D3Q19) with mesh spacing Dx , where the
discrete velocities c p connect mesh points to first and second neighbors [ 2 ]. The fluid
populations are advanced in a time step Dt through the following evolution equation:
f e p Þð
f p ð
x
þ
c p D
t
;
t
þ D
t
Þ¼
f p ð
x
;
t
Þoð
f p
x
;
t
Þþ
F p ð
x
;
t
Þ:
(10.1)
The right-hand side of ( 10.1 ) represents the effect of fluid-fluid molecular
collisions, through a relaxation towards a local equilibrium, typically a second-
order expansion in the fluid velocity of a local Maxwellian with speed u ,
c s I
u
c p
c s þ
uu
: ð
c p c p
Þ
f e p ¼
w p r 1
þ
;
(10.2)
2 c s
where c s ¼ 1 =
p is the speed of sound, w p is a set of weights normalized to unity,
and I is the unit tensor in Cartesian space. The relaxation frequency
o
controls the
. The kinetic moments of the
discrete populations provide the local mass density
c s D
1
1
2
kinematic viscosity of the fluid,
n ¼
t
o
r
( x , t )
¼ ∑ p f p ( x , t ) and
momentum
¼ ∑ p c p f p ( x , t ). The last term F p in ( 10.1 ) represents a momen-
tum source, given by the presence of suspended bodies, if RBCs are included in the
model, as discussed in the following sections. In the incompressible limit, ( 10.1 )
reduces to the Navier-Stokes equation
r
u ( x , t )
r
u
¼
0
;
(10.3)
@ u
@
1
r r
2 u
t þð
u
u
¼
P
þ nr
þ
F
;
where P is the pressure and F is any body force, corresponding to F p in ( 10.1 ).
The LB is a low-Mach, weakly compressible fluid solver and presents several
major advantages for the practical implementation in complex geometries. In particu-
lar, in hemodynamic simulations, the curved blood vessels are shaped on the Cartesian
mesh scheme via a staircase representation, in contrast to body-fitted grids that can be
employed in direct Navier-Stokes simulations. This apparently crude representation
of the vessel walls can be systematically improved by increasing the mesh resolution.
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