Biomedical Engineering Reference
In-Depth Information
confirm results on altered stresses. It is also important to note that there are many other com-
mercial CFD software packages available, such as FLUENT, FLOW-3D, and many others.
However, under certain circumstances, the large amount of information we can gain
from CFD simulation is not quite enough for us to predict biofluid behavior, because not
only fluid mechanics but also biochemical responses are involved. The best scenario would
be when we want to simulate the behavior of blood within a blood vessel. We know that
blood is composed of plasma and formed cellular elements, including red blood cells, white
blood cells, and platelets. In large or medium-sized blood vessels, we usually do not need to
consider the interactions between these formed elements and the fluid phase, as long as we
are not studying the cellular responses to the fluid induced stress. However, in small blood
vessels, especially in the microcirculation, the interaction between the particle phase and
the fluid phase becomes relatively important; therefore, the particle phase has to be consid-
ered. To model the cells as a second phase, discrete-phase, or even multi-phase models can
be used. Let us use the coronary artery model again to demonstrate how a discrete-phase
model can be coupled into our CFD model to provide us with more information.
From previous chapters, we have learned that platelets are extremely sensitive to blood
flow
induced shear stress and platelet biochemical responses are closely related to the
local flow conditions and shear stress distribution. To investigate how platelets respond to
shear stress at a smaller scale using the previous CFD model, we have modeled platelets
as discrete-phase particles. The fluid phase, blood, will be treated as a continuum by solv-
ing the time-averaged Navier-Stokes equations (as before), and then the behavior of plate-
lets can be calculated through the flow field at each time step. In these types of
simulations, the particle phase can exchange mass, momentum, and energy with the fluid
phase. The trajectories of the platelets can be determined by the fluid forces applied
to them. According to Newton's second law of motion, the transient acceleration (in the
x-direction) of a platelet (or any particle within a fluid) can be defined as
du p
dt 5F D ðu2u p Þ 1F x
ð
13
:
15
Þ
where u p is platelet velocity. F x is an additional acceleration term, which is the force
required to accelerate the fluid surrounding the particle. F D ( u2u p ) is the drag force per
unit mass of platelet (however, the gravity acting on platelets is neglected) and can be
defined as
3
ρju p 2ujC D
4
F D 5
ð
13
:
16
Þ
ρ p d p
where C D is the drag coefficient (depends on the shape of the particle),
ρ
and u are the
density and velocity of the fluid,
ρ p and u p are the density and velocity of the platelet, and
d p is the platelet diameter. Once the velocity and acceleration of one platelet is determined,
its trajectory can be obtained using the kinematic equation:
dx i p
dt 5u i p
ð
13
:
17
Þ
where the index i refers to the coordinates directions. To account for turbulent effects, a
stochastic model developed by Gosman and Ioannides can be used. Briefly,
the
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