Biomedical Engineering Reference
In-Depth Information
time-dependent motions, linear/non-linear models, and time-linearized models. We will
highlight some of the equations within these methods and discuss some of the current bio-
fluid research that uses FSI modeling as one of the tools to characterize the flow through a
system.
The easiest method to model a solid boundary movement is based on the pressure that
the fluid exerts on that boundary at a particular instant in time and space. This can be
quantified as
p5ρu @v
@t 1urv
ð
13
:
20
Þ
where
is the bulk fluid density, u is the average fluid velocity, and v is the fluid velocity
as a function of time and space. This model is fairly useful in most biofluid applications,
but it is not necessarily accurate when the average fluid velocity is relatively large (e.g.,
close to the speed of sound, which we typically do not approach in biofluids). Also, this
model is simple enough that it can be used as an initial approximation to determine if the
model parameters are accurate and the flow through the vessel looks reasonable. As the
simulation includes more parameters, such as the fluid viscosity and possibility for turbu-
lence to develop, the equations become much more complex (this is termed the full-
potential theory/equations). Using this method, a nonlinear wave equation would be
coupled o the structure movement. This wave equation is nonlinear due to the elastic and
viscous mechanical properties of the vessel wall. If one generalizes this formulation, con-
sidering that the motion of the body is relatively small and the pressure is uniform
throughout the wall (with an inviscid, irrotational fluid), a small displacement form of the
full-potential theory can be developed. Making these assumptions, the convected-wave
equation becomes linear and can be solved for any fluid property as
ρ
2
2
2
2
Φ2 @
@t 2 2u @
Φ
@x 2 1 @
Φ
@y 2 1 @
Φ
Φ
@z 2
2
r
5
0
ð
13
:
21
Þ
where
is the fluid property of interest (e.g., velocity, pressure, among others). There are
various approaches to make the solution of these equations simpler. The first would be to
solve a quasi-linear set of fluid equations that would govern the fluid flow through a sys-
tem that is completely rigid. This solution would be able to determine the flow properties
at each spatial location within the volume of interest. From this solution, the boundary is
given a small temporal disturbance and then the changes in the fluid properties are calcu-
lated. This gives a direct coupling between the fluid properties and the solid boundary;
however, with this simplification the fluid is not directly causing boundary motion and
only the boundary is affecting the fluid properties.
The direct coupling of the fluid properties to the solid boundary properties is the chal-
lenge of fluid-structure interaction modeling. For each time step within the computational
solution, the fluid properties must be considered in relation to the motion of the boundary
and changes in the boundary position induced by the fluid must also be considered. This
coupling is what makes the complexity of FSI so great, and there are many examples of
simplifications that have been used to obtain a solution. As with computational fluid
Φ
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