Biomedical Engineering Reference
In-Depth Information
length of each bifurcation. Weibel [7] developed an improved model that is used in
approximating network of repeatedly bifurcating tubes in the lungs.
The behavior of fluid flow in the regions where there is branching in the blood
vessels and airway, change in fluid properties, change in fluid velocity are intensely
investigated areas because of widespread applications. For example, understand-
ing the changes in fluid flow at the athereosclerotic sites are important to decipher
the etiology of the disease and developing therapies such as stents to normalize
flow patterns in that location. During
rouleaux
formation, the extent of aggrega-
tion is determined by opposing forces: the aggregation induced by the presence of
macromolecules and the disaggregation induced by the negative surface charge and
the flow-induced shear stress. Another example is inhalation of aerosols affecting
lung function. The Poiseuille equation is developed with the assumption that the
flow is at steady state, occurs in one direction in rigid tubes although many biologi-
cal conduits are not rigid during flow (but elastic, see Chapter 6), and the fluid is
Newtonian. However, at the entry to the tube before the laminar flow has become
fully developed, flow is not steady and variations in two or three directions exist.
A similar analysis can be performed to describe the fluid flow in 3D. Then for a
Newtonian incompressible fluid, one would obtain
Du
∂
P
μ
⎛
∂
2
u
∂
2
u
∂
2
u
⎞
the -component
x
ρ
=
ρ
g
−
+
+
+
(4.39)
⎜
⎟
x
2
2
2
Dt
∂
x
⎝
∂
xy
∂
∂
z
⎠
Dv
∂
P
μ
⎛
∂
2
v
∂
2
v
∂
2
v
⎞
the -component
y
ρ
=−+
ρ
g
++
(4.40)
⎜
⎟
y
2
2
2
Dt
∂
y
⎝
∂
xy
∂
∂
z
⎠
w
∂
P
μ
⎛
∂
2
w
ww
∂
2
∂
2
⎞
the -component
z
ρ
=−+
ρ
g
+ +
(4.41)
⎜
⎟
z
2
2
2
Dt
∂
z
⎝
∂
x
∂
y
∂
z
⎠
Equations (4.39) to (4.41) are called the Navier-Stokes equations in the Car-
tesian coordinate system and if the viscous terms are omitted, one would obtain
Euler equations. They can be developed for any other coordinate system that spans
the solution space. Alternatively, a transformation can be performed to the devel-
oped equations to acquire different coordinate systems from the Cartesian result.
These differential equations are difficult to solve analytically, and one could have
a number of possible solutions. However, the development of computational fluid
dynamics (CFD) has significantly helped understanding the flow fields in biologi-
cal systems. A number of solutions have been published for various applications.
Changes in fluid properties in time and space is obtained by solving incompress-
ible Navier-Stokes equations coupled with mass, and energy balances using finite
volume techniques. With the exponential increase in massive computer capabilities,
CFD is employed by several researchers to explore the nature of flow stagnation
patterns in various biological systems. For further discussion, the reader is referred
to [8].
