Biomedical Engineering Reference
In-Depth Information
oil). In principal, the two different fluid phases will have a different density, viscosity,
among others. To describe a two-phase flow, the same laws that will be developed in sub-
sequent chapters can be applied, but they must be applied separately to each phase. In
solid mechanics, if you were to analyze an axial loaded solid bar composed of one material
on the outside and a separate material on the inside, this would introduce a statically inde-
terminate problem. However, by writing the equations independently and then equating
variables that must be the same (e.g., displacement), this type of problem could be solved.
In two-phase flow you must take a similar approach. In general, you must equate the pres-
sure, velocity or shear force at the phase boundary so that a solution can be reached.
One common two-phase (or multi-phase) fluid is blood. The first phase to consider is
the plasma, which contains water, salts and proteins. The second phase is composed of the
blood cells. These cells can be considered a fluid because they are mostly composed of
water surrounded by an elastic membrane (cell membrane), which itself exhibits fluid
properties (fluid-mosaic model). Currently, most blood simulations do not account for the
different fluid properties of the cells; however, for a more accurate solution, these cells
should be accounted for because they can transfer energy to the fluid (see Chapter 13).
Blood cells comprise approximately 40% of the blood volume and, in small blood vessels
(see Part 3), the cells can group together, forming a type of plug flow region. In micro-
vessels, pockets of blood that is composed only of plasma followed by pockets of blood
which is composed mostly of cellular matter are found. Although we will not develop
solutions for general two-phase flows, a few examples will be used to illustrate this phe-
nomenon later in this textbook.
2.10 CHANGES IN THE FUNDAMENTAL RELATIONSHIPS
ON THE MICROSCALE
In the previous sections of this chapter, many relationships between various fluid
mechanics parameters were described. These relationships still hold within the microcircu-
lation; however, some of the parameters become more critical and can govern the blood
flow throughout the vascular beds. Within the microcirculation, the viscosity of blood
must be described by the apparent viscosity (
). This is because blood is no longer a
homogenous fluid within the microcirculation. The blood vessel diameter approaches the
size of red blood cells (approximately 8 mm) and this causes a redistribution of the cells
and the plasma components into distinct regimes. It is possible and likely to have a plasma
pocket followed by a cellular pocket within the capillary and, therefore, different viscosi-
ties are associated with these two distinct phases (
Figure 2.22
). The apparent viscosity
would represent the overall viscosity of these distinct components. It is not an average, or
a weighted average, of these components, but a best estimate of the viscosity.
As shown in
Equation 2.1
, conservation of mass must now include the movement of
fluid out of the blood vessel. In the macrocirculation, we will typically consider that what-
ever fluid enters the inlet must leave via the outlet (unless there is an increase or decrease
η
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