Microvascular Beds - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 198

Under the Eulerian approach, the average fluid properties for each phase would be
used independently to define the motion of the total fluid. All of the equations that have
been developed in previous chapters (for mass conservation, momentum conservation,
and Navier-Stokes, among others) apply to each phase individually. There is an extra
boundary condition that the interface of the phases must have the same fluid properties
(e.g., velocity, pressure, stress), so that there is a balance between the two phases. The dif-
ficulty with this approach would be defining where the interface between the two phases
occurs within the fluid regime. Also, for a problem where blood is the fluid, there may be
more than one interface, which makes the computations very time-consuming by hand.
For a 50-

m-diameter blood vessel, there can be a total of five red blood cells aligned
within one particular cross section (if they are flowing face-on). To solve this problem
using two-phase flow methods, the exact distance between each cell would need to be
known. Instead of using the Navier-Stokes equations to solve for the entire flow field
within the 50-

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m-diameter vessel, each section of the vessel would need to be solved inde-
pendently and the boundary conditions at each interface would need to be applied for the
solution of a different section, which would end up developing a number of coupled dif-
ferential equations.

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6.10 INT ERACTIONS BETWEEN CELLS AND THE VESS ELWALL

Within the microcirculation, cell-wall interactions need to be considered to accurately
depict the flow field. As discussed previously, red blood cells squeeze through the
capillaries, effectively shearing against the endothelial cell wall. In slightly larger vessels
(25 to 50 m in diameter), red blood cells can rebound off of the wall, changing the flow

profile of the cell and the blood. Also, it is common for white blood cells to stick to the
vascular wall and roll along it. This changes the cross-sectional area and affects where the
no-slip boundary condition would occur. In larger vessels, it is unlikely that any cell-wall
interaction will disturb the flow profile.
To model these interactions, we need to make a number of assumptions for the flow
around the cells. First, we will assume that there is a no-slip condition at the vessel wall
and along the particle. This means that the fluid velocity is matched along the cell wall; if
the cell is within the free fluid this leads to the imbalance along the cells, which pushes
them to the flow centerline. We will however make the assumption that the particles are
neutrally buoyant which means that the forces acting on the cell must summate to zero.
This allows us to compute a solution by hand instead of using numerical methods (see
Chapter 13).
In order to model these conditions, the mechanical properties of the cells should be
developed first. For red blood cells, it has been seen that the bending moment of the cell is
very important to allow the cell to squeeze through a small capillary. For instance, if the
red blood cell is too stiff to fit through a capillary, then it seals off the blood flow prevent-
ing other red cells from entering that capillary. If the red blood cell is too pliant, then
the flow stresses can easily shear the cell, preventing it from functioning properly. For

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