Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 509

The microscale models are typically built on Lagrangian tracking of red blood
cells, platelets, and other cellular matter within the blood flow. As such, they are
often closely related to blood rheology and cellular mechanics. Many of these
models are used as microscale components of more general hybrid multiscale
models that include also components from macroscopic continuum models. Here
we only focus on some of the methods often used at the microscopic scale. Although
the models are in principle very similar to each other, at least three major groups can
be distinguished among them.

Euler-LagrangianParticleTrackingmethods(ELPT) . Within this subclass of
microparticle tracking methods we classify those where Lagrangian methods
are used for tracing particles, while an independent Eulerian description is used
for the fluid flow field. The particles are typically only coupled to the fluid flow
by a one-way coupling scheme, i.e., particle trajectories depend on the fluid
velocity field, which however is not affected by the presence and motion of the
particles.

The fluid field is described by the Navier-Stokes like equations for incompress-
ible, but possibly non-Newtonian fluid:

r u D 0 (7.4)

@ u u

@t C u r

D r p C divT (7.5)

The stress tensor T is given by an appropriate rheological constitutive relation.
The fluid velocity u .x;t/ and pressure p.x;t/ are first computed (independently
of particles' motion) and further used to evaluate the forces acting on each particle
that follows the trajectory being governed by the Newtonian second law of motion:

dv
dt .x 0 ;t/D F D

C F P C F G

m p (7.6)

This means that the particle at position x 0 in time t having the velocity v.x 0 ;t/
and mass m p accelerates due to the action of the drag force F D , pressure gradient
force F P , and gravity force F G . These forces can be expressed in terms of particle
parameters and actual fluid velocity field. For spherical particles with diameter d p
and density p the drag force can be expressed as

1
8 p d p C D . u .x 0 ;t/ v.x 0 ;t//j u .x 0 ;t/ v.x 0 ;t/j

F D

D (7.7)

Here the fluid velocity at the position x 0 is denoted by u .x 0 ;t/while v.x 0 ;t/stands
for the velocity of the particle, as mentioned above. The drag coefficient C D depends