Hydrodynamics in Porous Media with Applications to Tissue Engineering - Porous Media: Applications in Biological Systems and Biotechnology - page 101

of PCL by fused deposition modeling (a rapid prototyping technique [Zein
et al . 2002]) present a periodic fibrous architecture with sucient simple geom-
etry well adapted to modeling. They constitute a bioresorbable and porous
structure made of a simple network displaying fully interconnected pores
(porosity of the order of 60%, pore and fiber diameters about a = h = 250

m).
A suciently simple three-dimensional geometry of the pore channel was then
defined taking into account the symmetries of the implant architecture: we
considered here that each constitutive fibrous layer has a fiber direction per-
pendicular to the previous one (as schematically shown in Figure 3.8). More
precisely, an elementary representative test section containing 10 cylindrical
fibers ( l Ch =2 . 5 mm) was studied. Liquid flows from left to right perpendicu-
larly to the inlet edge as seen on Figure 3.9, and the cell layer was considered
as a homogeneous thin monolayer attached around the fibers.
The reference values of entry data model's parameters, coming from exper-
imental conditions of our biologist colleagues or from the literature, are given
in Table 3.2.
As expected for such a flow dominated by viscous effects (Reynolds num-
ber of the order of 10 − 2 ), the structure of the velocity field reproduces the
waviness, periodicity, and symmetry of the substrate structure, as shown by
the flow field and spiraling streamlines given in Figure 3.9.
Despite this smallness of the Reynolds number, a secondary flow due to
the cross crenellation of the channel is generated by a tiny amount of inertial
effects as shown in the helicity of the streamlines and the vortical structure
of this secondary flow field.
The analysis of such a flow field is relevant of the fluid dynamics inside
slowly curved pipe, where the response depends upon the Dean number Dn
defined as (Dean 1927)

µ

a / r c 2

Dn = Re × (3.26)

where r c is the radius of curvature of the streamlines.
The resulting velocity, u t , characterizing the transverse secondary flow
takes then the following approximate value:

( a / r c ) 2

u t ≈ u 0 × Dn × (3.27)

Such a velocity is to compare with the apparent velocity u D for oxygen
diffusion as defined by

u D ≈ D O 2 / a (3.28)

The result is

( a / r c ) 2

u t / u D ≈ Pe × Dn ×

(3.29)

The field of this secondary transverse flow is illustrated by arrows in
Figure 3.10.
For the reference data concerning this model, the obtained magnitude
(around 10 − 6 m sec − 1 ) of such a recirculation flow velocity is slightly lower

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