Biomedical Engineering Reference
In-Depth Information
process that leads to the governing equations for local volume average quan-
tities (e.g., velocity, temperature, mass concentration). The next step is to
develop and to solve the closure problem for the spatial deviations of the
spatially smoothed quantities to determine the effective transport coecients
that appear in the spatially averaged equations. A comprehensive review of
this method was presented in the monograph “
The Method of Volume Aver-
aging
” by Whitaker (1999). In the present study, we consider a homogeneous,
isotropic porous medium containing a dilute suspension of gravitactic microor-
ganisms. The suspension is assumed to behave as a Newtonian incompressible
fluid while the microorganisms swim through the porous medium randomly
but with an average upward velocity,
V
c
. Furthermore, the pore size should
be significantly larger than the cell size so that the flow through the pores will
not cause the cells to tumble and drastically affect their ability to reorient
(Hill and Pedley 2005).
The objective of this part is to determine the critical conditions for the
onset of bioconvection in term of the gravitactic velocity,
V
c
, and to investigate
the postonset status. The problem under consideration is schematically shown
in Figure 14.2. Within the Boussinesq approximation (see Appendix), the
fluid flow and microorganism concentration may be described by the following
equations (Khaled and Vafai 2003; Nguyen-Quang et al. 2005; Nguyen-Quang
2006).
Continuity equation
V
∗
= 0
∇·
(14.1)
Darcy equation
µ
K
V
∗
+
gρ
= 0
P
∗
−
−
∇
(14.2)
Y
g
V
c
H
X
O
L
e
2
e
1
FIGURE 14.2
Geometry of the system.
Search WWH ::
Custom Search