Biomedical Engineering Reference
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(a)
(b)
Streamlines
Streamlines
v s
v s
FIGURE 10.13
Schematics of the flow around a solid sphere (a) versus a porous sphere (b).
In the latter case a partial throughflow also exits.
More recently, Bhattacharyya et al. (2006) have considered a circular cylin-
drical porous structure, and showed that porosity and permeability of the
aggregate drastically alter the patterns of streamlines, vorticity, and nutrient
transfer.
Because the aggregates have, in general, a complex shape, an ecient
LBM code has been developed (Liu and Khalili 2008, 2009), which has made
possible to treat not only spherical, but also arbitrarily-shaped porous domains
(Liu and Khalili 2010b). A comparison between the nutrient release from a
solid aggregate versus that of a porous aggregate solved by the model of Liu
and Khalili (2010b) can be seen in Figure 10.14. Furthermore, while current
calculations have assumed constant porosity, the LBM code can easily account
for spatially heterogeneous porosity.
For comparison, a complex geometry is given in Figure 10.15b which con-
sists of four different porous subdomains, a square, a rectangle, an oval, and
a circle, which all lie within the same viscous ambient fluid.
As shown in the figure, the flow past the aggregate partially passes through
the porous bodies and partially bypasses them. In the example shown, all
subdomains have the same fixed porosity of φ =0 . 993.
The literature discussed so far invariably assumed a homogeneous fluid
density. However, in lakes, oceans, and estuaries, vertical density gradients
within the water column are ubiquitous. In freshwater systems, density gradi-
ents are caused by a decrease of temperature with depth, while in the ocean
it is often salinity that increases with depth. The strength of the stratification
is quantified by the Brunt-Vaisala frequency, N =
( g/ρ 0 )( ∂ρ/∂z ), which
measures the natural frequency of a fluid parcel in a stable density gradient,
 
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