Biomedical Engineering Reference
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given by
D
∂φC
∂x
+
∂vφC
∂x
∂φC
∂t
∂
∂x
Zone 1:
=
D
∂φC
∂x
+
∂vφC
∂x
∂φC
∂t
∂
∂x
Zone 2:
=
+
S
(
x, t
)
(10.36)
D
∂φC
∂x
∂φC
∂t
∂
∂x
Zone 3:
=
where
C
is the solute concentration in a volume of pore water,
φ
is the porosity,
S
(
t,x
) represents the source of solute because of injection of overlying water
at feeding depth,
v
(
t, x
) is the velocity of the advectively recirculating water
and
D
(
x
) represent the apparent diffusion coecient in the sediment. Dividing
pumping rate by the area of advection column and sediment porosity, they
estimated the advective velocity used in equation (10.36).
A more recent model is presented by Meysman et al. (2006) for the same
bioturbator, called the two-dimensional pocket injection model, which was
regarded as the advective counterpart of Aller's “diffusive” two-dimensional
tube bioirrigation model. They started from Darcy-Brinkman-Forchheimer
equation as a general equation to model pore flow and neglected Brinkman-
Forchheimer effects because of the low-pore velocity and large-length scales
compared to the Brinkman layer involved in the problem, and finally employed
the momentum balance reduced to Darcy's law (10.37).
k
µ
(
v
d
=
−
∇
p
−
ρg
∇
x
)
(10.37)
In the above equation,
k
is the permeability,
µ
is the dynamic viscosity of
the pore water,
ρ
is the pore water density,
g
is the gravitational acceleration,
and
x
is the vertical coordinate. The Darcy velocity
v
d
is related to the actual
velocity of pore water as
v
d
=
φv
, where
φ
is the porosity. A commercially
available code (Comsol Multiphysics) was employed to solve (10.38) and the
results were used to solve the reactive transport equations for concentrations.
To date, no exclusive modeling is performed on bioturbation because of
U-shape burrows, nor is there any model to contribute the animal's motion
characteristics to the flow generated along the burrows and in the sediment.
Beside this issue, there exists still a good number of challenging questions.
One of the most prominent one is whether or not the pumping strength of
C. plumosus
is sucient to mediate an additional advective flow through
the burrow walls they construct. Some recent simple models are available
for this problem (Shull et al. 1995), however, a real evidence in the form of
a two-dimensional flow in a porous-fluid-burrow domain has not yet been
provided. Such a model is under development (Morad et al. 2010b) for the
geometry shown in Figure 10.19 (a) with a porous sediment having a fixed
tube with permeable walls underneath a fluid layer. Using high speed cameras,
the equation for the larva's motion has been obtained by digital analysis,
and inserted as an input to the model. By solving the momentum equations
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