4.6. ACCRETIV E G ROW T H
being absorbed upstream of the object. The absorption patterns show an
asymmetry in the absorption of nutrients at the sink nodes of the objects
for increasing Pe numbers. For the diffusion-limited case, the highest degree
of absorption is found at the tips of the object, while near the center hardly
any nutrient is being absorbed. The observation that the highest amount of
absorption is found at the upstream side of the object corresponds to the observation recorded in experimental work on the absorption offood-particles
by the stony coral Madracis mirabilis (see Fig. 2.5),where the highest amount
of food particle capture was found at the upstream side of the colony.
Within the given error bounds in Tables 4.4 and 4.5 from the standard
deviations sd; and sd, no particular flattening in the lobed and branching
forms can be detected. In both cases due to the effect of bidirectional flow
in the range of both the lobed and branching forms, objects with a roughly
radial symmetry develop.
The objects shown in Figs. 4.39 and 4.41 differ still in many aspects from
the growth forms shown in Fig. 1.1. One important difference between the
actual growth forms and the simulated ones, where the range of branching
forms in Fig. 4.41 seems to best approximate the actual forms, is the high
degree of irregularity in the formation of branches in Fig. 4.41. A simple
explanation for this irregularity is that in the model the highest deposition
velocities occur at the highest gradients of nutrients and are only slightly
limited by the deposition velocities in the previous layers. The main direction
of growth is not determined by the direction of growth in the previous
layer and influenced only by local nutrient gradients. Consequently the main
direction of growth may vary strongly after each deposition step, resulting
in a highly irregular form. In Fig. 4.37more regular forms were generated by
applying a growth axes mechanism in the accretive growth model using the
growth function (4.22), where the local growth velocity is also determined by
the direction of growth in the previous growth layers. As mentioned in the
first paragraph of this section, it might be plausible to apply a growth axes
mechanism in the accretive growth models .
An important issue, in the comparison with experimental work, is to
verify whether the simulated Pe numbers approximate the actual conditions.
In order to make the Pe numbers mentioned in the text above comparable to
Pe numbers in experimental work, all Pe numbers shown for the simulations
have to be multiplied by 144(the size of the simulation box, the characteristic
length scale L in (2.4». This yields an upper limit of Pe = 432 in our simulations. When we compare this result to the low flow velocity experiments
done by Sebens et al. (1997) , with an average flow velocity of 15 cm-s'" and
using a characteristic length scale of 10cm for a Madracis mirabilis colony,
this gives a value of D =3.5' 10- 5 rrr's'" in (2.4) for the diffusion coefficient of
the food particles. Assuming that there is a direct relation between the diffusion coefficient and the size of the diffusing particle (Vogel1988),this would
roughl y correspond to food particles with a size of about 5' 10- 3 m. This
size corresponds to the order of magnitude of food particles which Madracis
mirabilis colonies capture. For sponges the typical size of the food particles is
within the range 0.2' 106 m-50 ' 106 m (Brien et al. 1973), showing that the
Pe numbers in our simulations are below realistic Pe values for sponges. In
the paper by Ghorai and Hill (2000) an estimate for the value of D is provided
for Chlamydomonas nivalis cells. Suspensions of Chlamydomonas have been
used for culturing sponges under laboratory conditions (Brien et al. 1973).
143
being absorbed upstream of the object. The absorption patterns show an
asymmetry in the absorption of nutrients at the sink nodes of the objects
for increasing Pe numbers. For the diffusion-limited case, the highest degree
of absorption is found at the tips of the object, while near the center hardly
any nutrient is being absorbed. The observation that the highest amount of
absorption is found at the upstream side of the object corresponds to the observation recorded in experimental work on the absorption offood-particles
by the stony coral Madracis mirabilis (see Fig. 2.5),where the highest amount
of food particle capture was found at the upstream side of the colony.
Within the given error bounds in Tables 4.4 and 4.5 from the standard
deviations sd; and sd, no particular flattening in the lobed and branching
forms can be detected. In both cases due to the effect of bidirectional flow
in the range of both the lobed and branching forms, objects with a roughly
radial symmetry develop.
The objects shown in Figs. 4.39 and 4.41 differ still in many aspects from
the growth forms shown in Fig. 1.1. One important difference between the
actual growth forms and the simulated ones, where the range of branching
forms in Fig. 4.41 seems to best approximate the actual forms, is the high
degree of irregularity in the formation of branches in Fig. 4.41. A simple
explanation for this irregularity is that in the model the highest deposition
velocities occur at the highest gradients of nutrients and are only slightly
limited by the deposition velocities in the previous layers. The main direction
of growth is not determined by the direction of growth in the previous
layer and influenced only by local nutrient gradients. Consequently the main
direction of growth may vary strongly after each deposition step, resulting
in a highly irregular form. In Fig. 4.37more regular forms were generated by
applying a growth axes mechanism in the accretive growth model using the
growth function (4.22), where the local growth velocity is also determined by
the direction of growth in the previous growth layers. As mentioned in the
first paragraph of this section, it might be plausible to apply a growth axes
mechanism in the accretive growth models .
An important issue, in the comparison with experimental work, is to
verify whether the simulated Pe numbers approximate the actual conditions.
In order to make the Pe numbers mentioned in the text above comparable to
Pe numbers in experimental work, all Pe numbers shown for the simulations
have to be multiplied by 144(the size of the simulation box, the characteristic
length scale L in (2.4». This yields an upper limit of Pe = 432 in our simulations. When we compare this result to the low flow velocity experiments
done by Sebens et al. (1997) , with an average flow velocity of 15 cm-s'" and
using a characteristic length scale of 10cm for a Madracis mirabilis colony,
this gives a value of D =3.5' 10- 5 rrr's'" in (2.4) for the diffusion coefficient of
the food particles. Assuming that there is a direct relation between the diffusion coefficient and the size of the diffusing particle (Vogel1988),this would
roughl y correspond to food particles with a size of about 5' 10- 3 m. This
size corresponds to the order of magnitude of food particles which Madracis
mirabilis colonies capture. For sponges the typical size of the food particles is
within the range 0.2' 106 m-50 ' 106 m (Brien et al. 1973), showing that the
Pe numbers in our simulations are below realistic Pe values for sponges. In
the paper by Ghorai and Hill (2000) an estimate for the value of D is provided
for Chlamydomonas nivalis cells. Suspensions of Chlamydomonas have been
used for culturing sponges under laboratory conditions (Brien et al. 1973).
143