Environmental Engineering Reference
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ˆ 2
<
so that
0, with negligible but non-zero concentration beyond s tr . Focusing
on the top layer
[
s tr ,
]
1
where X
1, we view this as a perturbed version of the
monodisperse bifurcation. As
ˆ 0 is increased across
ˆ A , the average concentration
of lighter particles in
[
s tr ,
1
]
becomes large enough to produce a ridged solution.
Ridged B (
ˆ B 0 C ): If
ˆ 0 c , 2 ,
then
ˆ
is always monotone increasing
ˆ
(
0), which defines this second ridged regime, R B . The heavier particles still
settle to the substrate as in the R A case, so that
<
ˆ 2
<
0. Note that, in the absence
of tracer diffusivity (no mixing layer),
ˆ B
= ˆ c , 2 , unless X
=
1 exactly, leading to
ˆ B = ˆ c , 1 . This (discontinuous)
ˆ B closely approximates the actual curve in Fig. 4.4 ,
which curves due to the mixing of particle layers but retains the same endpoints.
There are two situations for
ˆ 0 c , 2 in which tracer diffusion can produce a ridged
solution. First, if
ˆ c , 2 , then even a small concentration of lighter
particles can perturb the otherwise settled solution in the heavier layer so that
ˆ 0 is very close to
ˆ >
0.
Second, if X 0 is close to 1 and
ˆ 0 c , 1 , then there is no well-defined settled layer
of heavier particles, so the ridged behaviour of the lighter layer ensures that
ˆ >
0.
Ridged C (
ˆ C
0 m ): For sufficiently large
ˆ 0 , the average concentration
of
ˆ 2 near s
=
0 (where X
0) is large enough to produce an initially increasing
solution in
ˆ 2 . Thus, distinct from R A and R B , the heavier particles tend to migrate
away from the substrate in this last ridged regime. However, the lighter particles
displace the heavier particles near the free surface so that
ˆ 2 is still
not monotone increasing—it eventually decreases sharply to nearly zero around s tr .
ˆ 2
0. Hence
4.4 Conclusions
The same pattern of transitions is also observed for fixed
ˆ 0 with varying X 0 and
ʱ
,
both experimentally and theoretically. As with increasing
has the
effect of altering the balance of fluxes to favour shear-induced migration, in this case
by reducing the normal component of gravity [ 15 ]. The previous discussion applies
to the
ˆ 0 , an increase in
ʱ
(
X 0 ,ʱ)
plane as well, and, in particular, there is a critical
ʱ A (
X 0 )
, analogous to
ˆ A (
, separating settled and ridged solutions. The predicted bifurcation is shown in
Fig. 4.5 along with experimental results [ 10 ] identifying settled or ridged behaviour.
Experiments to date have not measured the particle concentrations inside the thin
film and thus do not distinguish between different theoretically predicted types of
ridged behaviour. Overall, the current theory captures the bifurcation curve obtained
experimentally, although the critical angles predicted by the model are greater than
what is measured in the experiment by about five degrees. This discrepancy can
be attributed primarily to the value of the empirical parameter K c . While we based
the value of K c on [ 15 , 16 ], the types of beads used in [ 10 ] differ slightly in size
and texture from the previous experiments and warrant further experiments to better
estimate K c .
In this chapter, we derive a diffusive model of bidensity suspensions flowing down
an incline and use it to describe the normal equilibrium of the suspensions inside the
thin film. The mixture consists of the viscous fluid of density,
X 0 )
ˁ l , and two negatively
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