Environmental Engineering Reference
In-Depth Information
reduces Eq. (
4.3
)to
J
z
=
0 at leading order. This scaling indicates that particles must
rapidly equilibrate in the the
z
-direction [
16
]. Integrating
J
z
=
0 with respect to
z
and applying
J
z
(
z
=
0
)
=
0 yields
J
z
=
0, or
=
ˆ˃
+
˃ˆ
1
ˆ
ˆ
m
−
ˆ
0
+
c
1
+
c
0
(
1
−
ˆ)
ˁ
s
,
1
X
+
ˁ
s
,
2
(
1
−
X
)
,
(4.9)
where
X
K
c
are constants.
As expected, Eq. (
4.9
) exactly matches the monodisperse model of [
15
,
16
], when
X
is set to 0 (i.e.
≡
ˆ
1
/ˆ
, while
c
0
≡
2 cot
ʱ/(
9
K
c
)
and
c
1
≡
2
(
K
v
−
K
c
)/
0). For equilibrium inside the thin film, we
also require zero net flux of each particle species in the
z
-direction,
J
z
,
i
ˆ
1
=
0) or 1 (i.e.
ˆ
2
=
=
0, and set
J
z
,
1
ˆ
2
−
J
z
,
2
ˆ
1
=
0, which leads to
,
c
2
X
(
1
−
X
)
ˆ
m
ˆ
m
−
ˆ
X
=
(4.10)
˃
D
tr
where
c
2
=
9.
The Eqs. (
4.8
)-(
4.10
) form a system of ODEs for the unknowns:
2
(ˁ
s
,
2
−
ˁ
s
,
1
)
cot
ʱ/
ˆ
,
X
and
˃
.
Following [
16
], we define the scaled height
s
=
z
/
h
, where
h
is the dimensionless
film thickness, so that
ˆ(
),
X
h
; tildes are
subsequently dropped from the text. In addition, the average particle concentration
ˆ
0
and proportion of lighter particles
X
0
correspond to:
s
)
=
ˆ(
hs
(
s
)
=
X
(
hs
)
, and
˃(
s
)
=
˃(
hs
)/
1
0
ˆ(
1
1
ˆ
0
ˆ
0
=
)
,
X
0
=
(
)ˆ(
)
.
s
d
s
X
s
s
d
s
(4.11)
0
For given
ˆ
0
and
X
0
with 0
≤
ˆ
0
<ˆ
m
, the system has a unique solution for
s
. Solutions in Sect.
4.3
are computed via shooting in MATLAB, with an
inclination angle fixed at
∈[
0
,
1
]
30
ⓦ
unless otherwise noted.
ʱ
=
4.3 Results
We begin by briefly reviewing the monodisperse theory described by [
15
]. For the
monodisperse system which consists of (
4.9
) and (
4.8
) with
X
=
0 or 1, there
is a critical particle concentration
ˆ
c
such that
ˆ(
s
)
is monotone increasing (i.e.
ˆ
ˆ
>
0) when
ˆ
0
>ˆ
c
and monotone decreasing (i.e.
<
0) when
ˆ
0
<ˆ
c
.
The constant solution
ˆ
=
ˆ
c
separating the two regimes is an unstable equilibrium.
This bifurcation is illustrated in Fig.
4.2
.In[
15
], the two regimes are referred to as
'ridged' and 'settled', respectively. Physically, ridged solutions describe aggregation
of particles at the fluid surface, while a settled solution describes particles settling to
the substrate, which leaves a clear fluid layer above. As there are two particle species
to consider here, we denote as
ˆ
c
,
i
the critical concentration for the
i
th species in the
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