90
S. Lee et al.
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
0 = φσ
+ σ φ
1 + c 1
φ
φ m − φ
+ c 0 (1 − φ)
ρ s,1 X + ρ s,2 (1 − X )
,
(4.9)
where X ≡ φ 1 /φ, while c 0 ≡ 2 cot α/(9K c ) and c 1 ≡ 2(K v − K c )/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. φ 1 = 0) or 1 (i.e. φ 2 = 0). For equilibrium inside the thin film, we
also require zero net flux of each particle species in the z-direction, J z,i = 0, and set
J z,1 φ 2 − J z,2 φ 1 = 0, which leads to
X
= c 2
X (1 − X )
σ D tr
φ m
φ m − φ
,
(4.10)
where c 2 = 2(ρ s,2 − ρ s,1 ) cot α/9.
The Eqs. (4.8)–(4.10) form a system of ODEs for the unknowns: φ, X and σ .
Following [16], we define the scaled height s = z/ h, where h is the dimensionless
film thickness, so that
φ(s) = φ(hs), X (s) = X (hs), and σ (s) = σ (hs)/ h; tildes are
subsequently dropped from the text. In addition, the average particle concentration
φ 0 and proportion of lighter particles X 0 correspond to:
φ 0 =
1
0
φ(s)ds,
X 0 =
1
φ 0
1
0
X (s)φ(s)ds.
(4.11)
For given φ 0 and X 0 with 0 ≤ φ 0 < φ m , the system has a unique solution for
s ∈ [0, 1]. Solutions in Sect. 4.3 are computed via shooting in MATLAB, with an
inclination angle fixed at α = 30 ◦ unless otherwise noted.
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 φ c such that φ(s) is monotone increasing (i.e.
φ > 0) when φ 0 > φ c and monotone decreasing (i.e. φ < 0) when φ 0 < φ c .
The constant solution φ = φ 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 φ c,i the critical concentration for the ith species in the
S. Lee et al.
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
0 = φσ
+ σ φ
1 + c 1
φ
φ m − φ
+ c 0 (1 − φ)
ρ s,1 X + ρ s,2 (1 − X )
,
(4.9)
where X ≡ φ 1 /φ, while c 0 ≡ 2 cot α/(9K c ) and c 1 ≡ 2(K v − K c )/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. φ 1 = 0) or 1 (i.e. φ 2 = 0). For equilibrium inside the thin film, we
also require zero net flux of each particle species in the z-direction, J z,i = 0, and set
J z,1 φ 2 − J z,2 φ 1 = 0, which leads to
X
= c 2
X (1 − X )
σ D tr
φ m
φ m − φ
,
(4.10)
where c 2 = 2(ρ s,2 − ρ s,1 ) cot α/9.
The Eqs. (4.8)–(4.10) form a system of ODEs for the unknowns: φ, X and σ .
Following [16], we define the scaled height s = z/ h, where h is the dimensionless
film thickness, so that
φ(s) = φ(hs), X (s) = X (hs), and σ (s) = σ (hs)/ h; tildes are
subsequently dropped from the text. In addition, the average particle concentration
φ 0 and proportion of lighter particles X 0 correspond to:
φ 0 =
1
0
φ(s)ds,
X 0 =
1
φ 0
1
0
X (s)φ(s)ds.
(4.11)
For given φ 0 and X 0 with 0 ≤ φ 0 < φ m , the system has a unique solution for
s ∈ [0, 1]. Solutions in Sect. 4.3 are computed via shooting in MATLAB, with an
inclination angle fixed at α = 30 ◦ unless otherwise noted.
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 φ c such that φ(s) is monotone increasing (i.e.
φ > 0) when φ 0 > φ c and monotone decreasing (i.e. φ < 0) when φ 0 < φ c .
The constant solution φ = φ 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 φ c,i the critical concentration for the ith species in the