4 Equilibrium Theory of Bidensity Particle-Laden Flows on an Incline
93
Fig. 4.3 Dependence of the width w of the mixing region (where 0.05 < X < 0.95) on tan α; the
relationship, shown on the left, is approximately linear. The right panel shows the X profiles for
α = 10, 30, 50 ◦
Fig. 4.4 The left panel shows a phase diagram of solution regimes determined by total and relative
concentrations (X 0 , φ 0 ), separated by transition curves φ A , φ B and φ C . Profiles in each region are
shown below, for total concentration φ (left) and for the individual concentrations φ 1 , φ 2
Ridged A (φ A < φ 0 < φ B ): When φ c,1 < φ 0 < φ c,2 , the monodisperse theory
suggests that the lighter species φ 1 is more likely to aggregate on the free surface
(φ 1 → φ m as s → 1) due to shear-induced migration while φ 2 favours settling.
Therefore, in this regime, φ 1 is monotonically increasing (i.e. φ
1 > 0) with the
lighter particles mostly confined to a top region [s tr , 1]. The heavier particles settle
93
Fig. 4.3 Dependence of the width w of the mixing region (where 0.05 < X < 0.95) on tan α; the
relationship, shown on the left, is approximately linear. The right panel shows the X profiles for
α = 10, 30, 50 ◦
Fig. 4.4 The left panel shows a phase diagram of solution regimes determined by total and relative
concentrations (X 0 , φ 0 ), separated by transition curves φ A , φ B and φ C . Profiles in each region are
shown below, for total concentration φ (left) and for the individual concentrations φ 1 , φ 2
Ridged A (φ A < φ 0 < φ B ): When φ c,1 < φ 0 < φ c,2 , the monodisperse theory
suggests that the lighter species φ 1 is more likely to aggregate on the free surface
(φ 1 → φ m as s → 1) due to shear-induced migration while φ 2 favours settling.
Therefore, in this regime, φ 1 is monotonically increasing (i.e. φ
1 > 0) with the
lighter particles mostly confined to a top region [s tr , 1]. The heavier particles settle