Equilibrium Theory of Bidensity Particle-Laden Flows on an Incline - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 85

In the current work, we extend the equilibrium model of [ 15 ] to thin free surface
flows of bidensity suspensions. This equilibrium theory is a crucial component of the
dynamic problem, since the leading order dynamic equations have shock solutions
whose structure is determined by the equilibrium profiles [ 14 , 26 ]. This warrants
a careful study of the equilibrium problem before proceeding to the dynamic case,
analogous to the work of [ 16 ]. Distinct from the monodisperse case, tracer diffusivity
is included in the bidensity model and compared to recent experimental results by
[ 9 ]. This work provides an important theoretical framework for segregating particles
of different densities, which has industrial applications.
This chapter is organized as follows. In Sect. 4.2 , we introduce the governing
equations for bidensity suspensions and develop the equilibrium model by applying
lubrication approximations. In Sect. 4.3 , we obtain the solution to the equilibrium
model for varying parameters to validate previous experimental results. The chapter
concludes with the summary of results and discussion of future directions in Sect. 4.4 .

4.2 Problem Formulation

We consider the dynamics of a bidensity slurry flowing down an incline, in which
the mixture consists of a viscous fluid with density μ l and two

species of negatively buoyant particles (See Fig. 4.1 ). The two particle types have
uniform diameter d but variant densities,

ˁ l and viscosity

ˁ 1 and ˁ 2 , such that ˁ 2 >ˁ 1 >ˁ l .The

ˆ 2 , respectively, while
ˆ = ˆ 1 + ˆ 2 is the total volume fraction. By assuming a sufficiently small particle
size, the particle-fluid mixture is modelled as a continuum and is governed by the
following momentum equations:

ˆ 1 and

local volume fractions of each species are denoted as

u ) ˁ (∂ t u + u ·∇ u ) = ∇ · − p I + μ( ∇ u + ∇ + ˁ g , (4.1)

where u and p are the velocity vector and pressure, respectively, and g denotes the
gravitational acceleration vector. As in [ 15 , 16 ], the mixture density, ˁ , is given by

− ˆ/ˆ m ) − 2 , ˁ = ˁ 1 ˆ 1 + ˁ 2 ˆ 2 + ˁ l ( 1 − ˆ) , while effective viscosity μ = μ l ( 1

where ˆ m is the maximum volume fraction. In addition to momentum, we have mass

conservation of the mixture:

∂ t ˁ + ∇ · (ˁ u ) = 0 . (4.2)

The velocity satisfies the no-slip condition ( u 0) on the bottom of the channel,

while the stresses vanish both in normal and tangential directions on the free sur-
face: n

=

· −

u ) =

+ μ( ∇ + ∇

p I u 0. The free surface also satisfies the kinematic

boundary condition, n · u = 0.