Chapter 4
Equilibrium Theory of Bidensity
Particle-Laden Flows on an Incline
Sungyon Lee, Jeffrey Wong and Andrea L. Bertozzi
Abstract The behaviour of inhomogeneous suspensions in a viscous oil is relevant in
the context of oil spill and other oil-related disasters which may lead to the unwanted
mixture of sand grains and oil. This warrants the fundamental study of the dynamics
of solid particles in a thin film of viscous fluid. Specifically, sheared concentrated
suspensions in a viscous fluid are subject to a diffusive mechanism called shearinduced migration that consists of “drift diffusion” and “self or tracer diffusion”.
Drift diffusion causes particles to move from high to low concentrations, while tracer
diffusion dictates mixing between particles of the same size. The latter mechanism
becomes important in polydisperse slurries. In this chapter, we incorporate the effects
of shear-induced migration and sedimentation to develop a model for the gravitydriven thin film of bidensity suspensions. We use this mathematical model to validate
recent experimental results.
4.1 Introduction
Particle-laden flows are ubiquitous in nature and in industrial applications; however,
the nonlinear coupling between particles and fluid motion presents challenges in the
development of mathematical models. In the case of monodisperse slurries, there
have been advances both in experiments and modelling based on diffusive flux phenomenology [12] and, more recently, suspension balance approach [17]. In particular,
[12] developed a diffusive model to justify the behaviour of sheared monodisperse
S. Lee (B)
Departmant of Mechanical Engineering, Texas A&M,
College Station, TX 77843, USA
e-mail: sungyon.lee@tamu.edu
J. Wong · A.L. Bertozzi
Department of Mathematics and Applied Mathematics Laboratory,
University of California, Los Angeles, CA 90095, USA
e-mail: jtwong@math.ucla.edu
A.L. Bertozzi
e-mail: bertozzi@math.ucla.edu
© Springer International Publishing Switzerland 2015
M. Ehrhardt (ed.), Mathematical Modelling and Numerical Simulation of Oil
Pollution Problems, The Reacting Atmosphere 2, DOI 10.1007/978-3-319-16459-5_4
85
Equilibrium Theory of Bidensity
Particle-Laden Flows on an Incline
Sungyon Lee, Jeffrey Wong and Andrea L. Bertozzi
Abstract The behaviour of inhomogeneous suspensions in a viscous oil is relevant in
the context of oil spill and other oil-related disasters which may lead to the unwanted
mixture of sand grains and oil. This warrants the fundamental study of the dynamics
of solid particles in a thin film of viscous fluid. Specifically, sheared concentrated
suspensions in a viscous fluid are subject to a diffusive mechanism called shearinduced migration that consists of “drift diffusion” and “self or tracer diffusion”.
Drift diffusion causes particles to move from high to low concentrations, while tracer
diffusion dictates mixing between particles of the same size. The latter mechanism
becomes important in polydisperse slurries. In this chapter, we incorporate the effects
of shear-induced migration and sedimentation to develop a model for the gravitydriven thin film of bidensity suspensions. We use this mathematical model to validate
recent experimental results.
4.1 Introduction
Particle-laden flows are ubiquitous in nature and in industrial applications; however,
the nonlinear coupling between particles and fluid motion presents challenges in the
development of mathematical models. In the case of monodisperse slurries, there
have been advances both in experiments and modelling based on diffusive flux phenomenology [12] and, more recently, suspension balance approach [17]. In particular,
[12] developed a diffusive model to justify the behaviour of sheared monodisperse
S. Lee (B)
Departmant of Mechanical Engineering, Texas A&M,
College Station, TX 77843, USA
e-mail: sungyon.lee@tamu.edu
J. Wong · A.L. Bertozzi
Department of Mathematics and Applied Mathematics Laboratory,
University of California, Los Angeles, CA 90095, USA
e-mail: jtwong@math.ucla.edu
A.L. Bertozzi
e-mail: bertozzi@math.ucla.edu
© Springer International Publishing Switzerland 2015
M. Ehrhardt (ed.), Mathematical Modelling and Numerical Simulation of Oil
Pollution Problems, The Reacting Atmosphere 2, DOI 10.1007/978-3-319-16459-5_4
85