Particle dynamics and focusing in confined inertial shear and Poiseuille flows through lattice Boltzmann computations

We calculate the translational and rotational dynamics of a rigid particle undergoing inertial focusing in confined inertial

ow of an incompressible Newtonian fluid. A particle in confined inertial ow will migrate toward an equilibrium

position dependent on the type off flow, the flow parameters, and the particle dimensions. We compute the trajectory of a particle using the lattice Boltzmann method, a computational fluids dynamics technique for modeling

the dynamics of a particle in Newtonian fluid governed by the Navier-Stokes equation. Our simulations determine the rotational and translational dynamics of the particle, from which we can quantify the equilibrium position and

focusing time. First, we calculate the inertial lift force on and migration of a circular cylinder in two-dimensional, confined shear flow at finite particle Reynolds number. A pitchfork bifurcation of the equilibrium position of the cylinder occurs beyond a critical particle Reynolds number, which is dependent on the confinement ratio of the particle. Below the critical Reynolds number, the circular cylinder migrates to a single stable equilibrium position at the center of the channel. After the bifurcation, there are three equilibrium

positions for the particle: two stable equilibrium positions equidistant from the center, and a single unstable equilibrium position at the center. Next, we extend our calculations to the migration of a spherical particle in confined, planar shear ow at finite particle Reynolds number. A pitchfork bifurcation of the equilibrium position of the sphere similarly occurs beyond a critical particle Reynolds number, again dependent on the confinement ratio of the particle. The bifurcation produces three equilibrium positions above the critical Reynolds number, with one unstable and two stable equilibrium positions. An external body force applied to the particle such as gravity can alter the bifurcation, creating an imperfect bifurcation or breaking the bifurcation altogether based on the magnitude of the external force. The sphere undergoes additional translational dynamics in time-dependent ow above the critical particle Reynolds number, inducing particle migration between the equilibrium position and the center of the channel. Finally, we calculate the inertial migration of a sphere and spheroid in steady and oscillatory Poiseuille ow. A sphere focuses at an equilibrium position primarily dependent on the channel

Reynolds number, with only a weak dependence on the period of oscillation of the ow. A spheroid only reaches a time-averaged equilibrium position, as the rotation of the particle induces additional transverse migration from

the instantaneous orientation. The time-averaged equilibrium position of the spheroid is dependent upon the channel Reynolds number and the particle aspect ratio.