The emphasis in this book is on the sedimentation of particles which are small enough for inertial and unsteady effects to be neglected, but large enough to make Brownian motion neglible.

Particle suspensions are ubiquitous across a variety of engineering processes, and examples can be found in oilfield applications, industrial separations, microfluidics, and additive manufacturing. In these applications, it is crucial to control and predict the mobility of the particles -- that is, their motion through a fluid due to an applied force. Often, that applied force is due to gravity, and the process of interest involves the sedimentation of rigid, non-Brownian particles. In many industrially-relevant processes, the suspending fluid is a polymeric fluid, which can exhibit both viscous and elastic flow behavior. In this work, we examine the effect of fluid elasticity on the motion of settling particles. To do so, we use a combination of experiments and large-scale numerical simulations to build a fundamental understanding of how and why the settling motion of spherical particles changes in elastic fluids. We begin by studying the motion of a single spherical particle in a model elastic Boger fluid. Initially, we address the case where a shear flow is imposed in a plane perpendicular to the sphere's motion, referred to as cross (or orthogonal) shear. We study the settling motion in highly elastic fluids, where the sphere's motion and the shear flow both result in significant stretching of the polymers in solution. We find that the shear flow results in polymer tension along the fluid streamlines and creates regions of high polymer stretching in the wake of the sphere which extend into the shear flow direction. We observe that these viscoelastic wake structures, resembling wings, are linked to an increase in the pressure drag, which drives a dramatic decrease in the particle's settling rate in the presence of a cross shear flow. In a surprising extension to this work, we show that rotation of a spherical particle (around the axis aligned with its motion) in an otherwise quiescent elastic fluid can result in the opposite trend: an increase in the sphere's settling rate as a function of its rotation rate. In this case, we propose a mechanism based on the generation of hoop stresses around the rotating and sedimenting sphere. Returning to the case of a spherical particle settling through a sheared elastic fluid, we find that the coupling between the particle's motion and an external shear flow depends on the direction of the applied force relative to the shear flow. Interestingly, when the particle settles in either the shear gradient or shear flow direction, a lateral drift becomes apparent. Utilizing the understanding gained from our single particle studies, we subsequently address the settling motion of a suspension of rigid particles at finite volume fraction. When the fluid is quiescent, we observe a characteristically distinct settling behavior in a viscoelastic suspending fluid compared to a Newtonian fluid: in the viscoelastic fluid, we observe the formation of particle-rich regions which settle more quickly, resulting in an inhomogeneous settling behavior and an overall enhanced settling rate. We propose that this structural concentration instability is driven in part by the lateral drift of particles in elastic fluids due to local concentration variations following random mixing. Alternatively, when a cross shear flow is applied, a hindered settling rate is observed -- we attribute this result to both the effect of elasticity in a cross sheared viscoelastic fluid (as initially addressed for a single particle) and the mixing of the suspension structure due to the shear flow. These results have significant implications for engineering applications involving suspensions of particles settling in both quiescent and flowing polymeric fluids. In summary, we examine the nonlinear coupling between the settling motion of a particle and a surrounding flow field in elastic fluids through a number of fundamental examples. Using experiments and simulations, we infer the coupling relations (when possible) and propose mechanisms to describe them on a physical basis. We use this knowledge to then study the industrially-relevant problem of a suspension of settling particles, with or without an applied flow. This work provides a framework for better understanding and predicting the settling behavior of rigid particles in polymeric fluids.