Kinematics of a symmetrically confined cylindrical particle in a “Stokes-type” regime

Abstract

To contribute to the study of the influence of the hydrodynamic interactions governing the dynamics of solid particles such as fibers in nondilute regimes, we consider in this work a cylindrical particle confined between two parallel walls at low Reynolds numbers. The particle moves with its axis always parallel to both walls. Our numerical results, derived with a projection method and a finite-volume approach, turn out to be very accurate and enable us to solve different problems using the matrix resistance technique. The first problem considers the consequences of these interactions on the settling velocity of a particle. In such confined situations, the hydrodynamic interactions are expressed by a backflow leading to a decrease of the settling velocity when the confinement increases with an asymptotical behavior varying like ε5∕2, where ε stands for the gap between the plane walls and the cylinder. The second problem consists in the accurate determination of the actual velocity of a neutrally buoyant particle transported in a Poiseuille flow, with its axis perpendicular to the mean flow. The hydrodynamic interactions lead to the existence of a relative velocity between the free particle and the flow unperturbed by the particle. This result reconsiders the assumption commonly used in some studies (same velocity for the particle and the unperturbed fluid) to analyze the transportation of fibers in the processing of composite materials (extrusion, injection molding) for concentrated regimes. In the third problem describing the transportation of a non-neutrally buoyant particle in a vertical Poiseuille flow, as in a fluidized bed, for instance, we obtain two regimes depending on the relative magnitude of the sedimentation velocity to the mean Poiseuille flow, and this suggests a potential method to separate the particles according to their density or size. On this occasion, we give the various flow patterns arising with each problem.

Numerical Simulations of Particle Sedimentation Using the Immersed Boundary Method

“…We then consider the case of a circular particle falling in a bounded fluid domain between two vertical walls and then review several of the most commonly-used formulas for the "wall-correction factors" that have been obtained from either fitting to experimental data or using approximate analytical techniques. A fairly extensive overview of settling for cylindrical particles, including many of the wall correction formulas reported in the literature, is given by Champmartin and Ambari [11].…”

Wall effects on the transportation of a cylindrical particle in power-law fluids

“…This is due to the increase of the screen length for Newtonian and dilatant fluids. When the problem is confined geometrically, we showed that the drag force is not anymore zero due to the existence of the Stokes-type solution as established in the Newtonian case [2]. Otherwise, once the solution for the unbounded medium is established, we give a solution for the confined medium numerically and asymptotically.…”

“…1). If this cylindrical particle is very long and highly confined, it has been established that the end effects are negligible [2,22]. For pseudoplastic fluids, for which the Stokes' type solution exists, we study the influence of the fluidity ðnÞ and the backflow on the settling velocity.…”

“…Then, the first step to understand these interactions leads us to study the basic hydrodynamics related to the transportation of an individual circular cylindrical particle in a confined situation. The justification of the use of this model in Newtonian fluids at high concentration has been made by Champmartin and Ambari [2]. To do so, we have chosen a simple model constituted by an infinitely long circular cylinder of radius a, translating uniformly midway perpendicularly to its axis, between two parallel plane walls distant of 2b (Fig.…”

Toward application of conformal decomposition finite elements to non‐colloidal particle suspensions

“…The Faxen formula yields reasonably accurate results up to R=L 0.5 [26]. CDFEM is used to solve the settling disk problem similarly to [10], setting gravity g D 1.0, fluid viscosity D 1.0, and the density of the solid s D 1.0.…”