Lubrication analysis of interacting rigid cylindrical particles in confined shear flow

Abstract

Lubrication analysis is used to determine analytical expressions for the elements of the resistance matrix describing the interaction of two rigid cylindrical particles in two-dimensional shear flow in a symmetrically confined channel geometry. The developed model is valid for non-Brownian particles in a low-Reynolds-number flow between two sliding plates with thin gaps between the two particles and also between the particles and the walls. Using this analytical model, a comprehensive overview of the dynamics of interacting cylindrical particles in shear flow is presented. With only hydrodynamic interactions, rigid particles undergo a reversible interaction with no cross-streamline migration, irrespective of the confinement value. However, the interaction time of the particle pair substantially increases with confinement, and at the same time, the minimum distance between the particle surfaces during the interaction substantially decreases with confinement. By combining our purely hydrodynamic model with a simple on/off non-hydrodynamic attractive particle interaction force, the effects of confinement on particle aggregation are qualitatively mapped out in an aggregation diagram. The latter shows that the range of initial relative particle positions for which aggregation occurs is increased substantially due to geometrical confinement. The interacting particle pair exhibits tangential and normal lubrication forces on the sliding plates, which will contribute to the rheology of confined suspensions in shear flow. Due to the combined effects of the confining walls and the particle interaction, the particle velocities and resulting forces both tangential and perpendicular to the walls exhibit a non-monotonic evolution as a function of the orientation angle of the particle pair. However, by incorporating appropriate scalings of the forces, velocities, and doublet orientation angle with the minimum free fraction of the gap height and the plate speed, master curves for the forces versus orientation angle can be constructed.

Self-pinning of a nanosuspension droplet: Molecular dynamics simulations

“…A computation of forces on entrained particles during de-pinning indicated that, for the small nanoparticles studied here, drag forces due to liquid flow were below the resolution of the force calculation. This is in at least qualitative agreement with recent work [35] examining drag forces on immersed cylinders for varying separation distance between the cylinder and a substrate surface.…”

“…Though our flow geometry is different here from the one studied in Ref. 35, sufficient similarity exists that we may expect drag forces to be small, despite the high flow rates observed in Figs. 8 and 9.…”

“…8 and 9 indicate that non-trivial capillary and/or drag forces potentially manifest on the particle. Regarding drag on the particle, an important finding was recently advanced for shear flow in a channel past suspended cylinders [35] using both analytical and simulation approaches. Authors of that work found that, when the cylinder was positioned directly adjacent to the channel wall, forces due to drag diminished significantly.…”

Local drag of a slender rod parallel to a plane wall in a viscous fluid

“…Again the total force is determined by the integration of this result. These coefficients produce the correct lubrication behaviour as d → [41,42].…”

“…This issue is also present in the simple case of rod by a single plane wall. In this case there exists asymptotic solutions in the limit that the separation is much larger then all lengths of the rod [35,40], the separation is much larger than the thickness of the rod but much smaller than the length [37] and the separations is of order of the thickness [41,42]. Furthermore each of these solutions were found using different asymptotic techniques; Brenner used the method of reflections to determine the drag when the separation is much larger than all lengths of the rod [40], Katz et al represented the body as a line of point forces above a wall [43] to determine the drag when the separation is larger than the thickness but smaller than the length [37], and Jeffrey and Onishi used lubrication arguments to determine the flow when an infinite cylinder is very close to the wall [42].…”