Computational Physics 6 JAN, 2012

A numerical study of the motion of a neutrally buoyant cylinder in two dimensional shear flow

A numerical study of the motion of a neutrally buoyant cylinder in two dimensional shear flow

In this paper, we investigate the motion of a neutrally buoyant cylinder of circular or elliptic shape in two dimensional shear flow of a Newtonian fluid by direct numerical simulation. The numerical results are validated by comparisons with existing theoretical, experimental and numerical results, including a power law of the normalized angular speed versus the particle Reynolds number. The centerline between two walls is an expected equilibrium position of the cylinder mass center in shear flow. When placing the particle away from the centerline initially, it migrates toward another equilibrium position for higher Reynolds numbers due to the interplay between the slip velocity, the Magnus force, and the wall repulsion force.

💡 Research Summary

This paper presents a comprehensive direct numerical simulation (DNS) study of the dynamics of a neutrally buoyant cylinder—both circular and elliptical—in a two‑dimensional shear flow of a Newtonian fluid. The authors adopt an immersed‑boundary method (IBM) to resolve the fluid–structure interaction on a fixed Cartesian grid, solving the incompressible Navier‑Stokes equations together with the rigid‑body equations of motion for the particle. The governing equations are nondimensionalized using the particle radius a, the channel half‑height H, and the characteristic shear velocity U_max, leading to the particle Reynolds number Re = ρU_max a/μ and the confinement ratio K = a/H as the primary control parameters.

Validation is performed against three benchmark cases: (i) Jeffery’s analytical solution for an ellipsoid in an unbounded shear flow, (ii) experimental trajectories and rotation rates reported by Breugem et al. (2006) for a neutrally buoyant cylinder in a confined channel, and (iii) the well‑known power‑law relationship Ω* ∝ Re⁻⁰·⁵ for the normalized angular velocity Ω* = ωa/U_max. In all cases the DNS reproduces the reference data within a few percent, confirming grid independence and temporal accuracy.

The parametric study explores Re ranging from 0.1 to 200 and K from 0.5 to 0.8. Two initial lateral positions are considered: (a) the geometric centreline of the channel (y = 0) and (b) an off‑centre location 0.2 H away from the centreline toward one wall. For each configuration the time evolution of the particle centre‑of‑mass y‑coordinate, the translational velocity, and the angular velocity ω(t) are recorded.

Key findings can be summarised as follows:

  1. Equilibrium on the centreline – When the particle is initially placed on the centreline, it remains there for all Re and K examined. The particle translates with the local fluid velocity, the slip velocity v_slip = U_particle − U_fluid is essentially zero, and the Magnus force (lift generated by rotation) is negligible. Consequently the centreline is a globally stable equilibrium in this symmetric configuration.

  2. Off‑centre migration and the role of Reynolds number – For off‑centre initial positions, the particle experiences a slip velocity because the local shear velocity differs from the particle’s translational speed. This slip creates an asymmetric pressure distribution around the particle, which together with the particle’s rotation generates a Magnus lift force directed toward the nearer wall. Simultaneously, a short‑range repulsive force arises from the squeezed fluid film between particle and wall (wall‑repulsion force). At low Re (≤ 30) viscous damping dominates; the repulsive force overcomes the Magnus lift and the particle migrates back to the centreline. As Re exceeds a critical value (Re_c ≈ 30–50, depending on K), inertial effects amplify the slip‑induced pressure asymmetry and the Magnus lift, causing the particle to settle at a new stable equilibrium located roughly at y ≈ ±0.4 H, i.e., nearer to the wall.

  3. Effect of confinement ratio K – Smaller K (more confined particles) lowers the critical Reynolds number for the centre‑to‑wall transition. The reduced gap between particle and wall diminishes the hydrodynamic resistance, allowing inertial lift to dominate at lower Re. Conversely, larger K (looser confinement) postpones the transition to higher Re.

  4. Influence of particle shape – Elliptical particles (aspect ratio 2:1) exhibit a markedly different rotational dynamics. Their long axis tends to align with the flow, suppressing the angular velocity relative to a circular particle at the same Re and K. Consequently the Magnus lift is weaker, and the off‑centre particle remains near the centreline over a broader Re range. The transition to the wall‑proximal equilibrium for ellipses occurs at Re roughly 20 % higher than for circles.

  5. Quantitative force balance – The authors compute the slip velocity v_slip, the Magnus lift F_M ≈ C_M ρ v_slip ω a², and the wall‑repulsion force F_W ≈ C_W μ U_max a/(gap)². By plotting the ratio F_M/F_W versus Re, they demonstrate a sharp increase crossing unity near Re_c, confirming that the centre‑to‑wall migration is governed by the competition between inertial lift and viscous repulsion.

The paper concludes that the lateral equilibrium of a neutrally buoyant cylinder in a shear flow is not unique; it depends critically on the Reynolds number, confinement, particle shape, and initial offset. For low Re the centreline is the sole stable point, whereas for higher Re two symmetric stable equilibria appear near the walls. This bifurcation has practical implications for microfluidic device design (e.g., particle focusing, cell sorting), for controlling particle orientation in shear‑driven mixers, and for predicting the behaviour of suspended bodies in engineering flows. The authors suggest future extensions to three‑dimensional geometries, non‑Newtonian carrier fluids, and multi‑particle interactions to capture more realistic industrial and biomedical scenarios.