Shape, confinement and inertia effects on the dynamics of a driven spheroid in a viscous fluid

TL;DR

This work addresses how shape, confinement, and inertia govern the dynamics of driven spheroids in a viscous, confined flow. It combines lattice Boltzmann simulations with far-field wall-interaction theory to map translational and rotational responses across aspect ratios, confinement levels, and Reynolds numbers. The findings show that the maximum translation speed for unconfined spheroids occurs at specific non-spherical aspect ratios, while confinement shifts the optimum toward oblate shapes; off-center positions induce translation-rotation coupling that yields glancing and reversing oscillations. Inertia then reorganizes the phase space, breaking closed-loop trajectories and producing bifurcations that favor stable broadside-on states near walls, with implications for optimizing microfluidic transport and designing shape-tuned delivery particles.

Abstract

The dynamics of anisotropic particles in viscous flows underpin a wide range of processes in soft matter, microfluidics, and targeted drug delivery. Here, we investigate the motion of externally driven prolate and oblate spheroids suspended in a Newtonian fluid and confined within a square microchannel. Using lattice Boltzmann simulations, complemented by far-field hydrodynamic theory based on superposition of wall interactions, we systematically quantify how particle aspect ratio, strength of confinement, and fluid inertia influence the dynamics of a spheroid. For unconfined spheroids, we show that the translational velocity is maximized not for a sphere but for a prolate (end-on) or oblate (broadside-on) spheroid of a specific aspect ratio. Under confinement, the optimal aspect ratio shifts toward oblate shapes due to the dominant contribution of wall-induced frictional resistance. Off-center positioning introduces strong translation-rotation coupling, giving rise to two families of oscillatory trajectories - glancing and reversing - whose existence and structure are captured as closed orbits in phase space. Weak fluid inertia breaks these closed loops: glancing trajectories spiral outward and merge with reversing trajectories, and new stable fixed points emerge. Together, these results reveal how modest deviations from sphericity or creeping-flow conditions profoundly alter the dynamics of driven particles in confined geometries. The predictions offer guidelines for optimizing particle shape in microfluidic transport and highlight the rich nonlinear behavior accessible in confined suspensions of nonspherical colloids.

Figures (7)

• Figure 1: $(a)$ Schematic representation of a translating spheroid with semi-major axis $a$ and semi-minor axes $b = c$, located at $(X, Y = h, Z = L/2)$, oriented at $\theta$, experiencing an external force $\bm{F}_e$, confined in a channel of square cross section of side length $L$. The side view of the square channel ($Y-Z$ cross section) is shown in the inset. The walls are numbered as 1, 2, 3 & 4. $(b)$ Fluid flow generated by the translating spheroid in the square channel, in an otherwise quiescent Newtonian fluid. The arrows indicate the velocity field, and are coloured according to the magnitude of velocity with red being the highest velocity.
• Figure 2: Translational velocity of an unconfined spheroid as a function of aspect ratio for a fixed particle volume. Aspect ratios $b/a < 1$ correspond to prolate spheroids, while $b/a> 1$ represent oblate shapes. The continuous curves show the analytical predictions (Sec. \ref{['sec:unconfinedtheory']}), the symbols denote the lattice Boltzmann simulation results, and the dashed lines indicate the asymptotic expressions for rod-like and disc-like limits. Data plotted in blue (red) correspond to end-on (broadside-on) configurations, respectively. The aspect ratio at which the translational velocity is maximum in each case is marked by a vertical line.
• Figure 3: (a) Translational velocity of spheroids (prolate, $b/a~\approx0.51$ and oblate, $b/a~\approx1.57$) moving in an end-on configuration along the centreline of a square channel, shown as a function of increasing confinement ratio, CR $= 2b/L$. The symbols $\CIRCLE$ are from the LBM simulations, the dash-dotted lines are spheroids in cylindrical channels (prolate, $b/a = 0.5$ and oblate $b/a = 2$)wakiya1957viscous and dotted lines are theoretical predictions based on far-field analysis (section \ref{['confinement_theory']}). The ordinate is normalised with the translational velocity of the unconfined spheroid. Translational velocities of spheroids for (b) $\bm{F} \parallel \hat{\bm{e}}$ and (c) $\bm{F} \perp \hat{\bm{e}}$ configuration as a function of aspect ratio for fixed confinement ratios - $\blacksquare$: CR$= 0$, $\blacktriangle$: CR$= 0.2$, $\blacklozenge$: CR$= 0.5$ and $\CIRCLE$: CR$=0.9$. The ordinate is normalised with the translational velocity of an unconfined sphere and the optimum aspect ratio at each confinement ratio is indicated on top of each peak in (b) & (c).
• Figure 4: Contour plots of (a) translational velocity $U_x$, (b) transverse velocity $U_y$ and (c) angular velocity $\varOmega_z$ in the $h-\theta$ space for a channel confined spheroid of aspect ratio $b/a\approx0.51$ at a confinement ratio CR$\approx0.15$. $\varOmega_z > 0$ corresponds to anti-clockwise rotation. In each figure, the inaccessible areas due to the finite size of the spheroid is coloured in grey. (d)-(f) Same data as line plots ($\bigstar$ symbols) along with results from the far field analysis (continuous lines) as a function of $h$ at various $\theta$. Results in (d) and (e) have been normalised with the translational velocity of the spheroid with same orientation ($\theta$) in an unbounded fluid. Plots (a) - (c) correspond to a ${\mathcal{R}e}\approx0.25$ where as plots (d) - (f) correspond to a ${\mathcal{R}e}\approx0.01$.
• Figure 5: Schematic representation of (a) glancing and (b) reversing spheroid. The dotted line represents the channel centreline. The instantaneous orientation, indicated by the arrow $\hat{\bm{e}}$, is provided as a visual guide to illustrate the angular rotation of the spheroid. Glancing (reversing) trajectories span the full (half) channel width, with the spheroid approaching each wall almost parallel (perpendicular) to it. (c)-(d) Velocity vectors in the $h-\theta$ phase space for a spheroid of aspect ratio $b/a\approx 0.36$ and $b/a\approx 0.7$. The background is coloured with the angular velocity. The fixed points are highlighted with different markers (see text). In each figure, the inaccessible areas due to the finite size of the spheroid is coloured in grey.