Rising and Sinking in Resonance: Mass Distribution Critically Affects Buoyancy-Driven Spheres via Rotational Dynamics

We present experimental results for spherical particles rising and settling in a still fluid. Imposing a well-controlled center of mass offset enables us to vary the rotational dynamics selectively by introducing an intrinsic rotational timescale to the problem. Results are highly sensitive even to small degrees of offset, rendering this a practically relevant parameter by itself. We further find that, for a certain ratio of the rotational to a vortex shedding timescale (capturing a Froude-type similarity), a resonance phenomenon sets in. Even though this is a rotational effect in origin, it also strongly affects translational oscillation frequency and amplitude, and most importantly, the drag coefficient. This observation equally applies to both heavy and light spheres, albeit with slightly different characteristics for which we offer an explanation. Our findings highlight the need to consider rotational parameters when trying to understand and classify path properties of rising and settling spheres.

Figure 1

(a) Schematic of a sphere with c.m. offset. (b) The particle Frenet-Serret (TNB) coordinate system, with unit vectors $\mathit{T}$ (parallel to $\mathit{u}$), $\mathit{N}$ (pointing in the direction of curvature of the path), and $\mathit{B}$ (defined such that $\mathit{N}=\mathit{B}\times \mathit{T}$). The angles $\varphi$ (azimuth) and $\theta$ (elevation) uniquely define a vector in this space. (c) Explored parameter space. Grey shading indicates the resonance regime and $\mathcal{T}$ isocontours correspond to $I^{*}=1$.

Figure 2

(a) Characteristic trajectories of rising particles ($\mathrm{Ga}\approx 1800$ and $\mathrm{\Gamma }\approx 0.80$) as seen from the top for different values of $\mathcal{T}$. The length of the horizontal blue lines represent the corresponding amplitudes $\widehat{a}/D$. (b) Inset: $f$ (symbols) and $f_p$ (dashed lines) vs $\gamma$ for three different Ga values. Main figure: ratio $f/f_p$ vs $\mathcal{T}$ for the entire dataset. (c) Normalized histograms of the orientation of $\mathit{\omega }$ in the TNB coordinates for the cases corresponding to (a). The histograms contain data of all particles with nominally the same parameters.

Figure 3

Dependence on $\mathcal{T}$ for (a) amplitude of the path oscillations $\widehat{a}/D$, (b) particle vertical drag coefficient $C_d$, (c) particle rotational amplitude ${\widehat{\theta }}_z$, (d) time averaged angular velocity $⟨\omega ⟩$, (e) phase angle $\mathrm{\Delta }\mathrm{\Phi }$ between horizontal particle acceleration and Magnus lift force. All data points represent averages over multiple experiments with the same particle.

Figure 4

Particle drag coefficients for rising and settling spheres compiled from literature (black dots) [17, 19, 20, 21, 22, 25, 26, 57, 58, 59, 60, 61, 62, 63], and present data (color coded by $\mathcal{T}$) vs $\mathrm{Re}=⟨u_z⟩D/\nu$.