Inertial rotation of a small oblate spheroid in a simple shear flow

TL;DR

This work investigates how weak inertia and finite confinement influence the angular dynamics of a neutrally buoyant oblate spheroid in simple shear. By combining experiments, fully resolved immersed boundary simulations, and theory valid for small $Re_p$, the authors show that confinement slows the drift toward the log-rolling attractor while inertia accelerates it, with the net behavior depending on the confinement ratio $\kappa$ and particle Reynolds number $Re_p$. For oblate spheroids with $\lambda>0.14$, the log-rolling orbit is the attractor when confinement and inertia are properly accounted, and finite-domain effects must be included to reconcile observations with inertial theories derived for unbounded flows. The results highlight the importance of boundary conditions in dilute-to-semi-dilute suspensions and provide a framework for predicting particle orientations in microfluidic and industrial flows where inertia and confinement are non-negligible.

Abstract

We compare experiments and fully-resolved particle simulations designed to match the experimental conditions of a weakly inertial, neutrally buoyant, moderately oblate spheroid in shear flow under confinement. Experimental and numerical results are benchmarked against theory valid for asymptotically small particle Reynolds numbers and for unconfined systems. By considering the combined effects of confinement and inertia, sensitivity to initial conditions, and the time span of observation, we reconcile the findings of theory, experiments, and numerical simulations. Furthermore, we demonstrate that confinement significantly influences the orientational stability of log-rolling spheroids compared to weak inertia, with the primary consequence being a reduced drift rate towards the stable log-rolling orbit.

Figures (12)

• Figure 1: Oblate spheroid in a simple shear. (a) log-rolling (LR) orbit, (b) tumbling (T) in the flow-shear plane. The particle-symmetry axis is denoted by $\ve n$, and the flow vorticity by $\ve \Omega$. The basis vectors of the lab coordinate system are denoted by $\hat{\bf e}_1$, $\hat{\bf e}_2$, and $\hat{\bf e}_3$. (c) Definitions of the azimuthal and polar angles, $\phi$ and $\theta$.
• Figure 2: Comparison of the experimental time series (blue symbols) with the results of numerical simulations (orange solid lines) and theoretical predictions of Eq. (\ref{['eq:einarsson']}) (black solid lines) for the case of spheroids with Re$_p=$ 0.43, $\lambda=0.56$ and $\kappa=0.2$. The only difference among the different runs is the initial orientations of the particle. The plots in panel (a) refer to an experimental run (labeled here as Experiment 10 to follow the notation used in the Jupyter notebook). The plots in panel (b) refer to the experimental run labeled Experiment 9 in the Jupyter notebook. The plots in panel (c) refer to the experimental run labeled Experiment 4 in the Jupyter notebook. Left-hand panels show the full time evolution of particle orientation, while right-hand panels show a zoomed-in view to highlight the different behaviour of the curves. The grey shaded area indicates the theoretical predictions obtained with initial conditions perturbed by $n_3 \pm \sigma_{n_3}$, and is shown here to highlight the sensitivity of the theory to the initial orientation of the particle. See supplementary materials for the directory of the figure including the data, animation movies, and the Jupyter notebook, https://cocalc.com/share/public_paths/7775c0fbf82ac429d6a9a32ba46f99ed7b8ff788/figure_2.
• Figure 3: Quantification of the orbit drift. Panel (a): drift of the orbit constant $C$ as a function of time for different values of $\kappa$ for Re$_p=0.43$ (simulation results with the experimental setting). The dashed orange line shows the theoretical prediction obtained by numerical integration of Eq. (\ref{['eq:einarsson']}) for the same particle geometry and with the same initial particle orientation as in Experiment 4 (see figure \ref{['fig:comparison']}). The orange square in the inset is the corresponding value of $\gamma_{\rm LR}/\text{Re}_p$. A systematic scanning of the dependence of $\gamma_{\rm LR}/\text{Re}_p$ on $\kappa$ for Re$_p=0.30$ (squares), Re$_p=0.43$ (triangles, experimental setting), and Re$_p=0.60$ (stars), is shown in the inset. The black dashed line represents a linear fit with $\gamma_{\text{LR}}/\text{Re}_p = -0.0389 \kappa + 0.035$. Panel (b): drift of the orbit constant $C$ as a function of time for different values of Re$_p$ for $\kappa =0.2$ (simulation results with the experimental setting). A systematic scanning of the dependence of $\gamma_{\rm LR}/\text{Re}_p$ on Re$_p$ for $\kappa=0.2$ (squares, experimental setting), $\kappa=0.4$ (triangles), and $\kappa=0.6$ (stars) is shown in the inset, where the dashed orange line represents the asymptotic value $\gamma_{\text{LR}}/\text{Re}_p = 0.035$ that the linear fit yields as $\kappa \rightarrow 0$. See supplementary materials for the directory of the figure including the data and the Jupyter notebook, https://cocalc.com/share/public_paths/7775c0fbf82ac429d6a9a32ba46f99ed7b8ff788/figure_3.
• Figure 4: Normalized change in the orbit constant in a single Jeffery period, $\Delta C / (C^2+1) \times \text{Re}_p^{-1} \times \xi_0^2$, plotted as a function of $C/(C+1)$ for an oblate spheroid. Note that $C/(C+1)=0$ and $C/(C+1)=1$ correspond to the LR and T modes, respectively. Colored symbols refer to the simulation results at varying $\kappa$, shaded green symbols in Panel (a), or at varying Re$_p$, shaded blue symbols in Panel (b). The red dashed line shows the theoretical prediction in the slender limit einarsson2015adabade2016effect. See supplementary materials for the directory of the figure including the data and the Jupyter notebook, https://cocalc.com/share/public_paths/7775c0fbf82ac429d6a9a32ba46f99ed7b8ff788/figure_4.
• Figure A1: Comparison of experimental time series (filled symbols) with results of numerical simulations (thick solid lines) and theoretical predictions (thin solid lines) for the case Experiment 1. Panels on the left illustrate the full time series, while panels on the right show the corresponding zoomed-in views.