This work investigates the orbital dynamics of a fluid-filled deformable prolate capsule in unbounded simple shear flow at zero Reynolds number using direct simulations. The motion of the capsule is simulated using a model that incorporates shear elasticity, area dilatation, and bending resistance. Here the deformability of the capsule is characterized by the nondimensional capillary number $\Ca$, which represents the ratio of viscous stresses to elastic restoring stresses on the capsule. For a capsule with small bending stiffness, at a given $\Ca$, the orientation converges over time towards a unique stable orbit independent of the initial orientation. With increasing $\Ca$, four dynamical modes are found for the stable orbit, namely, rolling, wobbling, oscillating-swinging, and swinging. On the other hand, for a capsule with large bending stiffness, multiplicity in the orbit dynamics is observed. When the viscosity ratio $\lambda \lesssim 1$, the long-axis of the capsule always tends towards a stable orbit in the flow-gradient plane, either tumbling or swinging, depending on $\Ca$. When $\lambda \gtrsim 1$, the stable orbit of the capsule is a tumbling motion at low $\Ca$, irrespective of the initial orientation. Upon increasing $\Ca$, there is a symmetry-breaking bifurcation away from the tumbling orbit, and the capsule is observed to adopt multiple stable orbital modes including nonsymmetric precessing and rolling, depending on the initial orientation. As $\Ca$ further increases, the nonsymmetric stable orbit loses existence at a saddle-node bifurcation, and rolling becomes the only attractor at high $\Ca$, whereas the rolling state coexists with the nonsymmetric state at intermediate values of $\Ca$. A symmetry-breaking bifurcation away from the rolling orbit is also found upon decreasing $\Ca$. The regime with multiple attractors becomes broader as the aspect ratio of the capsule increases, while narrowing as viscosity ratio increases. We also report the particle contribution to the stress, which also displays multiplicity.

中文翻译：

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剪切流中可变形扁囊的稳定轨道的多重性

这项工作使用直接模拟研究了零雷诺数下无边界简单剪切流中充满流体的可变形扁长胶囊的轨道动力学。使用包含剪切弹性，面积膨胀和抗弯曲性的模型来模拟胶囊的运动。在此，胶囊的可变形性由无量纲的毛细管数$ \ Ca $表征，其代表了胶囊上的粘滞应力与弹性回复应力之比。对于具有较小弯曲刚度的太空舱，在给定的\\ Ca $下，其方向会随着时间的推移而朝着唯一稳定的轨道收敛，而与初始方向无关。随着$ \ Ca $的增加，为稳定轨道找到了四种动力学模式，即滚动，摆动，摆动-摆动和摆动。另一方面，对于具有大的弯曲刚度的胶囊，观察到轨道动力学的多样性。当粘度比$\lambda \lesssim \mathrm{1个}$，胶囊的长轴总是趋向于在流动梯度平面中的稳定轨道，无论是翻滚还是摆动，这取决于$ \ Ca $。什么时候$\lambda \gtrsim \mathrm{1个}$，无论初始方向如何，胶囊的稳定轨道都是在$ \ Ca $低的情况下翻滚运动。随着$ \ Ca $的增加，远离翻滚轨道的位置会出现对称破坏的分叉，并且观察到的太空舱会采用多种稳定的轨道模式，包括非对称进动和滚动，具体取决于初始方向。随着$ \ Ca $的进一步增加，非对称稳定轨道在鞍节点分叉处不复存在，并且在$ \ Ca $高的情况下，滚动成为唯一的吸引子，而在$ \ Ca的中间值时，滚动状态与非对称状态共存。 $。减小时，还会发现远离滚动轨道的对称破坏分叉$ \ Ca $。随着胶囊长宽比的增加，具有多个吸引子的范围变得更宽，而随着粘度比的增加，其范围变窄。我们还报告了颗粒对应力的贡献，这也显示了多重性。