The equations governing the dynamics of a periodically driven micro-spheroid in an unsteady viscous fluid at low Reynolds number are derived. Its oscillation properties in the presence/absence of memory forces are reported. The core part of the derivation is a perturbation analysis of motion of a sphere. The calculated solutions match with those available in the literature in the limiting case of a sphere. The dependence of the solutions on shape ( $$\alpha $$ α ), free oscillation frequency ( $$\omega _0$$ ω 0 ) and particle–fluid density ratio ( $$d_r$$ d r ) is calculated. The maximum amplitude of the oscillations of an oblate spheroid is greater than that of a prolate spheroid, showing that the velocity disturbance for an oblate spheroid is higher in the presence/absence of the memory force. The increase in $$\alpha $$ α leads to the enhancement(reduction) of amplitude peaks in the case of the oblate (prolate) spheroid in the presence and more dominantly in the absence of the force. There is also a reduction in the amplitude of spheroid oscillations of many multiples due to the presence of the memory force. Stronger oscillation variations are observed on changing $$\omega _0$$ ω 0 or $$d_r$$ d r compared to $$\alpha .$$ α . The variations of the value of the phase are similar for both the spheroids on varying $$\omega _0$$ ω 0 and $$d_r$$ d r , whereas they are reversed on varying $$\alpha .$$ α . The linear scaling of amplitude on $$\alpha $$ α observed for the spheroids may give insight into the physics, especially regarding the quantum of velocity disturbances due to particle size. The slopes are high in the absence of the force, confirming that the presence of the force increases the resistance of spheroid motion, largely. The dependencies of oscillations on the parameters can be utilized for better separation of particles or for characterizing the suspension. The novelty of the problem and its analytical solutions might have value as tests in software for more complicated and realistic systems and hence strikes a good balance between complication and tractability.

中文翻译：

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低雷诺数下不稳定粘性流中周期性受迫的略微偏心球体的振荡

推导了在低雷诺数下非定常粘性流体中控制周期性驱动微球体动力学的方程。报告了其在存在/不存在记忆力时的振荡特性。推导的核心部分是对球体运动的微扰分析。在球体的极限情况下，计算出的解决方案与文献中可用的解决方案相匹配。计算解决方案对形状 ($$\alpha $$ α )、自由振荡频率 ( $$\omega _0$$ ω 0 ) 和粒子-流体密度比 ( $$d_r$$ dr ) 的依赖性。扁球体振荡的最大振幅大于扁长球体的最大振幅，表明在存在/不存在记忆力的情况下，扁球体的速度扰动更高。$$\alpha $$ α 的增加导致在存在力的扁（扁长）球体的情况下振幅峰值的增强（减少），并且在没有力的情况下更显着。由于记忆力的存在，许多倍数的球体振荡幅度也有所降低。与 $$\alpha .$$ α 相比，在改变 $$\omega _0$$ ω 0 或 $$d_r$$ dr 时观察到更强的振荡变化。对于不同的 $$\omega _0$$ ω 0 和 $$d_r$$ dr 上的两个球体，相位值的变化是相似的，而在不同的 $$\alpha .$$ α 上则相反。对球体观察到的 $$\alpha $$ α 振幅的线性标度可以深入了解物理学，特别是关于由于粒子尺寸引起的速度扰动的量子。在没有力的情况下坡度很高，确认力的存在在很大程度上增加了球体运动的阻力。振荡对参数的依赖性可用于更好地分离颗粒或表征悬浮液。问题的新颖性及其分析解决方案可能具有作为更复杂和现实系统的软件测试价值，因此在复杂性和易处理性之间取得了良好的平衡。