多变形体低雷诺数下的非定常动力学问题

The project concetrates our attention on the theoretical investigation of the unsteady description for dynamical behaviors of a linear dissipative fluid-body system involved in many deformable bodies submerged in unsteady Stokes flows so as to investigate a full coupling of moving boundaries of multiple deforming bodies with the flow field. On such a system basis, the researchers hope to reveal the importance of fully-coupled translational and angular accelerations of all deforming bodies to unsteady description of dynamical behaviors of anyone, explore the transfer mechanism from deformation energy of the bodies into their translations and rotations by flow due to inhomogeneous mass distributed in the deforming bodies, confirm influences of the rates of deformation of the bodies and their deforming patterns upon their direction-changing manoeuvrable motions, and furthermore search effects of the phase-modulated and phase-coherent due to collective deformation of a sort of the deforming bodies on their colletive evading tactics in order to reveal nature of full hydrodynamic coupling of deforming bodies with the flow field and explain their high collective maneuver behaviors. To do this, the Basset seperation technique is theoretically adopted combined with successive offset functions and the generalized cyclic permutation in order to establish N dynamical equations of motion with coupled acceleration terms and unsteadily describe the behaviors of N deforming bodies. Moreover, the harmonic and biharmonic terms in integrands of the integral equations are numerically segregated from each other so as to give an unsteady description for motions of those bodies of complicated shapes.

本项目将对含多个变形体的流固线性耗散系统（非定常Stokes控制方程）动力学问题的非定常描述进行理论研究，探索多个变形运动边界和流场完全耦合的本质。基于此线性耗散系统，申请者希望通过研究揭示所有变形体的平动和转动加速度对于任意一个变形体动力学行为非定常描述的重要性；探讨质量分布不均匀的变形体通过流体将变形能转换成其平动和转动运动的机制；确认变形体变形速率及方式对其变向运动行为的影响。进而探寻同类变形体群体通过集体周期变形相位的调制和相干对群体整体规避运动策略的影响；加深对流场中多个变形体的群体机动运动现象的认识。为此我们将数学上采用Basset分离技术结合连续补函数和广义循环排序以得出含有N个加速度耦合的动力学方程组以便非定常地描述N个变形体的动力学行为；边界积分解法上将采用拆分被积函数中的调和函数项和双调和函数项的技术来完成对复杂形状变形体运动的非定常刻画。

项目针对非定常流动中变形体变速运动问题以及相应的基础流体物理问题进行了理论上的探讨，取得了较为重要进展。成果如下：.1.非定态描述物体在低雷诺数流动中变速运动方面的研究取得了理论上的进展。获得了内、外非定常Stokes流场分析表达式；根据此理论结果，得出了液滴在母液中的非定态迁移的运动表达式，从而讨论了可滑移边界条件下运动液滴内、外流场动量的瞬态扩散形式，为进一步揭示流场中物体的非定态运动奠定了分析基础。.2.理论上讨论了变形体在液体中的自驱动机制。证明了仅调整物体的质心和变形的耦合可以使物体轨迹包络实现任意形式的机动行为。.3.通过物体上附加环量建立非欧空间旋转场中物体间的水动力相互作用形式，得出了水动力相互作用下动力学常微分方程组的分析解，从而揭示了相互作用包含了静止有心力和运动有心力等动力学特征。.4.建立了一种肿瘤细胞在人体间质组织液流中转移的动力学模型，研究表明力学因素和生化因素共同制约着整个肿瘤细胞迁移过程。此项工作为进一步研究癌细胞在组织中的转移提供了力学-生化耦合机制提供了分析基础。