Shen Xinhui, Fu Henry C
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore.
Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah 84112, USA.
Phys Rev E. 2017 Mar;95(3-1):033105. doi: 10.1103/PhysRevE.95.033105. Epub 2017 Mar 9.
The influence of a plane boundary on low-Reynolds-number swimmers has frequently been studied using image systems for flow singularities. However, the boundary effect can also be expressed using a boundary integral representation over the traction on the boundary. We show that examining the traction pattern on the boundary caused by a swimmer can yield physical insights into determining when far-field multipole models are accurate. We investigate the swimming velocities and the traction of a three-sphere swimmer initially placed parallel to an infinite planar wall. In the far field, the instantaneous effect of the wall on the swimmer is well approximated by that of a multipole expansion consisting of a force dipole and a force quadrupole. On the other hand, the swimmer close to the wall must be described by a system of singularities reflecting its internal structure. We show that these limits and the transition between them can be independently identified by examining the traction pattern on the wall, either using a quantitative correlation coefficient or by visual inspection. Last, we find that for nonconstant propulsion, correlations between swimming stroke motions and internal positions are important and not captured by time-averaged traction on the wall, indicating that care must be taken when applying multipole expansions to study boundary effects in cases of nonconstant propulsion.
平面边界对低雷诺数游泳者的影响经常通过用于流动奇点的图像系统进行研究。然而，边界效应也可以用边界上牵引力的边界积分表示来表达。我们表明，研究游泳者在边界上产生的牵引力模式可以为确定远场多极模型何时准确提供物理见解。我们研究了最初与无限平面壁平行放置的三球体游泳者的游泳速度和牵引力。在远场中，壁对游泳者的瞬时影响可以很好地用由力偶极和力四极组成的多极展开来近似。另一方面，靠近壁的游泳者必须用反映其内部结构的奇点系统来描述。我们表明，通过检查壁上的牵引力模式，无论是使用定量相关系数还是通过目视检查，都可以独立识别这些极限及其之间的过渡。最后，我们发现对于非恒定推进，游泳冲程运动与内部位置之间的相关性很重要，并且壁上的时间平均牵引力无法捕捉到这一点，这表明在应用多极展开来研究非恒定推进情况下的边界效应时必须谨慎。
