Lynn Thomas F, Smith Lachlan D, Ottino Julio M, Umbanhowar Paul B, Lueptow Richard M
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Phys Rev E. 2019 Mar;99(3-1):032204. doi: 10.1103/PhysRevE.99.032204.
We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for nonorthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Nonorthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the nonorthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell.
我们使用分段等距（PWI）理论框架，研究了由绕两个水平轴交替旋转驱动的半球形壳的切割与重排动力学。先前对区域切割与重排方式的限制得以放宽，以允许非正交旋转轴，这为PWI增加了一个新的自由度。一种使用并行处理高效执行切割与重排的新计算方法，使得能够进行广泛的参数扫描，并研究产生低混合度的混合协议。非正交旋转轴打破了一些导致正交轴混合不佳的对称性，平均而言增加了整体混合度。与正交轴情况一样，由旋转角度的有理比及其交点产生的阿诺德舌，是导致非正交轴参数空间中许多低混合度协议的原因。沿着系统的一个基本对称平面的阿诺德舌交点揭示了一类新的、意想不到的协议，其动力学是周期性的，例外集形成半球形壳的多边形平铺。
