Study on side-branching structure of dynamical pattern formation in diffusion field.

1.我们对二维扩散限制聚集（DLA）进行了模拟，发现了分形维数与核维数之间的新关系。将连续的DLA离散化，并参考相似维数，得到了该公式。提出了一些公式，但我们的公式比那些更精确。我们在更高的Eucledian维中进行模拟，并将推广公式。我们已经进行了模拟的枝晶，密集的分支形态和DLA与耦合映射格子模型。我们发现这些图案与表面张力的各向异性，表面动力学的各向异性和表面噪声的影响之间的关系。我们还明确了花样转变的物理机制，发现在密集分支形貌区出现了双枝花样，双枝花样与一对不对称的枝晶一起生长，并且比枝晶生长得快.利用Hele-Shaw池进行了粘性指进实验，研究了枝晶的整体包络。我们发现，在包层中存在三个区域：早熟侧支、竞争侧支和独立侧支。我们发现了一个有趣的结果，即终止其他侧枝的竞争侧枝可以比茎尖生长得更快。我们推测，竞争的侧枝生长为双金发。

1. We have performed simulation of two-dimensional diffusion-limited aggregation (DLA) and found a new formulation between the fractal dimension and Eucledian dimension. The formulation is obtained by making continuous DLA discrete and referring similarity dimension. Some formulations are proposed, however, our formulation is more precise than those. We are performing the simulation in higher Eucledian dimensions and are going to generalize the formulation.2. We have performed simulation of dendrite, dense-branching morphology and DLA with a coupled map lattice model. We have found the relation between those patterns and the anisotropy of surface tension, the anisotropy of surface kinetics and the influence of surface noise. We have also made the physical mechanism of transition of the patterns clear and found doublon pattern in dense-branching morphology region which grow with a pair of asymmetric dendrite and can grow faster than dendrite.3. We have performed the experiment of viscous fingering with Hele-Shaw cell and studied the global envelope of dendrite. We have found that there exist three regions in the envelope ; premature side-branches, competing side-branches and independent side-branches. We have found an interesting result that the competing side-branches which cease other side-branches can grow faster than the tip of the stem. We conjecture that the competing side-branches grow as doublons.