Low dissipative configuration in flow networks subject to constraints

The homothetic relationships for the design of dendritic networks are examined in terms of minimal size under constant flow resistance, and minimum flow resistance under constant size. Based on a comprehensive methodology, we offer a general approach for the homothety ratios of diameters and lengths that apply to different flow regimes and size constraints. In addition, scaling laws for diameters-lengths, sizes-diameters, and resistances-lengths are provided. Since the dendritic trees designed based on size homothety ratios have prefractal characteristics, a methodology for determining prefractal dimensions in terms of fluid flow and size constraint characteristics is also offered. Among the findings, we show that the homothety ratios are the same regardless of whether the functions we selected are used as the constraint or the cost function. The approaches presented and literature data were compared, and a significant degree of agreement was found.

The findings presented here not only serve as a tool for the design of microfluidic chip devices but also deepen our understanding of natural networks such as the ones that support the life of mammals.