Engineering of branched fluidic networks that minimise energy dissipation

Power minimisation of fluid transport in branched fluidic networks has become of paramount importance for microfluidics, additive manufacturing and hierarchical functional materials. For fully developed laminar flow of Newtonian fluids, Murray's theory provides a solution for the channel and network dimensions that minimise power consumption. However, design and optimisation of networks that transport complex fluids is still challenging. Here, we generalise Murray's theory towards fluid rheologies, including non-Newtonian (power-law) and yield-stress fluids (Bingham, Herschel-Bulkley, Casson). A straightforward graphical approach is presented that provides the optimal radii in a branching network, and the angles between these branches. The wall shear stress is found to be uniform over the entire network, and the velocity profile is self-similar. Furthermore, the effect of non-optimal channel radii on the power consumption of the network is investigated. Finally, examples illustrate how this approach applies to a wide variety of systems.

Automated summary

Power minimisation in fluid transport is crucial for microfluidics, additive manufacturing, and hierarchical functional materials. The traditional Murray's theory is extended to include non-Newtonian and yield-stress fluids, providing a graphical approach for determining optimal radii and angles in branching networks. The study reveals a uniform wall shear stress and self-similar velocity profile. The impact of non-optimal channel radii on power consumption is investigated, and the approach is demonstrated using various examples.