Critical review of Murray's theory for optimal branching in fluidic networks

Abstract

Murray's theory of constrained minimum-power branchings is critically reviewed in a generalised framework for a range of cases: channels with arbitrary cross-section shape, laminar flows of Newtonian and non-Newtonian fluids, and low and high Reynolds-number turbulent flows of Newtonian fluids. The theory states that the sum of hydraulic and metabolic power is minimised if and only if all channels satisfy the same relation between flow rate and effective radius. This relation leads to a generalised form of Murray's law. It is shown that, satisfying Murray's law is a necessary requirement for power minimisation, but not a sufficient requirement. The generalisation of Kamiya & Togawa's law that holds for minimum-volume branchings, also holds for minimum-power branchings. It is a necessary requirement but not a sufficient requirement for both minimum-power and minimum-volume branchings. For symmetric branchings the two generalised laws of Murray and Kamiya & Togawa become identical.