In biophysical fluid dynamics, Murray's law is a potential relationship between radii at junctions in a network of fluid-carrying tubular pipes. Its simplest version proposes that whenever a branch of radius splits into two branches of radii and , then all three radii should obey the equation If network flow is smooth and leak-free, then systems that obey Murray's law minimize the resistance to flow through the network. For turbulent networks, the law takes the same form but with a different characteristic exponent . Murray's law is observed in the vascular and respiratory systems of animals, xylem in plants, and the respiratory system of insects. In principle, Murray's law also applies to biomimetic engineering, but human designs rarely exploit the law. Murray's law is named after Cecil D. Murray, a physiologist at Bryn Mawr College, who first argued that efficient transport might determine the structure of the human vascular system. Assumptions Murray's law assumes material is passively transported by the flow of fluid in a network of tubular pipes, and that said network requires energy both to maintain flow and structural integrity. Variation in the fluid viscosity across scales will affect the Murray's law exponent, but is usually too small to matter. At least two different conditions are known in which the cube exponent is optimal. In the first, organisms have free (variable) circulatory volume. Also, maintenance energy is not proportional to the pipe material, but instead the quantity of working fluid. The latter assumption is justified in metabolically active biological fluids, such as blood. It is also justified for metabolically inactive fluids, such as air, as long as the energetic "cost" of the infrastructure scales with the cross-sectional area of each tube; such is the case for all known biological tubules. In the second, organisms have fixed circulatory volume and pressure, but wish to minimize the resistance to flow through the system. Equivalently, maintenance is negligible and organisms with to maximize the volumetric flow rate. Although most derivations of Murray's law assume a steady state flow field, the same results apply for flow in tubes that have a width moderate-to-small relative to the flow wavelength. Derivation in laminar, mass-conservative networks with non-negligible upkeep Murray's original derivation uses the first set of assumptions described above. She begins with the Hagen–Poiseuille equation, which states that for fluid of dynamic viscosity , flowing laminarly through a cylindrical pipe of radius and length