Murray's law
In biophysical fluid dynamics, Murray's law is a potential relationship between radii at junctions in a network of fluid-carrying tubular pipes.
For turbulent networks, the law takes the same form but with a different characteristic exponent α. Murray's law is observed in the vascular and respiratory systems of animals, xylem in plants, and the respiratory system of insects.
Murray's law assumes material is passively transported by the flow of fluid in a network of tubular pipes,[1] and that the network requires energy to maintain both flow and structural integrity.
[2] Variation in the fluid viscosity across scales will affect the Murray's law exponent, but is usually too small to matter.
Also, maintenance energy is not proportional to the pipe material, but instead the quantity of working fluid.
[4] 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.
[5] 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.
She begins with the Hagen–Poiseuille equation, which states that for fluid of dynamic viscosity μ, flowing laminarly through a cylindrical pipe of radius r and length l, the volumetric flow rate Q associated with a pressure drop Δp is
Minimizing this quantity depends on precisely which variables the organism is free to manipulate, but the minimum invariably occurs when the two terms are proportional to each other.
[8] In that minimal case, the proportionality determines a relationship between Q and r. Canceling common factors and taking a square root, That is, when using as little energy as possible, the mass flowing through the pipe must be proportional to the cube of the pipe's radius.
The same law would apply to a direct-current electrical grid composed of wires of only one material, but varying diameter.
In general, networks intermediate between diffusion and laminar flow are expected to have characteristic exponents between 2 and 3, at least approximately.
[15][16] Mice genetically engineered to lack the blood-vessel-wall protein elastin have smaller and thinner blood vessels, but still obey Murray's law.
[18]Insects do not have a fully-fledged circulatory system, instead relying on passive diffusion through the haemocoel.
For those networks, Murray's law predicts constant cross-sectional area, which is observed.
Plant xylem is known to exhibit that scaling except in scenarios where the passages double as structural supports.
[19][20] The first phenomenon now recognized as Murray's law is Young's rule for circulatory systems, which states that two identical subcapillaries should combine to form a capillary with radius about 1.26≈3√2 times larger, and dates to the early 19th century.
[21] Bryn Mawr physiologist Cecil D. Murray published the law's modern, general formulation in 1926,[22][21] but it languished in a disciplinary no-man's-land for the next fifty years: too trivial for physicists and too complicated for biologists.
[21] In circulatory system governed by Murray's law with α=3, shear stress on vessel walls is roughly constant.
[23] Murray's law rarely applies to engineered materials, because man-made transport routes attempt to reduce flow resistance by minimizing branching and maximizing diameter.
[24] Materials that obey Murray's law at the microscale, known as Murray materials, are expected to have favorable flow characteristics, but their construction is difficult, because it requires tight control over pore size typically over a wide range of scales.
[14][25] Lim et al propose designing microfluidic "labs on a chip" in accord with Murray's law to minimize flow resistance during analysis.
Conventional lithography does not support such construction, because it cannot produce channels of varying depth.
[26] Seeking long-lived lithium battery electrodes, Zheng et al constructed Murray materials out of layers of sintered zinc oxide nanoparticles.
[14] Because power plant working fluids typically funnel into many small tubules for efficient heat transfer, Murray's law may be appropriate for nuclear reactor design.
