When applied to a branching network, Murray’s law states that the optimal branching of vascular networks is achieved when the cube of the parent channel radius is equal to the sum of the cubes of the daughter channel radii. It is considered integral to understanding biological networks and for the biomimetic design of artificial fluidic systems. However, despite its ubiquity, we demonstrate that Murray’s law is only optimal (i.e. maximizes flow conductance per unit volume) for symmetric branching, where the local optimization of each individual channel corresponds to the global optimum of the network as a whole. In this paper, we present a generalized law that is valid for asymmetric branching, for any cross-sectional shape, and for a range of fluidic models. We verify our analytical solutions with the numerical optimization of a bifurcating fluidic network for the examples of laminar, turbulent and non-Newtonian fluid flows.
The optimal branching of fluidic networks has been the subject of numerous studies owing to its importance in understanding the behaviour of biological vessels and for the biomimetic design of artificial systems. Much of the research stems from Murray’s law , who posited that there were two main energy requirements for blood flow through a cylindrical vessel of radius R: (i) the energy required to overcome viscous drag and drive the flow and (ii) the energy required to metabolically maintain the fluid and vessel. Assuming the flow to be laminar, Newtonian, steady, incompressible and fully developed, Murray used the Hagen–Poiseuille equation to model the driving power requirement (i.e. the driving power is proportional to R−4), and assumed that the maintenance power was proportional to the volume of the channel (i.e. proportional to R2). Applying the principle of minimum work to the total power requirement, Murray surmised that in an optimized cylindrical channel, the volumetric flow rate Q is proportional to R3. By applying mass conservation over a branching network, and assuming that local pressure losses through the junction (owing to bends and channel contractions) are negligible compared with the pressure losses over the channel lengths, this principle is most commonly expressed as a power law between a parent channel and N daughter channel branches 1.1where the subscripts p and di denote the parent and ith daughter, respectively. Although originally targeting blood transport through the cardiovascular system [2–9], experimental data have shown Murray’s law to be a decent approximation for a number of other biological networks, e.g. in the bronchial trees of humans and dogs [10–12]; in the chick embryo , and in the leaf veins of plants [14–16].
Whole blood (plasma and cells) is a non-Newtonian fluid that exhibits shear-thinning behaviour, i.e. its viscosity decreases with increased shear-strain rate. To more accurately model vascular networks, Murray’s law has been applied to non-Newtonian fluids [17,18] using the popular power-law fluid model [19,20]. In both studies, it was found that the optimal radius relation is unaffected by shear-thinning or shear-thickening fluid behaviour, and equation (1.1) is maintained for the whole range of non-Newtonian fluids. Murray’s law has also been applied to turbulent flows [21–23], which can be found in the upper airways of the lung  and, under some circumstances, in blood flow through the aorta , as well as in a number of hydraulic and pneumatic civil engineering applications. For fully rough-wall turbulent flow, the flow rate was found to be proportional to R7/3, leading to the relation 1.2for branching networks.
While Murray’s law has been most often applied to circular or elliptical  cross sections (owing to the shape of biological networks), optimized branching is also useful for the design of artificial systems , such as for fuel cells  or heat exchangers [27,28], which are often constrained to certain shapes by manufacturing procedures. To this end, Murray’s law has been adapted for networks of rectangular and trapezoidal cross sections .
Although originally derived from the principle of minimum work, it has been noted that the application of other optimization principles results in the same relationship between flow rate and channel radius: minimizing the total mass of the channel , minimizing volume for a constant pressure drop and flow rate , minimizing pumping power , maintaining a constant shear stress in all channels , and minimizing flow resistance for a constant volume [5,23,33].
However, despite many developments to Murray’s law, we submit that it is, in fact, suboptimal for asymmetric branching. In this paper, we derive a generalized law that is applicable to symmetric and asymmetric branching, for any cross-sectional shape, and for a range of fluidic models (e.g. Newtonian and non-Newtonian, laminar and turbulent).
2. Analytical solutions
The conditions for optimal branching can be generalized as a maximization of flow conductance per unit volume through each branch of the network, for a variety of constraint combinations. As with Murray’s law, we assume the flow to be steady and fully developed. For a two-level network (consisting of a single parent channel branching into multiple daughter channels), this can be expressed as 2.1where Q is the volumetric flow rate through the parent channel, is the set of pressure drops ΔPi over each of the N network branches—from the inlet of the parent to the outlet of the corresponding daughter—V is the network volume, and 2.2is the jth daughter–parent cross-sectional area ratio. Note, for each constraint option, two parameters are fixed and the third parameter is optimized (volume minimization, flow-rate maximization and pressure-drop minimization, respectively). Therefore, all three constraint options lead to an identical optimal relation because 2.3regardless of the constraints chosen. Note, noting that the last term means that d(ΔPi)/dΓj=0 for all i. For our optimization, we assume that the channel lengths L are large compared with the size of the parent–daughters junction, so that (i) the localized pressure losses at the junction are negligible compared with the pressure drops over individual channels (as in Murray’s law), and (ii) the volume of the network can be considered to be the sum of the channel volumes 2.4If we consider the optimization of the jth daughter channel, then inserting equation (2.4) into the fitness function of equation (2.1), and noting equation (2.3), gives 2.5The pressure drop over the parent and each of the i daughter channels can be expressed in terms of the volumetric flow rate 2.6and 2.7where Ψi=Qdi/Q is the fraction of the total flow rate taken by the ith daughter channel and k is flow resistance per unit length, e.g. kp=ΔPp/(QLp). The pressure drop over each network branch ΔPi=ΔPp+ΔPdi is then 2.8Differentiating equation (2.8) with respect to Γj, and noting equation (2.3), gives 2.9Substituting equation (2.9) into equation (2.5), via the chain rule, gives our generalized optimal area ratio 2.10which, for brevity, will be referred to as the generalized law to distinguish it from Murray’s law. It should be noted that the subscript j is not present in equation (2.10), so this relationship is not specific to a particular daughter channel; it relates the properties of all daughter channels to that of the parent.
This generalized law is valid for any cross-sectional shape, for any fluid (e.g. non-Newtonian) and for any Reynolds number (e.g. for turbulent flow). It is also valid for flows through nanoscale networks where the fluid is dominated by velocity slip at the walls; however, in the paper, we restrict our attention to the continuum-flow limit. We now consider some important cases where A can be expressed easily as an analytical function of k.
(a) Laminar flow
The steady-state incompressible Navier–Stokes momentum equation describes laminar flow through a long channel with an arbitrary cross-sectional shape, i.e. 2.11where μ is the dynamic viscosity (constant for a Newtonian fluid) and u is the streamwise channel velocity. This can be non-dimensionalized using ΔP/L, μ and cross-sectional area A, such that 2.12where 2.13and tilde denotes a dimensionless quantity or operator. The axes of the cross-sectional plane are defined as y,z and 2.14Provided the boundary conditions are fixed (which is the case for the continuum-flow limit, where the no-slip boundary condition applies), the solution of equation (2.12), , is independent of A, ΔP, L and μ, and is thus a property of the cross-sectional shape alone. Similarly, so is 2.15An expression for the volumetric flow rate is obtained by integrating the fluid momentum over the cross-sectional area 2.16Substitution of equations (2.13)–(2.15) into (2.16) gives the volumetric flow rate through a channel with an arbitrary cross-sectional shape 2.17and flow resistance per unit length 2.18It is assumed that the pressure drop over the network is small such that the viscosity and density are constants for a Newtonian fluid. For cylindrical channels S=1/(8π), equation (2.17) becomes the Hagen–Poiseuille flow rate. If the cross-sectional shape is constant throughout the network, i.e. S=const., substituting (2.18) into the generalized law (2.10) and cancelling the constant terms gives 2.19From equation (2.7), it can be seen that 2.20for all daughter channels. Combining equations (2.18) and (2.20) produces the cross-sectional area relationship between the ith and jth daughter channels 2.21where 2.22is the pressure-gradient ratio between the jth and ith daughter channels. Note that, because the shape property S cancels in equation (2.21), the generalized law will be independent of the cross-sectional shape of the channels. Substituting equation (2.21) into equation (2.19) and rearranging for Γj as defined by equation (2.2) gives 2.23Equation (2.23) relates the area of the parent channel to the area of the jth daughter channel in an optimized two-level network of laminar flow. It is valid for any cross-sectional shape, provided the shape is constant through the network. Equation (2.23) is only equivalent to Murray’s law (which is when posed in terms of an area ratio) if the daughter channels branch symmetrically, i.e. Ψi=Ψj=1/N and Φij=1. By inserting these constraints into equation (2.23), the symmetric generalized law for laminar flow is 2.24This means that for symmetric branching, Murray’s law is valid for any cross-sectional shape, not just circles. The reason Murray’s law produces a suboptimal result for asymmetric branching is that it was derived to optimize a single channel in isolation. However, as shown above (and verified later), for asymmetric branching, the global optimum is not the same as the optimum for each channel considered separately. One important reason for this is that the result of Murray’s single-channel optimization (Q∝R3) is independent of the pressure drop; so when applied to a branching network, the relative pressure drops over the daughter channels are not considered, and the optimization is under-constrained. Murray’s original principle, over time, has been misinterpreted as a general branching law (for symmetric and asymmetric configurations), leading to the prevalence of the incorrect form shown in equation (1.1). This misinterpretation has endured in subsequent literature regarding turbulent flow  and non-Newtonian fluids [17,18], as we shall now demonstrate.
(b) Turbulent flow
Turbulent flow is described by the phenomenological Darcy–Weisbach equation, which relates the pressure drop to the mean velocity for an incompressible fluid in a channel of arbitrary cross-sectional shape 2.25where f is the Darcy friction factor, is the mean streamwise velocity, is the hydraulic diameter, and is the wetted perimeter. Note that equation (2.25) is applicable to gravity-driven open channels, e.g. rivers, as well as closed pipes. In a river, the pressure drop is a function of the channel slope . Making the substitutions (which is a property of the cross-sectional shape, like S) and , equation (2.25) can be rewritten as 2.26and the flow resistance per unit length k is 2.27For turbulent flows, the pressure drop is proportional to the square of the volumetric flow rate, so k is a function of Q. However, for all constraint options of the optimization described by equation (2.1), dQ/dΓj=0. For the sake of deriving an analytical expression comparable to Murray’s law, we restrict our interest to fully rough-wall turbulent flow, where the friction factor is also approximately constant1 —i.e. it is independent of the Reynolds number and the volumetric flow rate. In this regime, the main applications are civil engineering hydraulic and pneumatic systems. So, when the shape is constant through the network, substituting equation (2.27) into the generalized law (equation (2.10)) gives 2.28Combining equations (2.20) and (2.27) produces the cross-sectional area relationship between the ith and jth daughter channels 2.29Substituting equation (2.29) into equation (2.28) and rearranging for the daughter–parent area ratio gives 2.30Equation (2.30) relates the area of the parent channel to the area of the jth daughter channel in an optimized two-level network for turbulent flow. It is valid for channels of any cross-sectional shape, provided the shape is constant through the network, and is only equivalent to the turbulent Murray’s law  (equation (1.2)) for symmetric branching, i.e. Ψi=Ψj=1/N and Φij=1, where (2.30) reduces to 2.31This also agrees with the turbulent flow symmetric branching results from previous studies [22,23]. Comparing equations (2.24) and (2.31) shows that, for symmetric branching, the optimal daughter–parent area ratio is smaller for turbulent flow than it is for laminar flow.
(c) Non-Newtonian fluid flow
Non-Newtonian fluids are typically characterized by a nonlinear relationship between shear stress and shear-strain rate. The power-law constitutive model [19,20] is one of the most popular, enabling a wide range of engineering problems to be solved analytically. For fluid flow through a circular channel, this is 2.32where m is the flow consistency index, dd/dr is the shear strain rate and n is the flow behaviour index. This relationship leads to an effective viscosity of 2.33Power-law fluids can be divided into three classes based on their flow behaviour index: (i) pseudo-plastic (shear thinning) fluids (n<1) exhibit a decrease in viscosity with increased shear strain rate; (ii) Newtonian fluids (n=1) exhibit a constant viscosity; and (iii) dilatant (shear thickening) fluids (n>1) exhibit an increase in viscosity with increased shear strain rate. Owing to the difficulty of applying the power-law model in two dimensions, the optimal branching of non-Newtonian fluids are considered here only for circular cross sections.
In cylindrical coordinates, the steady-state Navier–Stokes equation for incompressible laminar flow through a circular cross section is 2.34where u is the streamwise velocity (the radial and swirl velocity components are assumed to be zero). Substituting equation (2.33) into (2.34) and integrating with respect to r gives 2.35where C1 is a constant. At the midpoint of the cross section, when r=0, the velocity is at a maximum and thus dd/dr=0; therefore, C1=0. Integrating with respect to r once more produces 2.36where C2 is another constant. Inserting the no-slip condition at the wall into equation (2.36) produces the non-Newtonian velocity profile 2.37The volumetric flow rate is obtained by integrating the fluid momentum over the cross-sectional area, which, in cylindrical coordinates, is 2.38where θ is the azimuth. Substituting equation (2.37) into (2.38) gives 2.39By setting n=1 in equation (2.39) the Hagen–Poiseuille equation is recovered. Noting that A=πR2, the flow resistance per unit length is 2.40For non-Newtonian fluids, there is a nonlinear relationship between the pressure gradient and the volumetric flow rate, so k is again a function of Q. As explained previously, dQ/dΓj=0 for all the constraint options in the optimization described by equation (2.1), so substituting equation (2.40) into the asymmetric generalized law (equation (2.10)) gives 2.41Combining equations (2.20) and (2.40) produces the cross-sectional area relationship between the ith and jth daughter channels 2.42Substituting equation (2.42) into equation (2.41), and rearranging for the daughter–parent area ratio gives 2.43Equation (2.43) relates the area of the parent channel to the area of the jth daughter channel in an optimized two-level network of circular channels transporting a non-Newtonian fluid. By setting n=1, the asymmetric generalized law for Newtonian fluid flows (equation (2.23)) is retrieved. Equation (2.43) shows that Γj is dependent on the flow behaviour index n, contrary to results from previous studies on non-Newtonian branching flows that used Murray’s law [17,18]. The optimal daughter–parent area ratio is independent of n only for symmetric branching, i.e. Ψi=Ψj=1/N and Φij=1: 2.44This is exactly the same as equation (2.24) for symmetric Newtonian flows and agrees with previous studies [17,18] for symmetric non-Newtonian flows. Equation (2.43) can also be used to determine the optimal area ratio for the theoretical limits of the flow behaviour index n. For the shear-thickening-fluid limit, when , equation (2.43) reduces to 2.45Interestingly, equation (2.45) is independent of the daughter–daughter pressure-gradient ratio Φij and is exactly the same as Murray’s law for asymmetric branching (1.1) when posed in terms of areas. For the shear-thinning-fluid limit, when , equation (2.43) reduces to 2.46For this limit, the optimal area ratio is independent of the daughter flow-rate fraction Ψj, so when the daughter channel pressure gradients are equal, i.e. Φij=1, the daughter channel areas are also equal and Γ=N−2/3.
3. Numerical verification and discussion
In this section, we construct a numerical model of a two-level branching network which adopts the same fluid-physics assumptions used in Murray’s original paper, its subsequent extensions for turbulent flows and non-Newtonian fluid flows, and our own generalized law. These are (i) the flow through each channel is steady-state, incompressible and fully developed; (ii) the pressure is continuous throughout the network; and (iii) the pressure linearly varies over the entire length of each channel from inlet/outlet to a common branching point. The purpose of using the same physical model for the fluidic network is solely to demonstrate that its optimization does not lead to Murray’s law, but to the generalized law we derived above; verification of Murray’s fluid-physics assumptions is beyond the scope of this paper.
For clarity, we present the numerical model in a form specific to laminar flow through cylindrical channels (as per Murray’s original case), and refer the reader to appendix A for a more general description. The flow through each channel of the fluidic network is determined by momentum conservation, and is treated as being positive if it flows towards the point of branching—i.e. flow through the parent channel will be positive and flow through daughters will be negative. In the specific case of laminar flow through a cylindrical channel, and given the previously stated fluid-physics assumptions, this is the Hagen–Poiseuille law 3.1where qi is the volumetric flow rate through the ith channel in the network, ai is the cross-sectional area of the ith channel, pi is the pressure at the end of the ith channel (i.e. the inlet of the parent channel or the outlet of a daughter channel), li is the length of the ith channel and pB is the pressure at the point of branching (which is common to all channels). Note, here, unlike in our analytical derivation, the subscript i could denote either the parent channel (i=1) or one of the daughter channels (i=2,3…,M, where M is the total number channels that comprise the network). The model is completed by mass continuity at the branching point, i.e. 3.2calculation of the total network volume v, i.e. 3.3and the definition of the cross-sectional area ratio between the (i+1)th and ith channels 3.4
The system of equations (3.1)–(3.4) can be solved for the mass flow rates q(1:M) if the following are fixed: pressure at boundaries p(1:M), channel lengths l(1:M), fluid viscosity μ, network volume v, and the channel cross-sectional area ratios γ(1:M−1). In this paper, the solution is obtained using the trust-region dogleg algorithm  in MATLAB.
To enable a comparison with Murray’s law and our generalized law, we now optimize the network model (i.e. equations (3.1)–(3.4)) using a brute-force approach. For otherwise fixed properties (e.g. fixed volume, boundary pressures, etc.), the cross-sectional area ratio between the first daughter channel and the parent channel area γ1 is varied, through all physically viable values, to locate that which maximizes the volumetric flow rate through the network. This result can then be compared directly with Murray’s law and the generalized law, as the definition of γ1 is equivalent to that of Γd1.
(a) Laminar flow
The first set of optimization results demonstrate that Murray’s law is suboptimal for asymmetrically branching networks of any cross-sectional shape, even for laminar flow. To demonstrate that our generalized law is valid for any cross-sectional shape (as long as the shape remains constant through the network), the numerical verification is performed for three different cross sections: circular, square, and rectangular with an aspect ratio α=5. For circles, S=1/(8π), and for rectangles, an accurate approximation of S is calculated from simulations, using a standard central-difference solution of the laminar Navier–Stokes equations (2.12).2 For Murray’s law, mass conservation provides the closure , which leads to 3.5
In figure 1, to induce asymmetry, the daughter flow-rate fraction Ψj is varied whereas the daughter–daughter pressure-gradient ratio is kept constant at Φij=1. All solutions that the greater the fraction of flow through the daughter channel, the greater the optimum daughter’s area (relative to the parent); as is intuitive. The results confirm the finding that Murray’s law is only optimal for symmetric bifurcations (Ψ=0.5); for a flow-rate percentage of 10% (Ψ=0.1), Murray’s law under-predicts the optimum daughter area by as much as 26%. In contrast, the generalized law is accurate for all values of Ψj for all cross-sectional shapes tested. This confirms the analytical finding that Murray’s law has been mistakenly applied to asymmetrically branching networks, where the optimized result for each individual channel is not optimal for the network as a whole.
This can be further demonstrated by inducing asymmetry by varying the daughter–daughter pressure-gradient ratio Φij, and maintaining a constant daughter flow-rate fraction Ψj=0.5, as shown in figure 2. Murray’s law does not consider Φij to be a variable that affects the optimal daughter–parent area ratio Γj and shows a notable departure from the numerical optimization results; for a pressure-gradient ratio of Φij=2, Murray’s law over predicts the optimum daughter area by 24%. In contrast, the generalized law is accurate for all values of Φij, for all cross-sectional shapes and, as expected, shows that the optimal area of the jth daughter channel is smaller when it has a larger pressure gradient relative to the other daughter channel, as the flow rate flowing through each daughter is equal (Ψj=Ψi=0.5). This result is the same whether the pressure gradient is altered by varying the relative daughter channel lengths or the pressure drops. Figures 1 and 2 both show that as the extent of asymmetry increases, Murray’s law provides a poorer estimate of the optimal area ratio.
(b) Turbulent flow
The next set of optimization results are for fully rough-wall turbulent flow through an asymmetrically bifurcating network of channels with arbitrary, but constant, cross-sectional shape. For Murray’s law, the closure is provided by mass conservation, which leads to 3.6
The optimization results shown in figures 3 and 4 have asymmetry induced by varying the daughter flow-rate fraction Ψj and daughter–daughter pressure-gradient ratio Φij, respectively. Again, there is excellent agreement between the numerical optimization and the turbulent generalized law for all values of Ψj and Φij. Both figures 3 and 4 show that, except in the case of large asymmetries, the optimal daughter–parent area ratio for laminar flow is larger than the optimal area ratio for turbulent flow. This trend can broadly be explained by considering the equations for volumetric flow rate for laminar and turbulent flow (equations (2.17) and (2.26), respectively). For laminar flow Q∝A2, whereas for turbulent flow Q∝A5/4. Considering these relationships for the parent and the jth daughter channel, then and for laminar and turbulent flows, respectively (this is confirmed by the generalized laws for laminar flow (2.23) and turbulent flow (2.30)). As Ψj is always less than one, and thus the optimal area ratio for laminar flow will generally be larger than optimal area ratio for turbulent flow for the same fixed parameters.
Murray’s law proves to be a more accurate approximation for asymmetrically branching turbulent flows (compared with laminar flows), but still errs by 10% when Ψj=0.1 (and Φij=1), and by 19% when Φij=2 (and Ψj=0.5). For symmetric branching (Ψ=0.5 and Φij=1), the numerical and analytical solutions both agree with Murray’s law  and the results by [22,23].
(c) Non-Newtonian fluid flow
The final set of optimization results are for non-Newtonian fluid flow through an asymmetrically bifurcating network of circular channels. For Murray’s law, equation (3.5) is used. In figure 5, asymmetry is induced by varying Ψj, whereas Φij=1 is constant. The results demonstrate that the optimal daughter–parent area ratio Γj is dependent on the flow behaviour index n, contrary to the results of previous studies based on Murray’s law [17,18].
The Newtonian fluid case (n=1) is highlighted with a filled marker, and it is noted that this solution is the same as that shown in figure 1. It is observed that, for all n, the gradient (dΓj/dΨj) increases monotonically with increasing n. As , the fluid approaches the shear-thickening-fluid limit (equation (2.45)) where Murray’s law is correct for all Ψj. For smaller values of n, Murray’s law is correct only for symmetric bifurcations (Ψj=0.5) and becomes increasingly inaccurate as n decreases; for a flow-rate percentage of 10% (Ψj=0.1), Murray’s law under predicts Γj by 66% for n=10−4. The increasing error in the Murray’s law solution as n decreases is also shown in figure 6, where asymmetry is induced by varying Φij and Ψj=0.5 is fixed. Here, for a pressure-gradient ratio of Φij=2, Murray’s law over predicts the optimum daughter area by 172% when n=10−4. In contrast, the generalized law is accurate for all values of Ψj and n. The plot for n=0.74 is an approximation of the optimal area ratio for the cardiovascular system, based on the measurements of a falling-ball viscometer . As , the fluid approaches the shear-thinning-fluid limit (equation (2.46)) and Γj becomes independent of Ψj.
The reason for the shear-thickening- and shear-thinning-fluid limits can be found by examining the volumetric flow rate of a non-Newtonian fluid. When , by raising all terms to the power of n, equation (2.39) becomes 1=ΔPR/(2Lm) and the area ratio is only a function of the pressure gradient; hence, Γj does not vary with Ψj. When the daughter–daughter pressure-gradient ratio Φij decreases, the area must increase (and vice versa), as shown in figure 6. Conversely, when , equation (2.39) becomes Q=πR3/3 and the optimal area ratio is only a function of the volumetric flow rate; hence, Γj does not vary with Φij. This expression, with Q∝R3, is equivalent to Murray’s law (equation (1.1)), where the local optimization is the same as the global optimization, as shown in figures 5 and 6.
We have derived a generalized optimization principle that leads to analytical expressions for the optimum daughter–parent area ratio Γ for asymmetrically branching networks of any cross-sectional shape and for a range of fluidic models. This new optimal relation will enable deeper understanding of biological network behaviour and provide a generalized biomimetic design principle that can be applied to a variety of artificial branching systems to maximize their efficiency.
We have verified analytical solutions using a numerical optimization procedure and shown that, for symmetric branching of laminar and Newtonian fluids, our generalized law is equivalent to Murray’s law. However, when applied to an asymmetrically branching network, Murray’s law is suboptimal, as the global optimization of the entire network is not equal to the local optimization of each individual channel, which Murray’s law presumes. We further demonstrate that this mistake in the application of Murray’s law to asymmetric branching networks has endured for non-Newtonian fluids (e.g. in the cardiovascular system) and turbulent flows (e.g. in hydraulic or pneumatic civil engineering applications).
In non-Newtonian fluidic networks, Γ is dependent on the flow behaviour index n for asymmetric branching, contrary to what previous studies based on Murray’s law have stated. Murray’s law is only retrieved for non-Newtonian fluid networks at the shear-thickening limit, when and Γ is no longer dependent on the relative pressure gradients over the daughter channels. At the shear-thinning limit, when , Γ becomes independent of the relative flow rates through each daughter channel.
D.S. performed verification simulations, contributed to the analytical solutions and drafted the manuscript. D.A.L. derived the generalized law, provided the main analytical contribution, and helped draft the manuscript.
We have no competing interests.
This work is financially supported in the UK by EPSRC Programme Grants EP/I011927/1 and EP/N016602/1, and Grant EP/K038664/1.
Appendix A. General volumetric flow rate expression for the numerical optimization procedure
The equation for volumetric flow rate through the ith straight channel can be generally expressed as A 1where b1, b2 and b3 are flow-model-dependent constants. Table 1 shows the values of the constants for laminar flow through a channel of arbitrary cross-sectional shape, fully rough-wall turbulent flow through a channel of arbitrary cross-sectional shape, and non-Newtonian fluid flow through a channel with a circular cross section.
↵1 A solution for the whole range of Reynolds numbers can be obtained by using an empirical relation between the friction factor and the Reynold’s number, e.g. fitting to the Moody diagram.
↵2 Our numerical solver is an in-house code written in MATLAB, which uses matrix inversion to calculate the volumetric flow rates. All simulations use a 100×100 mesh, which has been shown (via a grid resolution study) to provide volumetric flow rate solutions to within 1% of the values obtained using a 200×200 and 300×300 mesh.
- Received June 6, 2016.
- Accepted June 23, 2016.
- © 2016 The Authors.
Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.