Generalizing Murray's law: An optimization principle for fluidic networks of arbitrary shape and scale

Stephenson, David B.; Patronis, Alexander; Holland, David M.; Lockerby, Duncan A.

doi:10.1063/1.4935288

Cited by 23 publications

(23 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This allowed us to investigate the effects of expansion independent of shear stress, which is identical at 2.4 dyne/cm 2 under typical operating pressure across all designs. The second major change was a branching inlet pattern using the planar adaptation of Murray’s law 42 . This modification dramatically reduced the occurrence of channel inlet blockade.…”

Section: Resultsmentioning

confidence: 99%

Biomimetic post-capillary venule expansions for leukocyte adhesion studies

Benson

Myers

et al. 2018

Sci Rep

View full text Add to dashboard Cite

Leukocyte adhesion and extravasation are maximal near the transition from capillary to post-capillary venule, and are strongly influenced by a confluence of scale-dependent physical effects. Mimicking the scale of physiological vessels using in vitro microfluidic systems allows the capture of these effects on leukocyte adhesion assays, but imposes practical limits on reproducibility and reliable quantification. Here we present a microfluidic platform that provides multiple (54–512) technical replicates within a 15-minute sample collection time, coupled with an automated computer vision analysis pipeline that captures leukocyte adhesion probabilities as a function of shear and extensional stresses. We report that in post-capillary channels of physiological scale, efficient leukocyte adhesion requires erythrocytes forcing leukocytes against the wall, a phenomenon that is promoted by the transitional flow in post-capillary venule expansions and dependent on the adhesion molecule ICAM-1.

show abstract

Section: Resultsmentioning

confidence: 99%

Biomimetic post-capillary venule expansions for leukocyte adhesion studies

Benson

Myers

et al. 2018

Sci Rep

View full text Add to dashboard Cite

show abstract

“…was computed from Murray's law (1), giving diameter ratio d j+1 /d j = 2 −1/3 , and length ratio L j+1 /L j = 2 −1/2 . This ratio was used in a majority of the studies listed in Table 1, possibly because the length scaling L j+1 /L j = 2 −1/3 produces overlapping networks starting from some generation order [36][37][38].…”

Section: Of 23mentioning

confidence: 99%

“…The corresponding cross-section ratio of the adjacent channels was related as A j+1 /A j = 2 −2/3 . In real rectangular channels [37,38] the ratio…”

Section: Of 23mentioning

confidence: 99%

Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells

Kizilova

Sauermoser

Kjelstrup

et al. 2020

Entropy

View full text Add to dashboard Cite

The fractal-type flow-fields for fuel cell (FC) applications are promising, due to their ability to deliver uniformly, with a Peclet number Pe~1, the reactant gases to the catalytic layer. We review fractal designs that have been developed and studied in experimental prototypes and with CFD computations on 1D and 3D flow models for planar, circular, cylindrical and conical FCs. It is shown, that the FC efficiency could be increased by design optimization of the fractal system. The total entropy production (TEP) due to viscous flow was the objective function, and a constant total volume (TV) of the channels was used as constraint in the design optimization. Analytical solutions were used for the TEP, for rectangular channels and a simplified 1D circular tube. Case studies were done varying the equivalent hydraulic diameter (Dh), cross-sectional area (DΣ) and hydraulic resistance (DZ). The analytical expressions allowed us to obtain exact solutions to the optimization problem (TEP→min, TV=const). It was shown that the optimal design corresponds to a non-uniform width and length scaling of consecutive channels that classifies the flow field as a quasi-fractal. The depths of the channels were set equal for manufacturing reasons. Recursive formulae for optimal non-uniform width scaling were obtained for 1D circular Dh -, DΣ -, and DZ -based tubes (Cases 1-3). Appropriate scaling of the fractal system providing uniform entropy production along all the channels have also been computed for Dh -, DΣ -, and DZ -based 1D models (Cases 4-6). As a reference case, Murray’s law was used for circular (Case 7) and rectangular (Case 8) channels. It was shown, that Dh-based models always resulted in smaller cross-sectional areas and, thus, overestimated the hydraulic resistance and TEP. The DΣ -based models gave smaller resistances compared to the original rectangular channels and, therefore, underestimated the TEP. The DZ -based models fitted best to the 3D CFD data. All optimal geometries exhibited larger TEP, but smaller TV than those from Murray’s scaling (reference Cases 7,8). Higher TV with Murray’s scaling leads to lower contact area between the flow-field plate with other FC layers and, therefore, to larger electric resistivity or ohmic losses. We conclude that the most appropriate design can be found from multi-criteria optimization, resulting in a Pareto-frontier on the dependencies of TEP vs TV computed for all studied geometries. The proposed approach helps us to determine a restricted number of geometries for more detailed 3D computations and further experimental validations on prototypes.

show abstract

“…where r = r(x) ≥ r 0 > 0 is a prescribed function that models the isotropic background permeability of the medium, and I ∈ R d×d is the unit matrix. Again, (19) is equipped with the no-flux boundary condition…”

Section: Murray's Law For the Continuum Limit Model On Rectangular Gridsmentioning

confidence: 99%

Murray’s law for discrete and continuum models of biological networks

Haškovec

Markowich

Pilli

2019

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray's law for its linearly stable steady states.

show abstract

Generalizing Murray's law: An optimization principle for fluidic networks of arbitrary shape and scale

Cited by 23 publications

References 45 publications

Biomimetic post-capillary venule expansions for leukocyte adhesion studies

Biomimetic post-capillary venule expansions for leukocyte adhesion studies

Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells

Murray’s law for discrete and continuum models of biological networks

Contact Info

Product

Resources

About