A Fluid Dynamics-based Modified Murray Law for Hierarchical Vein Networks in Lotus Leaves: Geometry, Transport Mechanism and Biological Implications

[1]          A. Marmur, The Lotus Effect:  Superhydrophobicity and Metastability, Langmuir, Vol. 20, No. 9, pp. 3517-3519, 2004/04/01, 2004.

[2]          C. Neinhuis, W. Barthlott, Characterization and Distribution of Water-repellent, Self-cleaning Plant Surfaces, Annals of Botany, Vol. 79, No. 6, pp. 667-677, 1997/06/01/, 1997.

[3]          B. Bhushan, Y. C. Jung, Natural and biomimetic artificial surfaces for superhydrophobicity, self-cleaning, low adhesion, and drag reduction, Progress in Materials Science, Vol. 56, No. 1, pp. 1-108, 2011/01/01/, 2011.

[4]          T.-S. Wong, S. H. Kang, S. K. Y. Tang, E. J. Smythe, B. D. Hatton, A. Grinthal, J. Aizenberg, Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity, Nature, Vol. 477, No. 7365, pp. 443-447, 2011/09/01, 2011.

[5]          S. Achanta, J.-P. Celis, Nanotribology of MEMS/NEMS,  in: E. Gnecco, E. Meyer, Fundamentals of Friction and Wear on the Nanoscale, Eds., pp. 631-656, Cham: Springer International Publishing, 2015.

[6]          E. Flinn, Lotus leaf yields slick idea for MEMS, Aerosp. Am, Vol. 5, pp. 24-5, 2005.

[7]          J.-H. He, PERIODIC SOLUTION OF A MICRO-ELECTROMECHANICAL SYSTEM, 2024, pp. 12, 2024-07-31, 2024.

[8]          J.-H. He, Q. Bai, Y.-C. Luo, D. Kuangaliyeva, G. Ellis, Y. Yessetov, P. Skrzypacz, Modeling and numerical analysis for MEMS graphene resonator, Frontiers in Physics, Vol. Volume 13 - 2025, 2025-April-25, 2025. English

[9]          D. Tian, X.-X. Li, J.-H. He, Geometrical potential and nanofiber membrane’s highly selective adsorption property, Adsorption Science & Technology, Vol. 37, No. 5-6, pp. 367-388, 2019.

[10]        Y. Zhang, H. Wu, X. Yu, F. Chen, J. Wu, Microscopic Observations of the Lotus Leaf for Explaining the Outstanding Mechanical Properties, Journal of Bionic Engineering, Vol. 9, No. 1, pp. 84-90, 2012/03/01, 2012.

[11]        Q. He, Y. Wang, H. Gu, J. Feng, H. Zhou, Dynamic crushing analysis of a circular honeycomb with leaf vein branched characteristic, Mechanics of Materials, Vol. 153, pp. 103566, 2021/02/01/, 2021.

[12]        Q. He, Y. Wang, X. Shi, X. Jing, Y. Jiang, Crushing behavior on the cylindrical tube based on lotus leaf vein branched structure, Mechanics of Materials, Vol. 165, pp. 104205, 2022/02/01/, 2022.

[13]        K. Guo, M. Liu, D. Vella, S. Suresh, K. J. Hsia, Dehydration-induced corrugated folding in Rhapis excelsa plant leaves, Proceedings of the National Academy of Sciences, Vol. 121, No. 17, pp. e2320259121, 2024.

[14]        M. Tian, J. Shuai, B. A. Bishop, W. Zhang, J. Chen, X. Wang, Plant cellulose-based biomimetic artificial Small-Diameter vascular materials enabled by gradient Dual-Network entanglement, Chemical Engineering Journal, Vol. 476, pp. 146751, 2023/11/15/, 2023.

[15]        W. Wu, R. M. Guijt, Y. E. Silina, M. Koch, A. Manz, Plant leaves as templates for soft lithography, RSC Advances, Vol. 6, No. 27, pp. 22469-22475, 2016.

[16]        B. Feng, T. Sun, W. Wang, Y. Xiao, J. Huo, Z. Deng, G. Bian, Y. Wu, G. Zou, W. Wang, T. Ren, L. Liu, Venation-Mimicking, Ultrastretchable, Room-Temperature-Attachable Metal Tapes for Integrated Electronic Skins, Advanced Materials, Vol. 35, No. 8, pp. 2208568, 2023.

[17]        C. D. Murray, The Physiological Principle of Minimum Work, Proceedings of the National Academy of Sciences, Vol. 12, No. 3, pp. 207-214, 1926.

[18]        J. Haskovec, P. Markowich, G. Pilli, Murray’s law for discrete and continuum models of biological networks, Mathematical Models and Methods in Applied Sciences, Vol. 29, No. 12, pp. 2359-2376, 2019.

[19]        D. Jing, S. Song, Y. Pan, X. Wang, Optimal fractal tree-like microchannel networks with slip for laminar-flow-modified Murray’s law, Beilstein Journal of Nanotechnology, Vol. 9, pp. 482-489, //, 2018.

[20]        C. Hou, L. Li, R. Lv, Z. Tian, X. Chen, Mixing Performance of Micromixers with Fractal Obstacles Based on Murray's Law, Chemical Engineering & Technology, Vol. 44, No. 12, pp. 2220-2227, 2021.

[21]        A. M. Talkington, R. B. Davis, N. C. Datto, E. R. Goodwin, L. A. Miller, K. M. Caron, Dermal Lymphatic Capillaries Do Not Obey Murray's Law, Frontiers in Cardiovascular Medicine, Vol. Volume 9 - 2022, 2022-April-12, 2022. English

[22]        C.-H. He, C. Liu, FRACTAL DIMENSIONS OF A POROUS CONCRETE AND ITS EFFECT ON THE CONCRETE’S STRENGTH, 2023, pp. 14, 2023-04-10, 2023.

[23]        C.-H. He, H.-W. Liu, C. Liu, A FRACTAL-BASED APPROACH TO THE MECHANICAL PROPERTIES OF RECYCLED AGGREGATE CONCRETES, 2024, pp. 14, 2024-07-31, 2024.

[24]        T. Miao, A. Chen, L. Zhang, B. Yu, A novel fractal model for permeability of damaged tree-like branching networks, International Journal of Heat and Mass Transfer, Vol. 127, pp. 278-285, 2018/12/01/, 2018.

[25]        S. YANG, H. FU, B. YU, FRACTAL ANALYSIS OF FLOW RESISTANCE IN TREE-LIKE BRANCHING NETWORKS WITH ROUGHENED MICROCHANNELS, Fractals, Vol. 25, No. 01, pp. 1750008, 2017.

[26]        T. Miao, A. Chen, Y. Xu, S. Yang, B. Yu, Optimal structure of damaged tree-like branching networks for the equivalent thermal conductivity, International Journal of Thermal Sciences, Vol. 102, pp. 89-99, 2016/04/01/, 2016.

[27]        J. Fan, X. Yang, Y. Liu, Fractal calculus for analysis of wool fiber: Mathematical insight of its biomechanism, Journal of Engineered Fibers and Fabrics, Vol. 14, pp. 1558925019872200, 2019.

[28]        L. Li, B. Yu, Fractal analysis of the effective thermal conductivity of biological media embedded with randomly distributed vascular trees, International Journal of Heat and Mass Transfer, Vol. 67, pp. 74-80, 2013/12/01/, 2013.

[29]        B. Zhou, Q. Cheng, Z. Chen, Z. Chen, D. Liang, E. A. Munro, G. Yun, Y. Kawai, J. Chen, T. Bhowmick, K. K. Padmanathan, L. G. Occhipinti, H. Matsumoto, J. W. Gardner, B.-L. Su, T. Hasan, Universal Murray’s law for optimised fluid transport in synthetic structures, Nature Communications, Vol. 15, No. 1, pp. 3652, 2024/05/07, 2024.

[30]        G. Feng, A CIRCULAR SECTOR VIBRATION SYSTEM IN A POROUS MEDIUM, Facta Universitatis, Series: Mechanical Engineering, 2023.

[31]        Y.-P. LIU, C.-H. HE, K. A. GEPREEL, J.-H. HE, CLOVER-INSPIRED FRACTAL ARCHITECTURES: INNOVATIONS IN FLEXIBLE FOLDING SKINS FOR SUSTAINABLE BUILDINGS, Fractals, Vol. 0, No. 0, pp. 2550041.

[32]        Y.-P. Liu, J.-H. He, M. H. Mahmud, Leveraging Lotus Seeds’ Distribution Patterns For Fractal Super-Rope Optimization, FRACTALS (fractals), Vol. 33, No. 03, pp. 1-11, 2025.

[33]        Y. Cheng, C.-L. Luo, C. Zhong, H. Lin, D. Marinkovic, J.-H. He, Differential equation-driven intelligent control: Integrating AI, Quantum computing, and adaptive strategies for next-generation industrial automation, Advances in Differential Equations and Control Processes, Vol. 32, No. 1, pp. 3096, 04/24, 2025.

[34]        S. Zhang, H. Zhang, Y. Wang, Z. Li, Dynamic properties and numerical simulations of a fractional phytoplankton-zooplankton ecological model, Networks and Heterogeneous Media, Vol. 20, No. 2, pp. 648-669, 2025.

[35]        C. Han, Y.-L. Wang, Z.-Y. Li, A high-precision numerical approach to solving space fractional Gray-Scott model, Applied Mathematics Letters, Vol. 125, pp. 107759, 2022/03/01/, 2022.

[36]        H. Che, Y.-L. Wang, Z.-Y. Li, Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative, Mathematics and Computers in Simulation, Vol. 202, pp. 149-163, 2022/12/01/, 2022.

[37]        K. A. McCulloh, J. S. Sperry, F. R. Adler, Water transport in plants obeys Murray's law, Nature, Vol. 421, No. 6926, pp. 939-942, 2003/02/01, 2003.

[38]        M. Majumder, N. Chopra, R. Andrews, B. J. Hinds, Enhanced flow in carbon nanotubes, Nature, Vol. 438, No. 7064, pp. 44-44, 2005/11/01, 2005.

[39]        Y. Zhang, Y.-W. Tan, H. L. Stormer, P. Kim, Experimental observation of the quantum Hall effect and Berry's phase in graphene, Nature, Vol. 438, No. 7065, pp. 201-204, 2005/11/01, 2005.

[40]        S. Stankovich, D. A. Dikin, G. H. B. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, R. S. Ruoff, Graphene-based composite materials, Nature, Vol. 442, No. 7100, pp. 282-286, 2006/07/01, 2006.

[41]        S. Iijima, T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter, Nature, Vol. 363, No. 6430, pp. 603-605, 1993/06/01, 1993.

[42]        A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C. N. Lau, Superior Thermal Conductivity of Single-Layer Graphene, Nano Letters, Vol. 8, No. 3, pp. 902-907, 2008/03/01, 2008.

[43]        P. Sundqvist, F. J. Garcia-Vidal, F. Flores, M. Moreno-Moreno, C. Gómez-Navarro, J. S. Bunch, J. Gómez-Herrero, Voltage and Length-Dependent Phase Diagram of the Electronic Transport in Carbon Nanotubes, Nano Letters, Vol. 7, No. 9, pp. 2568-2573, 2007/09/01, 2007.

[44]        J.-H. He, A New Resistance Formulation for Carbon Nanotubes, Journal of Nanomaterials, Vol. 2008, No. 1, pp. 954874, 2008.

[45]        H. Ha, S. Müller, R.-P. Baumann, B. Hwang, PEAKFORCE QUANTITATIVE NANOMECHANICAL MAPPING FOR SURFACE ENERGY CHARACTERIZATION ON THE NANOSCALE: A MINI-REVIEW, 2024, pp. 12, 2024-04-01, 2024.

[46]        M. A. Ilgamov, A. A. Aitbaeva, I. S. Pavlov, S. V. Dmitriev, CARBON NANOTUBE UNDER PULSED PRESSURE, 2024, pp. 18, 2024-07-31, 2024.

[47]        N. Anjum, J.-H. He, Geometric potential in nano/microelectromechanical systems: Part I mathematical model, International Journal of Geometric Methods in Modern Physics, Vol. 0, No. 0, pp. 2440027.

[48]        R. Zhu, K. Li, D. Wang, J. Fei, J. Yan Tan, S. Li, J. Zhang, H. Li, S. Fu, Biomimetic optimized concept with Murray networks for accelerated solar-driven water evaporation, Chemical Engineering Journal, Vol. 467, pp. 143383, 2023/07/01/, 2023.

[49]        D. Ma, G. Wang, Y. Ma, C. Zhu, X. Tang, Hydrodynamics behavior and mass transfer performance of gas–liquid two-phase flow in the honeycomb fractal microreactor, Chemical Engineering Journal, Vol. 462, pp. 142228, 2023/04/15/, 2023.

[50]        J. Liao, J. Nie, B. Sun, T. Jiao, M. Zhang, S. Song, A cellulose composite filter with multi-stage pores had high filtration efficiency, low pressure drop, and degradable properties, Chemical Engineering Journal, Vol. 482, pp. 148908, 2024.