Numerical study of Darcy's law of yield stress fluids on a deep tree-like network

Stéphane Munier, Alberto Rosso
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Abstract

Understanding the flow dynamics of yield stress fluids in porous media presents a substantial challenge. Both experiments and extensive numerical simulations frequently show a non-linear relationship between the flow rate and the pressure gradient, deviating from the traditional Darcy law. In this article, we consider a tree-like porous structure and utilize an exact mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Specifically, we adapt an algorithm recently introduced by Brunet et al. [Europhys. Lett. 131, 40002 (2020)] to simulate exactly the tip region of branching random walks with the help of a spinal decomposition, to accurately compute the flow on extensive trees with several thousand generations. Our results confirm the asymptotic predictions proposed by Schimmenti et al. [Phys. Rev. E 108, L023102 (2023)], tested therein only for moderate trees of about 20 generations.
深层树状网络上屈服应力流体的达西定律数值研究
了解屈服应力流体在多孔介质中的流动动力学是一项巨大的挑战。实验和大量的数值模拟经常显示流速与压力梯度之间存在非线性关系,偏离了传统的达西定律。在本文中,我们考虑了树状多孔结构,并利用有向聚合物(DP)与 Cayleytree 上无序键能的精确映射。具体来说,我们调整了 Brunet 等人[Europhys. Lett. 131, 40002 (2020)]最近引入的算法,在脊柱分解的帮助下精确模拟分支随机漫步的顶端区域,从而精确计算数千代广泛树上的流动。我们的结果证实了 Schimmenti 等人[Phys.Rev. E 108, L023102 (2023)]提出的渐进预测,但他们只对约 20 代的中等树进行了测试。
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