AI 中文总结
本文建立弹性边界细长多孔介质的流体驱动压实模型，推导孔隙度的主导阶非线性扩散方程，揭示流态与弹性参数的关联，得到亚达西、超达西流等不同流态的区分及紧凑表达式。
AI 中文摘要
生物组织或水凝胶等软材料的灌注对于器官功能以及色谱柱、生物反应器等实验室系统的运行至关重要。受这些应用启发，我们对由不可渗透弹性膜约束的细长圆柱形多孔介质中流体驱动的压实过程进行建模，研究流态与弹性参数的关联。采用达西流的拉格朗日公式，结合孔隙度依赖渗透率和弹性模量的小应变线性弹性理论，我们在小纵横比极限下进行渐近约化，得到孔隙度的主导阶非线性扩散方程并对其进行数值求解。刚性边界会产生压实平台，而柔性壁在最大程度上仅呈现中间平台；当施加压力与膜刚度和初始孔隙度的乘积相当时，流动会在中间平台后增加。当膜刚度低于多孔介质刚度时，流速可超过刚性介质的预期值。参数空间图区分了出现平台和突破的流态、稳态流速低于不可变形介质预测值（亚达西流）或高于该预测值（超达西流）的流态，并界定了理论适用的小应变范围。针对可忽略重力的渐近解捕捉了偏离平台的情况，得到了有效渗透率和流速的紧凑表达式。
英文摘要
Perfusion of soft materials such as biological tissue or hydrogels is essential for the functioning of organ and laboratory systems such as chromatographic columns and bioreactors. Inspired by these applications, we model fluid-driven compaction in a long, thin cylindrical porous medium bounded by an impermeable elastic membrane and study how flow regimes relate to elastic parameters. Using a Lagrangian formulation of Darcy flow coupled to small-strain linear elasticity with porosity dependent permeability and elastic moduli, we perform an asymptotic reduction in the small aspect ratio limit and obtain a leading-order nonlinear diffusion equation for the porosity, which we solve numerically. Whereas rigid boundaries produce a compaction plateau, compliant walls exhibit, at most, an intermediate plateau beyond which the flow increases once the imposed pressure becomes comparable to the product of membrane stiffness and initial porosity. When the membrane is less stiff than the porous medium, flow rate can exceed that expected for a rigid medium. A parameter space map distinguishes regimes where plateau and breakthrough occur, where the steady flow rate is below (sub-Darcy) or above (super-Darcy) the undeformable-medium prediction, and delineates the small-strain domain in which the theory applies. An asymptotic solution for negligible gravity captures the departure from the plateau and yields compact expressions for effective permeability and flow rate.
Comments47 pages, 10 figures
