通过重覆均质化方法得到的绝对刚体的双孔隙度模型
Double Porosity Models for Absolutely Rigid Body via Reiterated Homogenization
AI总结：
本文通过重覆均质化方法，推导了绝对刚体中液体过滤的双孔隙度模型，得到了长短时间过程对应的不同控制方程。
摘要：
本文从均质化理论出发，推导了绝对刚体中液体过滤的双孔隙度模型。微观层面的流体动力学控制方程由占据裂隙-孔隙空间的可压缩粘性流体的Stokes方程组构成，该空间由裂隙（裂缝）和孔隙两个独立系统组成。假设孔隙的无量纲尺寸δ依赖于裂隙的无量纲尺寸ε，满足δ=ε^r（r>1）。当裂隙的无量纲尺寸趋于零且固体为几何周期结构时，对均质化过程进行了严格论证。结果表明，对于长时间过程，可推导出裂隙中液体的常规Darcy过滤方程，而孔隙中的液体被阻塞且静止；对于短时间过程，均质化系统由声学方程组成，描述具有三个独立参数的双速度连续介质：孔隙中的液体速度、裂隙中的液体速度以及共同压力。该证明基于G. Allaire和M. Briane提出的重覆均质化方法。
英文摘要：
Double porosity models for the liquid filtration in an absolutely rigid body is derived from homogenization theory. The governing equations of the fluid dynamics on the microscopic level consist of the Stokes system for a slightly compressible viscous fluid, occupying a crack -- pore space. In turn, this domain is a union of two independent systems of cracks (fissures) and pores. We suppose that the dimensionless size $δ$ of pores depends on the dimensionless size $\varepsilon$ of cracks: $δ=\varepsilon^{r}$ with $r>1$. The rigorous justification is fulfilled for homogenization procedure as the dimensionless size of the cracks tends to zero, while the solid body is geometrically periodic. As a result, for the long-time process we derive the usual Darcy equations of filtration for the liquid in cracks, while the liquid in pores is blocked and unmoved. For the short-time processes the homogenized system consists of acoustic equations, describing a two-velocity continuum with three independent parameters: the liquid velocity in pores, the liquid velocity in cracks and the common pressure. The proofs are based on the method of reiterated homogenization, suggested by G. Allaire and M. Briane.
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