AI 中文总结
该研究针对含小颗粒的变形管道内黏性不可压缩流体稳态运动，建立带混合边界条件的Navier-Stokes方程模型，用均匀化能量方法分析小参数趋于零时的渐近行为，证明有效方程含额外Brinkman项，并通过反证法得到规定通量问题的一致界。
AI 中文摘要
针对含若干直径为$\boldsymbol{\u03b5^3}$、相互间距为$\boldsymbol{\u03b5}$的小颗粒的变形管道内黏性不可压缩流体的稳态运动，采用带混合边界条件的Navier-Stokes方程进行建模。除颗粒表面的非齐次Dirichlet边界条件外，边界条件还涉及管道进出口处的伯努利压力与切向速度，同时沿管道规定了横向通量率或压降。应用均匀化理论中的能量方法，本文研究了当$\boldsymbol{\u03b5 \to 0}$时上述系统解的渐近行为，对数据大小无任何限制，并证明有效方程会额外出现一个Brinkman项。本研究的一个重要特点是所需的一致界：在规定通量问题的情形下，通过基于定常Euler方程解的伯努利定律的反证法得到了该一致界。
英文摘要
The steady motion of a viscous incompressible fluid in a distorted pipe, containing several small particles of diameter $\eps^3$ and mutual distance $\eps$, is modeled through the Navier-Stokes equations with mixed boundary conditions. Apart from inhomogeneous Dirichlet boundary conditions on the particles, these involve the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems as $\eps \to 0$, without any restriction on the magnitude of the data, and show that the effective equations display an additional Brinkman term. An important feature of the present work concerns the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.
DOI：10.13140/RG.2.2.18110.45123
