论文标题
在无粘性极限内穿孔域中Navier-Stokes方程的均匀化
Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit
论文作者
论文摘要
我们将解决方案$ u_ \ varepsilon $研究到$ \ mathbb r^3 $中的Navier-Stokes方程，由以$（\ VAREPSILON \ MATHBB Z）为中心的小粒子穿孔，在粒子处，没有滑滑边界条件。我们研究小$ \ varepsilon $的$ u_ \ varepsilon $的行为，具体取决于直径$ \ varepsilon^α$，$α> 1 $的粒子和粘度$ \ varepsilon^γ$，$γ$，$γ> 0 $。当粒子处的本地雷诺数可以忽略不计时，我们在所有制度中都证明了$ u_ \ varepsilon $的定量收敛结果。然后，颗粒大约在流体上施加线性摩擦力。获得的有效宏观方程取决于集体摩擦的数量级。我们获得a）批判性政权中的欧拉·布林克曼方程，b）超临界制度中的欧拉方程和超临界政权中的c）达西定律。
We study the solution $u_\varepsilon$ to the Navier-Stokes equations in $\mathbb R^3$ perforated by small particles centered at $(\varepsilon \mathbb Z)^3$ with no-slip boundary conditions at the particles. We study the behavior of $u_\varepsilon$ for small $\varepsilon$, depending on the diameter $\varepsilon^α$, $α> 1$, of the particles and the viscosity $\varepsilon^γ$, $γ> 0$, of the fluid. We prove quantitative convergence results for $u_\varepsilon$ in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain a) the Euler-Brinkman equations in the critical regime, b) the Euler equations in the subcritical regime and c) Darcy's law in the supercritical regime.
