arXiv 2103.059132021-03math.AP

无限大空间周期管道中的双周期粘性流动

Double periodic viscous flows in infinite space-periodic pipes

Hugo Beirao da Veiga, Jiaqi Yang

备注：: 28 pages

摘要：

研究了不可压缩流体在$n+1$维无限长管道$\,\La\,$中沿$z=x_{n+1}$方向的运动.我们设置了$\,x=(x_1,x_2,\cdots,x_{n})$和$z=x_{n+1}$。我们用$Σ_z$表示管道在水平$z\,,$处的横截面，用$v_z$表示速度的$n+1$分量。流体运动由演化Stokes或Navier-Stokes方程以及无滑移边界条件$\bv=\,0\,$描述。我们寻找解决方案$\bv(x,z,t)$与给定的，任意的，$T-$时间周期总通量$\,\int_{Σ_z} \,v_z(x,z,t)\,dx=g(t)\,,$，这应该是同时$T$ -周期的时间和$L$ -周期的$z\,.$我们证明了上述问题的解决方案的存在性和唯一性。所得结果推广了文献\cite{B-05}中的结果，其中截面与$z$无关。这个论证是通过一系列步骤提出的。我们首先考虑线性，平稳，$z-$周期斯托克斯问题。然后我们研究了双周期发展的Stokes方程，这是问题的核心。最后，我们推广到完整的Navier-Stokes方程.

英文摘要：

We study the motion of an incompressible fluid in an $n+1$-dimensional infinite pipe $\,\La\,$ with an $L$-periodic shape in the $z=x_{n+1}$ direction. We set $\,x=(x_1,x_2,\cdots,x_{n})$, and $z=x_{n+1}$. We denote by $Σ_z$ the cross section of the pipe at the level $z\,,$ and by $v_z$ the $n+1$ component of the velocity. Fluid motion is described by the evolution Stokes or Navier-Stokes equations together with the non-slip boundary condition $\bv=\,0\,$. We look for solutions $\bv(x,z,t)$ with a given, arbitrary, $T-$time periodic total flux $\,\int_{Σ_z} \,v_z(x,z,t)\,dx=g(t)\,,$ which should be simultaneously $T$-periodic with respect to time and $L$-periodic with respect to $z\,.$ We prove existence and uniqueness of the solution to the above problems. The results extend those proved in reference \cite{B-05}, where the cross sections were independent of $z$. The argument is presented through a sequence of steps. We start by considering the linear, stationary, $z-$periodic Stokes problem. Then we study the double periodic evolution Stokes equations, which is the heart of the matter. Finally, we end with the extension to the full Navier-Stokes equations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文