发表机构
Department of Mathematics and Statistics, University of Ottawa（渥太华大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究二维或三维广义斯托克斯方程解的结构，提出新方法，将速度与压力解耦，通过求解类似亥姆霍兹向量方程和边界方程求解，最后给出两种数值求解方法及结果证明有效性。
AI 中文摘要
我们证明了二维或三维中具有狄利克雷边界条件的广义斯托克斯方程的解具有特定结构。速度\(u\)是两个无散度且求解类似向量亥姆霍兹方程的速度场\(\omega\)和\(\theta\)的叠加，\(u = \omega + \theta\)。这些方程的右侧涉及外力亥姆霍兹分解的旋转分量和一个称为固体压力的调和函数\(q\)。同样，流体压力\(p\)是外力亥姆霍兹分解的势\(\pi\)和固体压力\(q\)的叠加，\(p = \pi + q\)。结果表明，无论固体压力\(q\)如何，\(\omega + \theta\)都满足广义斯托克斯方程，除了狄利克雷边界条件的法向分量。固体压力解决一个具有自伴强制算子的线性边界方程，其作用是约束\(\omega + \theta\)满足狄利克雷边界条件的法向分量。该方法从数值角度来看很有吸引力。它将速度与压力解耦，不直接求解不可压缩性约束。它只需要求解在边界上耦合的类似亥姆霍兹向量方程和一个边界方程。在最后一节中，我们提出了两种数值求解广义斯托克斯方程的方法，并给出了一些结果来证明我们方法的有效性。
英文摘要
We show that the velocity solution $u$ to the generalized Stokes equations with Dirichlet boundary conditions in dimension two and three is the superposition of two divergence-free velocity fields $ω$ and $\thetaup$, $u=ω+\thetaup$. Both $ω$ and $\thetaup$ solve screened Poisson vector equations with right hand side respectively the rotational component in the Helmholtz decomposition of the external force, and the gradient of a harmonic scalar function $q$ called ``solid pressure". Similarly, the fluid pressure $p$ is the superposition of the potential $π$ in the Helmholtz decomposition of the external force, and of $q$, $p=π+q$. It turns out that regardless the pressure $q$, $ω+\thetaup$ is divergence-free and satisfies the generalized Stokes equations, except the normal component of the Dirichlet boundary condition. The role of $q$, which solves a boundary linear equation with a self-adjoint definite positive operator, is to constrain $ω+ \thetaup$ to satisfy even the normal component of the Dirichlet boundary condition. The method is attractive for solving numerically the time dependent Navier-Stokes equations in two and three dimensions. It decouples the velocity from the pressure and solves the incompressibility constraint as a byproduct of unconstrained screened Poisson vector equations for $ω$ and $\thetaup$, equipped with appropriate boundary conditions. In the two last sections we present two methods for solving numerically the generalized Stokes equations, and some results to demonstrate the efficiency of our approach.
Comments26 pages, 4 figures (each with 4 images)
