In this article, we develop a least–squares/fictitious domain method for direct simulation of fluid particle motion with Navier slip boundary condition at the fluid–particle interface. Let $\Omega$ and $B$ be two bounded domains of ${\mathbb{R}}^d$ such that $\overline{B}\subset \Omega$. The motion of solid particle $B$ is governed by Newton’s equations. Our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full $\Omega$, followed by a well–chosen correction over $B$ and corrections related to translation velocity and angular velocity of the particle. This method is of the virtual control type and relies on a least–squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Since the fully explicit scheme to update the particle motion using Newton’s equation is unstable, we propose and implement an explicit–implicit scheme in which, at each time step, the position of the particle is updated explicitly, and the solution of Navier-Stokes equations and particle velocities are solved by the the least–squares/fictitious domain method implicitly. Numerical results are given to verify our numerical method.

在本文中，我们开发了一种最小二乘法/虚拟域方法，用于在流体 - 粒子界面处使用 Navier 滑动边界条件直接模拟流体粒子运动。让$\Omega$ 和 $乙$ 是两个有界域 ${\mathbb{电阻}}^d$ 以至于 $\overline{乙}\subset \Omega$. 固体粒子的运动$乙$受牛顿方程支配。我们的目标是开发一种虚构的领域方法，在该方法中，可以完整地解决原始问题的变体$\Omega$，然后是精心挑选的更正 $乙$以及与粒子的平移速度和角速度相关的修正。这种方法是虚拟控制类型，依赖于最小二乘公式，使问题可以通过在精心选择的控制空间中运行的共轭梯度算法来解决。由于使用牛顿方程更新粒子运动的完全显式方案是不稳定的，我们提出并实现了一个显式-隐式方案，其中，在每个时间步，粒子的位置都被显式更新，并且 Navier-Stokes 方程的解和粒子速度由最小二乘/虚拟域方法隐式求解。给出了数值结果来验证我们的数值方法。