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.

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