The Fictitious Domain Method Based on Navier Slip Boundary Condition for Simulation of Flow-Particle Interaction
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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 Ω and B be two bounded domains of ℝ^d such that B⊂Ω. 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 Ω, 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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