An Unconditionally Energy Stable Penalty Immersed Boundary Method for Simulating the Dynamics of an Inextensible Interface Interacting with a Solid Particle

In this paper, a novel penaltymethod based on the immersed boundary formulation is proposed for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow toweaken the incompressibility condition, relating the surface divergence to the elastic tension σ to relax the interface’s inextensibility, and connecting the particle surface-velocity with the particle surface force F to regularize the particle’s rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns ph, σh and Fh directly, so that we just need to solve a smaller linear system of the velocity approximation uh . This advantage is preserved and even enhanced when such approach is applied to the Po-Wen Hsieh was partially supported by the National Science Council of Taiwan under the Grant NSC 102-2115-M-033-007-MY2. Ming-Chih Lai was partially supported by the National Science Council of Taiwan under the Grant NSC 101-2115-M-009-014-MY3. Suh-Yuh Yang was partially supported by the National Science Council of Taiwan under the Grant NSC 101-2115-M-008-008-MY2. P.-W. Hsieh Department of Applied Mathematics, Chung Yuan Christian University, Jhongli City 32023, Taoyuan County, Taiwan e-mail: [email protected] M.-C. Lai Center of Mathematical Modeling and Scientific Computing, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan e-mail: [email protected] S.-Y. Yang (B) · C.-S. You Department of Mathematics, National Central University, Jhongli City 32001, Taoyuan County, Taiwan e-mail: [email protected] C.-S. You e-mail: [email protected] 123 Author's personal copy