A fractional step immersed boundary method for Stokes flow with an inextensible interface enclosing a solid particle

In this paper, we develop a fractional step method based on the immersed boundary (IB) formulation for Stokes flow with an inextensible (incompressible) interface enclosing a solid particle. In addition to solving for the fluid variables such as the velocity and pressure, the present problem involves finding an extra unknown elastic tension such that the surface divergence of the velocity is zero along the interface, and an extra unknown particle surface force such that the velocity satisfies the no-slip boundary condition along the particle surface. While the interface moves with local fluid velocity, the enclosed particle hereby undergoes a rigid body motion, and the system is closed by the force-free and torque-free conditions along the particle surface. The equations are then discretized by standard centered difference schemes on a staggered grid, and the interactions between the interface and particle with the fluid are discretized using a discrete delta function as in the IB method. The resultant linear system of equations is symmetric and can be solved by fractional steps so that only fast Poisson solvers are involved. The present method can be extended to Navier-Stokes flow with moderate Reynolds number by treating the nonlinear advection terms explicitly for the time integration. The convergent tests for a Stokes solver with or without an inextensible interface are performed and confirm the desired accuracy. The tank-treading to tumbling motion for an inextensible interface enclosing a solid particle with different filling fractions under a simple shear flow has been studied extensively, and the results here are in good agreement with those obtained in literature.