An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

Keywords: Incompressible Stokes equations Singular force Immersed interface method CG-Uzawa method Deformable interface Front tracking Irregular domains Rigid boundaries a b s t r a c t We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrang-ian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deform-able interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains. The low Reynolds number flow with interfaces in complex geometries is of interest in many engineering and physiological applications, for example, multi-phase flows in various fluid components within fuel cell, biological cell trapping and manipulation in microfluidic device, and droplet motion in confined geometries. Flow problems involving deformable interfaces and complex geometries often pose numerical difficulties and challenges in computational fluid dynamics. One of the difficulties and challenges in these problems is that the fluid motion, the motion of the deformable interface and the interaction with the rigid boundaries must be computed simultaneously. This is necessary in order to account for the complex interaction between the fluid, the interfaces and the rigid boundaries. The other difficulty is the accuracy of the fluid domain computation, and this can be improved partially by implementing moving mesh techniques as in [10,47]. Fig. 1 shows an illustration of such flow problems involving the rigid boundary and fixed/deformable interface embedded in a uniform …