A boundary condition-enhanced direct-forcing immersed boundary method for simulations of three-dimensional phoretic particles in incompressible flows

In this paper we propose an improved three-dimensional immersed boundary method coupled with a finite-difference code to simulate self-propelled phoretic particles in viscous incompressible flows. We focus on the phenomenon of diffusiophoresis which, using the driving of a concentration gradient, can generate a slip velocity on a surface. In such a system, both the Dirichlet and Neumann boundary conditions are involved. In order to enforce the boundary conditions, we propose two improvements to the basic direct-forcing immersed boundary method. The main idea is that the immersed boundary terms are corrected by adding the force of the previous time step, in contrast to the traditional method which relies only on the instantaneous forces in each time step. For the Neumann boundary condition, we add two auxiliary layers inside the body to precisely implement the desired concentration gradient. To verify the accuracy of the improved method, we present problems of different complexity: The first is the pure diffusion around a sphere with Dirichlet and Neumann boundary conditions. Then we show the flow past a fixed sphere. In addition, the motion of a self-propelled Janus particle in the bulk and the spontaneously symmetry breaking of an isotropic phoretic particle are reported. The results are in very good agreements with the data that are reported in previously published literature.</p