Abstract: We present the development of a fast and accurate boundary integral equation (BIE) method for the computation of two-dimensional incompressible Stokes flow over superhydrophobic (SH) surfaces. These surfaces are composed of alternating solid portions, grooves, or air pockets, leading to enhanced slip, and are particularly relevant for microfluidic device design. While BIE methods offer advantages in SH flow computations, problems involving SH surfaces present challenges due to flow singularities resulting from complex surface microstructures. These challenges include mixed boundary conditions and geometric corners, causing standard quadrature rules for smooth integrals to suffer from severe accuracy loss. Although often used adaptive mesh refinement can mitigate this issue, it tends to grow the size of discretization significantly, and has difficulties in achieving satisfactory accuracy due to linear system ill-conditioning. To resolve these issues, we combine the Recursively Compressed Inverse Preconditioning (RCIP) method with a scaling technique, kernel-split quadratures, and the Fast Multipole Method. This incorporation yields a fast and accurate numerical scheme for SH flow computations. We demonstrate the effectiveness of our method through several illustrative examples. This research is joint work with Shidong Jiang (Flatiron Institute) and Michael Siegel (New Jersey Institute of Technology).

Contact: Robert Krasny