Keywords: boundary singularity method, Stokeslet, direct simulation Monte Carlo, partial slip, micro- and nano- flows, coarse-grain parallelization.

The goal of this paper is to develop a combined continuum and molecular methodology to compute micro- and nano- scale flows for which slip boundary conditions are important. A novel hybrid method combining the boundary singularity method (BSM) for solving Stokes equations for the entire domain and the direct simulation Monte Carlo (DSMC) for the area near rigid boundaries is proposed, tested and implemented using parallel computers.

The idea of coupling is to use the DSMC method to generate physically accurate boundary conditions for higher values of Kn numbers in which heuristic partial slip formulas are either no longer accurate or invalid. These boundary conditions are used by the BSM method. In turn, velocity field obtained by BSM is used as boundary conditions for DSMC at outer boundary of DSMC zone. To initiate the coupling process, the BSM is used to provide the DSMC method with the boundary conditions at the outermost layer. At the outer boundary of DSMC domain the DSMC velocity is sampled stochastically around mean BSM velocity. Once the DSMC procedure is completed, the velocities at the centres of the innermost layer of cells are used as the boundary conditions for the BSM. The BSM and DSMC are coupled after every BSM and DSMC iteration until a prescribed tolerance is reached.

The numerical validation and optimization of the DSMC procedure includes the choice of the number density of the particles and the size of the computational cell. The optimization of the BSM includes the choice of the location of the singularities outside of computational domain to minimize the matrix condition number and thus maximize the convergence rate of linear system. The correct set-up of the BSM-DSMC coupling includes selection of the size of the DSMC zone, determination of the needed number of couplings and equalization of the statistical velocity obtained by the DSMC with the deterministic velocity obtained by BSM. Applications considered include Taylor-Couette flow and flow over multiple cylindrical fibers. The parallel computing implementation of the BSM-DSMC method for the flow over set of filtration fibers is considered in detail. The DSMC procedure is computationally expensive. To reduce the computational time when flows over multiple fibers are modelled, the code developed is parallelized using MPI. The efficient BSM developed in prior work of the authors requires much less computer time than the DSMC so the BSM was performed in a single processor. To follow the coupling algorithm described in the previous section, when the BSM procedure is completed, the boundary conditions for the DSMC procedure are passed to a number of processors (equal to number of fibers if the processors are available), which carry out the DSMC calculation. After completion of the DSMC step, the processors send boundary conditions for the BSM to the first processor. Efficient parallelization of the coupled BSM-DSMC method was implemented and tested in two multi-processor computers.

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