A far-field boundary condition for measuring drag force on micro/nano particles

Predicting the flow around airborne particles smaller than 1 µm, which is essential for many air-quality control applications, is challenging because the flow is both rarefied (due to the particle size approaching the mean free path of the gas) and characterised by very low Reynolds numbers (due to the small velocities and length scales involved). The accurate measurement of the drag force on these particles requires the solution of the Boltzmann equation, typically performed using the Direct Simulation Monte Carlo (DSMC) method. In the conventional formulation of these simulations, an accurate value of drag force can only be obtained if the domain boundaries in the DSMC simulation are placed at very large distances from the particle; using small domains can significantly over-predict the drag. The computational cost of DSMC simulations scales with the cube of the domain size for a general 3D flow, making it computationally intractable to measure drag on particles of arbitrary shape in the transition regime. In this work, we propose a boundary condition that emulates the flow conditions at a finite distance from the particle, rather than the free-stream conditions commonly used in previous studies. Our approach exploits the fact that the flow disturbance far from an arbitrary particle, travelling at very low Reynolds number, has a known analytical form that is given by the fundamental solution to the Stokes equations. Employing this boundary condition enables accurate simulations of drag prediction using much smaller domain sizes than otherwise possible, decreasing the computational cost by up to three orders of magnitude. The proposed approach allows the first accurate and tractable calculation of the drag force on slow-moving arbitrary-shaped 3D particles in the transition regime.