Efficient numerical method for Stokes flows in unbounded domains with informative boundary condition using axial Green function method

Abstract

Flow calculations in an unbounded domain have limitations and challenges due to its infiniteness. A common approach is to impose a far-field asymptotic condition to determine a unique flow. The leading behavior of the flow is identified at the far field, and then an unknown coefficient is assumed for the second behavior. This allows us to propose an efficient numerical method to solve two-dimensional steady Stokes and potential flows in a truncated domain along with the coefficients. The second term provides crucial hydrodynamic information for the flow and is referred to as the informative boundary condition. The truncation creates artificial boundaries requiring boundary conditions for the approximate solution. The axial Green function method (AGM), combined with a specific one-dimensional Green function over a semi-infinite axis-parallel line extended to infinity, allows us to implement the informative boundary condition in the truncated domain. AGMs, designed for complicated domains, are now applied to infinite domain cases because AGMs' versatility enables implementing the informative boundary condition by changing only the axial Green function. This approach's efficiency, accuracy, and consistency are investigated through several appealing Stokes flow problems including potential ones in infinite domains.