SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers

A working Smoothed Particle Hydrodynamics (SPH) formalism for solving the equations of motion of a viscous fluid is presented. The method is based on a standard symmetrized SPH expression for the viscous forces, which involves only first-order derivatives of the kernel through a direct evaluation of the viscous stress tensor. Therefore, the interpolation can be performed using low-order kernels of compact support without compromising the accuracy and stability of the results. In principle, the scheme is suitable for treating compressible fluids with arbitrary shear and bulk viscosities. Here, we demonstrate that when it is combined with the pressure-gradient correction proposed by Morris et al., the method is also suitable for solving the Navier-Stokes equations for incompressible flows without any further assumptions. Simulations using the method show close agreement with the analytic series solutions for plane Poiseuille and Hagen-Poiseuille flows at very low Reynolds numbers. At least for these specific tests, the results obtained are essentially independent of employing either a cubic or a quintic spline kernel