Semi-implicit operator splitting for the simulation of Herschel–Bulkley flows with smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) has become a popular numerical framework of choice for simulating free-surface flows, mainly for Newtonian fluids. The topic regarding the simulation of non-Newtonian free-surface flows, however, remains relatively untouched due to difficulties regarding the computation of viscous forces. In previous approaches, the viscous forces acting on each SPH particle were computed explicitly. Non-Newtonian fluids such as Herschel–Bulkley fluids, the effective viscosity between yielded and unyielded regions can differ by several orders of magnitudes; imposing severe time step restrictions for the simulation for explicit methods. Numerically, this can be seen as a stiff problem. We propose a semi-implicit time-stepping approach where the viscous forces are computed implicitly, within the context of SPH. We demonstrate the convergence of the method via a simple 2D test case.