Construction of a peridynamic model for viscous flow

Abstract

We derive the Eulerian formulation for a peridynamic (PD) model of Newtonian viscous flow starting from fundamental principles: conservation of mass and momentum. This formulation is nonlocal, different from viscous flow models that utilize numerical methods like, e.g., the so-called “peridynamic differential operator” to approximate solutions of the classical Navier-Stokes equations. We show that the classical continuity equation is a limiting case of the PD one, assuming certain smoothness conditions. The PD model for viscous flow is calibrated by enforcing linear consistency for the viscous stress term with the classical Navier-Stokes equations. Couette and Poiseuille flows, and incompressible fluid flow past a regular lattice of cylinders are used to verify the new formulation, at low Reynolds numbers. The constructive approach in deriving the model allows for a seamless coupling with peridynamic models for corrosion or fracture for simulating complex fluid-structure interaction problems in which solid degradation takes place, such as in erosion-corrosion, hydraulic fracture, etc. Moreover, the new formulation sheds light on the relationships between local and nonlocal models.