Laplace Form Of Navier-Stokes Equations: A Safe Path Or A Wrong Way?

Abstract

The Navier-Stokes Equations written in Laplace form are often the departure point for the
simulation of viscous newtonian flows and some studies of numerical stability. Researchers may not be
fully aware that the “physical traction boundary conditions” are not the “natural boundary conditions” of
the Laplace form of the Navier-Stokes Equations. This is not a problem per se, as long as one manages
to rigurously incorporate the physical boundary conditions into the formulation. However, we have
discovered that if some seemenly harmless assumptions are made, like using pseudo-tractions (i.e the
natural boundary conditions of the Laplace form) or neglecting viscous terms on the free-surfaces, the
resulting formulation violates a basic axiom of continuum mechanics: the principle of objectivity. In the
present article we give an accurate account about these topics. We also show that unexpected differences
may sometimes arise between Laplace discretizations and Divergence discretizations.