  Weakly-compressible particle-based discretization methods, utilized for the
solution of the subsonic Navier-Stokes equation, are gaining increasing
popularity in the fluid dynamics community. One of the most popular among these
methods is the weakly-compressible smoothed particle hydrodynamics (WCSPH).
Since the dynamics of a single numerical particle is determined by fluid
dynamic transport equations, the particle per definition should represent a
homogeneous fluid element. However, it can be easily argued that a single
particle behaves only pseudo-Lagrangian as it is affected by volume partition
errors and can hardly adapt its shape to the actual fluid flow. Therefore, we
will assume that the kernel support provides a better representative of an
actual fluid element. By means of non-equilibrium molecular dynamics (NEMD)
analysis, we derive isothermal transport equations for a kernel-based fluid
element. The main discovery of the NEMD analysis is a molecular stress tensor
which may serve to explain current problems encountered in applications of
weakly-compressible particle-based discretization methods.
