  Discrete mechanics is presented as an alternative to the equations of fluid
mechanics, in particular to the Navier-Stokes equation. The derivation of the
discrete equation of motion is built from the intuitions of Galileo, the
principles of Galilean equivalence and relativity. Other more recent concepts
such as the equivalence between mass and energy and the Helmholtz-Hodge
decomposition complete the formal framework used to write a fundamental law of
motion such as the conservation of accelerations, the intrinsic acceleration of
the material medium, and the sum of the accelerations applied to it. The two
scalar and vector potentials of the acceleration resulting from the
decomposition into two contributions, to curl-free and to divergence-free,
represent the energies per unit of mass of compression and shear.
  The solutions obtained by the incompressible Navier-Stokes equation and the
discrete equation of motion are the same, with constant physical properties.
This new formulation of the equation of motion makes it possible to
significantly modify the treatment of surface discontinuities, thanks to the
intrinsic properties established from the outset for a discrete geometrical
description directly linked to the decomposition of acceleration. The treatment
of the jump conditions of density, viscosity and capillary pressure is
explained in order to understand the two-phase flows. The choice of the
examples retained, mainly of the exact solutions of the continuous equations,
serves to show that the treatment of the conditions of jumps does not affect
the precision of the method of resolution.
