In the following the non-dimensional version of the
incompressible Navier-Stokes equations under influence of
a gravitational force will be presented.
Conservation of mass of an incompressible fluid flow can be
Conservation of momentum of a viscous fluid flow with
a gravitational acceleration in the z direction can
be written as
Re is the Reynolds number, and Fr the Froude number, .
By introducing a new variable for the pressure, which includes the both the
static and the hydrostatic component of the pressure,
the equation of momentum, Eq.(2), can be written as
The Freud number will now enter explicity in the boundary condition
instead of the governing equations.