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Conservation of mass and momentum

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 written  begin{equation}frac{partial u}{partial x_i}=0end{equation}


Conservation of momentum of a viscous fluid flow with a gravitational acceleration in the z direction can be written as  begin{eqnarray}frac{partial u_1}{partial t}+ u_jfrac{partial u_1}{parti... ...} frac{partial^2 u_3}{partial x_i partial x_i}- frac{1}{Fr}end{eqnarray}


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.