Similar derivations may be demonstrated for the y and z direction. This would make a good exercise to better understand this material. There are thus three different momentum equations that together comprise the Navier-Stokes Equations. The momentum fluxes into the control volume cancel with the first part of the momentum fluxes out of the control volume. We now turn our attention to the right hand side of the equation, where the forces on the control volume are represented.
Body forces act on the entire control volume.
The most common body force is that due to gravity. Electromagnetic phenomena may also create body forces, but this is a rather specialized situation. The stress on a surface of the control volume acts in the outward direction, and is given the symbol s ij , with two subscripts. The first subscript i indicates the normal direction of the face on which the stress acts, while the second subscript j indicates the direction of the stress. The force due to the stress is the product of the stress and the area over which it acts.