Momentum
In basic physics, momentum is the mass of a body times its velocity and is a vector quantity, whereas energy is scalar, lacking a direction. In hydraulics, the use of this concept is due mainly to the implication of Newton’s second law, that the resultant of all forces acting on a body causes a change in momentum. The momentum equation in hydraulics is similar in form to the energy equation and, when applied to many flow problems, can provide nearly identical results. However, knowledge of fundamental differences in the two concepts is critical to modeling certain hydraulic problems. Conceptually, the momentum approach should be thought of as involving forces on a mass of flowing water, instead of the energy state at a particular location. Friction losses in momentum relate to the force resistance met by that mass with its boundary, whereas in the energy concept, losses are due to internalenergy dissipation (Chow 1959).
The momentum equation can have advantages in modeling flow over weirs, drops, hydraulic jumps, and junctions, where the predominate friction losses are due to external forces, rather than internal energy dissipation.
Interpreted for open channel, Newton’s second law states that the rate of momentum change in this short section of channel equals the sum of the momentum of flow entering and leaving the section and the sum of the forces acting on the water in the section. Since momentum is mass times velocity, the rate of change of momentum is the mass rate of change times the velocity. The momentum equation may be written considering a small mass or slug of flowing water between two sections 1 and 2 and the principle of conservation of momentum.
The left side of the equation is the momentum entering and leaving, and the right side is the pressure force at each end of the mass, with Wsinθ being the weight of the mass, θ being the angle of the bottom slope of the channel, and Ffr being the resistance force of friction on the bed and banks.