You have an open channel, so, at the top of the fluid layer, the pressure is atmospheric pressure. At depth z, the gauge pressure is γw z. If you integrate this gauge pressure variation over section A, you get the first term on the right hand side of the equation. Atmospheric pressure contributes on all the surfaces of the control volume, so it cancels out. The hydrostatic pressure force at section 2 is negligible, because the pressures on both sides of the layer are atmospheric, and the gauge pressures are zero.Feodalherren said:
The conservation of momentum equation in fluid dynamics is a fundamental principle that states that the total momentum of a closed system remains constant over time, unless acted upon by external forces. It is represented by the equation: ρu + ρ(v+w) = C, where ρ is the density of the fluid, u, v, and w are the velocities in the x, y, and z directions respectively, and C is a constant.
The conservation of momentum is important in fluid dynamics because it allows us to predict and analyze the motion of fluids in a variety of scenarios. It is a fundamental principle that governs the behavior of fluids and is used in many applications, such as designing airplanes, ships, and other vehicles that move through fluids.
The conservation of momentum equation and the Bernoulli's equation are both important principles in fluid dynamics, but they are used for different purposes. The conservation of momentum equation applies to a closed system, where the total momentum remains constant. On the other hand, Bernoulli's equation applies to an open system, where the fluid is flowing and there is a change in pressure due to changes in velocity.
The conservation of momentum equation is closely related to Newton's laws of motion. In fact, it can be derived from Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In fluid dynamics, this translates to the change in momentum of a fluid being equal to the net force acting on it.
Yes, the conservation of momentum equation can be applied to both liquids and gases. This is because both liquids and gases are considered fluids and they both follow the same principles of fluid dynamics. However, the equation may need to be modified depending on the properties of the fluid, such as density and viscosity.