3. Moment of Momentum Equation

Now, similarly, there is something called moment of momentum equation, we are not going into detail, but just to show that here the B, capital B for deriving moment of momentum equation we can assume, m r into V moment of momentum or m gamma into V, this r is actually nothing but gamma and b will be m r into V / m, so, moment of momentum per unit mass.

The moment of momentum equation is a principle derived from the conservation of momentum, specifically applied to rotating systems or fluids. The equation takes into account the momentum (mass x velocity) and its relation to a point of rotation or an axis. In the equation, 'B' represents the total moment of momentum, which is calculated by multiplying mass (m) by the radius (r) from the axis of rotation and the velocity (V). Meanwhile, 'b' represents the moment of momentum per unit mass, essentially normalizing the value by dividing it by the mass. This formulation helps in analyzing fluid flows as they interact with different geometries, such as turning flows around bends or changes in channel geometry.

Consider a figure skater who pulls their arms inward while spinning. As they do this, they spin faster due to the conservation of angular momentum. In this scenario, the skater's mass, its distance from the rotation center, and their spinning speed act analogously to the moment of momentum equation. The key takeaway is that as the skater reduces their radius (similar to reducing 'r' in the equation), their rotational speed increases, illustrating how momentum must conserve in a rotating system.

This is going to be 0, we are not going into detail therefore this is. So, this is for steady state because this is delta t equals 0, we are not going to do the analysis.

The moment of momentum equation becomes particularly relevant under steady-state conditions, where there is no change over time (Δt = 0). In this context, the forces and movements are balanced, and the changes in momentum will sum to zero. Essentially, this means that whatever momentum enters a system must equal the momentum that exits, supporting the idea of conservation of momentum. By focusing on steady states, we can simplify complex fluid dynamics problems and effectively analyze systems without considering transient behaviors.

Think of a busy highway. When cars enter and exit, the total number of cars (momentum) remains relatively constant over time if the traffic flow is steady. If 10 cars enter the highway during a specific minute, then out, in a steady-state situation, 10 cars must exit the highway during that same minute. This analogy helps to visualize how momentum works within a defined system, maintaining balance as cars flow in and out, just like how momentum is conserved in fluid systems.

--