Question

Consider the general form of the Reynolds transport theorem (RTT). Let Bₛᵧₛ be the angular momentum H = r ×m V of a system of fluid particles, where r is the moment arm. We know that for a system, conservation of angular momentum is ∑ M = d/dt Hₛᵧₛ where ∑ M is the net moment applied to the system. Use the RTT and the above equation to derive the equation of conservation of angular momentum for a control volume.

Answer 1

Applying the Reynolds transport theorem to the conservation of angular momentum for a control volume, one must consider the angular momentum H = r × mV and the sum of external torques ∑M. By equating ∑M to the time rate of change of angular momentum within the control volume and net outflow across the control surface, we derive the conservation equation for a fluid system.

Step-by-step explanation:

To derive the equation of conservation of angular momentum for a control volume using the Reynolds transport theorem (RTT), we start with the angular momentum H of a system of fluid particles which is given by the vector product H = r × mV, where r represents the moment arm and V the velocity of the particle with mass m. Conservation of angular momentum for a system dictates that the sum of the external torques ∑M is equal to the time rate of change of the system's angular momentum. This can be written as ∑M = d/dt Hsys.

When applying RTT to a control volume (CV), the conservation of angular momentum can be expressed as:

∑M = d/dt (∫CV ρ r × V dV) + ∫CS ρ r × V (V · n) dA,

where the first integral on the RHS represents the rate of change of angular momentum within the control volume, and the second integral represents the net outflow of angular momentum across the control surface (CS).

Making use of RTT allows us to analyze problems such as the conservation of angular momentum in systems extending beyond point particles and rigid bodies, like fluids with continuously distributed mass.