Reynolds Transport Theorem

Consider a fluid in motion. The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property in a system and its change within a control volume.

Extensive properties such as mass, momentum, and energy depend on the total amount of the fluid under consideration.

In contrast, intensive properties like velocity, temperature, and acceleration represent these quantities per unit mass.

The total amount of extensive property in a system is the sum of the amounts associated with each infinitesimal fluid particle's mass, expressed as the integral. Its time rate of change is determined by differentiating the integral over the system volume.

Within a control volume, it is determined by the time derivative of the integral over the control volume.

The Reynolds transport theorem combines the system and control-volume perspectives, relating the rate of change of an extensive property to the sum of its accumulation within the control volume and the net flux across the control surface.

When the control volume and the system overlap, extensive property values may differ due to flux across the control surface or relative motion of boundaries.