Considerations on the transport-theorem for extensive flow properties

Two methods are presented for derivation of the transport theorem for volume and line properties, particularly the Reynolds transport and Thomson circulation theorems. The substantial time derivatives of the velocity field (including the density), the fluid volume, mass, momentum, kinetic and internal energies, enthalpy, and entropy, and a constant mass line that continually flows with the fluid are defined. The volume integral is then used to obtain the Reynolds transort theorem on a time differencing basis, allowing integration to be performed for a volume fixed in space. Alternatively, the volume can be treated as constant and the mass can be integrated over time within the bounds of the integration. For the Thomson theorem, the substantial time derivatives are volume properties determined from the line integral of the velocity of the circulation along a closed material line, or of the mass over a length of the flow.