The derivation of reynolds transport equation for arbitrary motion control volume based on the dynamic boundary calculus

Strictly speaking, all conservation laws in hydrodynamics are for a material system (or a fluid system), such as the conservation laws of mass, momentum, momentum moment and energy. The Lagrange method is used to describe and characterize the motion behavior of the fluid particle system. If the motion and the conservation laws of the material system are put into the space coordinate system, the Euler method is often used. Therefore, for the observed (followed) fluid material system, each conservation law makes a transformation from the material system to the control volume, which is the famous Reynolds transport equation. In this paper, the Reynolds transport equations for different velocity control volumes are derived based on the boundary calculus. The physical significance of various transport equations is also discussed.