Remarks on the Lie derivative in fluid mechanics

It is known that the invariance properties derived from Noether’s theorem or associated with the Lie derivative can be adapted to fluid mechanics. The associated invariance groups are easily analyzed in a four-dimensional reference space representing the Lagrangian variables. A calculation method uses the Lie derivative associated with the velocity quadrivector in space–time. An interpretation of this derivative connects the space–time and the reference space; it allows to analyze the notion of tensor moving with the fluid and to find conservation laws and many invariance theorems in fluid mechanics. •Invariance properties associated with the Lie derivative in fluid mechanics.•Tensors moving with the fluid in conservative motions.•Conservation laws and invariance theorems in fluid mechanics.