Conservation of acceleration and dynamic entanglement in mechanics

Within a flow of a viscous fluid the effects of compression and viscosity are closely coupled with inertia. Discrete mechanics gives a physical sense of these interactions by showing that they exist only in a dynamic vision where the variation of the velocity in time generates the entanglement of the effects of compression and shearing. These two phenomena are described in the form of a Helmholtz–Hodge decomposition by two orthogonal terms within the discrete equation of motion, the first curl-free and the second divergence-free. They can only exchange mechanical energy if the acceleration is nonzero. This entanglement, which occurs at all spatial scales, is a function of longitudinal and transverse celerities. After a presentation of the formal framework, simple examples allow to understand the temporal shifts of direct and induced flows in accordance with the causality principle.