Analysis on the Invariant Properties of Constitutive Equations of Hydrodynamics in the Transformation between Different Reference Systems

TL;DR

This paper investigates the invariant properties of constitutive equations in hydrodynamics under reference system transformations, proving invariance of divergence, stress, and heat flux density, and establishing the tensor nature of stress.

Contribution

It demonstrates that certain hydrodynamic quantities are invariant or variant under reference transformations and clarifies the tensorial nature of stress in different reference systems.

Findings

Divergence of velocity is invariant under reference transformations.

Stress tensor components transform as a second-order tensor.

Invariance of the divergence of velocity, stress, and heat flux density.

Abstract

The velocities of the same fluid particle observed in two different reference systems are two different quantities and they are not equal when the two reference systems have translational and rotational movements relative to each other. Thus, the velocity is variant. But, we prove that the divergences of the two different velocities are always equal, which implies that the divergence of velocity is invariant. Additionally, the strain rate tensor and the gradient of temperature are invariant but, the vorticity and gradient of velocity are variant. Only the invariant quantities are employed to construct the constitutive equations used to calculate the stress tensor and heat flux density, which are objective quantities and thus independent of the reference system. Consequently, the forms of constitutive equations keep unchanged when the corresponding governing equations are transformed between different reference systems. Additionally, we prove that the stress is a second-order tensor since its components in different reference systems satisfy the transformation relationship.