Non-classical continuum theory for fluids incorporating internal and Cosserat rotation rates

This paper presents a non-classical continuum theory for fluent continua in which the conservation and balance laws are derived by incorporating both internal rotation rates arising from the velocity gradient tensor and the rotation rates of the Cosserats. Specifically, in this non-classical continuum theory we have (1) the usual velocities (\(\bar{ \pmb {\varvec{v }}}\)), (2) the three internal rotation rates (\({}_i^t\bar{ \pmb {\varvec{\Theta }}}\)) about the axes of a fixed triad whose axes are parallel to the x-frame arising from the velocity gradient tensor \((\bar{ \pmb {\varvec{L }}})\) that are completely defined by the antisymmetric part of the velocity gradient tensor, and (3) three additional rotation rates (\({}_e^t\bar{ \pmb {\varvec{\Theta }}}\)) about the axes of the same triad located at each material point as additional three unknown degrees of freedom, referred to as Cosserat rotation rates. This gives rise to \(\bar{ \pmb {\varvec{v }}}\) and \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) as six degrees of freedom at a material point. The internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\), often neglected in classical fluid mechanics, exist in all deforming fluent continua as these are due to velocity gradient tensor. When the internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) are resisted by deforming fluent continua, conjugate moment tensor arises that together with \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) may result in energy storage and/or dissipation, which must be considered in the conservation and balance laws. The Cosserat rotation rations \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) also result in conjugate moment tensor that together with \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) may also result in energy storage and/or dissipation. The main focus of this paper is a consistent derivation of conservation and balance laws for fluent continua that incorporate the aforementioned physics and associated constitutive theories for thermofluids using the conditions resulting from the entropy inequality. The material coefficients derived in the constitutive theories are clearly defined and discussed.