A More Complete Thermodynamic Framework For Fluent Continua

Abstract

Polar decomposition of the changing velocity gradient tensor in a deforming fluent continua into pure stretch rates and rates of rotations shows that a location and its neighboring locations can experience different rates of rotations during evolution. Alternatively, we can also consider decomposition of the velocity gradient tensor into symmetric and skew symmetric tensors. The skew symmetric tensor is also a measure of pure rates of rotations whereas the symmetric tensor is a measure of strain rates. The measures of the internal rates of rotations due to deformation in the two approaches describe the same physics but in different forms. Polar decomposition gives the rate of rotation matrix and not the rates of rotation angles whereas the skew symmetric part of the velocity gradient tensor yields rates of rotation angles that are explicitly defined in terms of velocity gradients. These varying rates of rotations at neighboring locations arise due to varying deformation of the continua, hence are internal to the volume of matter and are explicitly defined by deformation. If the internal varying rates of rotations are resisted by the continua, then there must exist internal moments corresponding to these. The internal rates of rotations and the corresponding moments can result in additional rate of energy storage or rate of dissipation. This physics is all internal to the deforming continua and exists in all deforming isotropic, homogeneous fluent continua but is completely neglected in the presently used thermodynamic framework for fluent continua. In this paper we present derivation of a more complete thermodynamic framework in which the derivation of the conservation and balance laws consider additional physics due to varying rates of rotations. The currently used thermodynamic framework for fluent continua is a subset of the thermodynamic framework presented in this paper. The continuum theory presented here considers internal varying rates of rotations and the associated conjugate moments in the derivation of conservation and balance laws, thus the theory presented in this paper can be called “a polar continuum theory” but is different than micropolar continuum theories published currently in which material points have six external degrees of freedom i.e. the rotation rates are additional external degrees of freedom. In the remainder of the paper we refer to this new thermodynamic framework as ‘a polar continuum theory’. The continuum theory presented here only accounts for internal rotation rates and associated moments that exist as a consequence of deformation but are neglected in the present theories hence this theory results in a more complete thermodynamic framework. The polar continuum theory and the resulting thermodynamic framework presented in this paper is suitable for compressible as well as incompressible thermoviscous fluent continua such as Newtonian, Power law, Carreau-Yasuda fluids etc. and thermoviscoelastic fluent continua such as Maxwell, Oldroyd-B, Giesekus etc. The thermodynamic framework presented here is applicable to all isotropic, homogeneous fluent continua. Obviously the constitutive theories will vary depending upon the choice of physics. These are considered in subsequent papers.