Material Coefficients in Non-Classical Viscous Fluids

The paper begins by highlighting the limitations of classical continuum theories for fluent continua. These theories primarily utilize velocity (v) and the symmetric part of the velocity gradient tensor (D) to describe deformation, neglecting the antisymmetric part (W) which represents internal rotation rates. This omission, the authors argue, leads to an incomplete thermodynamic framework. They introduce the concept of a non-classical continuum theory that explicitly incorporates internal rotation rates arising from the full velocity gradient tensor (L). The paper positions itself within the context of previous research on couple stress theories and micro-theories, differentiating itself by its foundational approach and rationale. The goal is to develop a thermodynamically consistent framework for thermoviscous compressible fluids without memory, incorporating both D and W to describe the complete deformation physics. The paper promises to establish restrictions on material coefficients in derived constitutive relations using the entropy inequality and representation theorem, acknowledging the inherently nonlinear nature of these constitutive theories.

This section delves into the kinematics of deformation within the context of non-classical continuum theory. It emphasizes the crucial role of both the velocity (v) and the velocity gradient tensor (L) in defining deformation physics. The decomposition of L into symmetric (D) and antisymmetric (W) parts is explored, with D representing strain rates and W representing pure rotation rates. Alternatively, the polar decomposition of L is introduced, providing different yet equivalent descriptions of strain rates (tSr, tSl) and rotation rates (tR). The paper explicitly defines internal rotation rates as those arising from the antisymmetric part of L, highlighting their inherent presence in all deforming isotropic and homogeneous fluent continua. The inclusion of the full L in the thermodynamic framework is justified by incorporating the additional physics introduced by these internal rotation rates, which are absent in classical theories. The use of internal rotation rates and their gradients forms the foundation of the proposed non-classical continuum theory.

This section addresses the implications of including internal rotation rates on the standard conservation and balance laws used in classical continuum mechanics. It raises the fundamental question: Are existing conservation laws (conservation of mass, balance of linear and angular momenta, the first and second laws of thermodynamics) sufficient to ensure equilibrium in a non-classical continuum with internal rotation rates? The authors refer to the work of Yang et al. [31] and Surana et al. [29, 30], which suggests the necessity of an additional balance law—the balance of moments of moments—to ensure equilibrium. This arises due to the presence of a Cauchy moment tensor independent of forces. The introduction of this new balance law, along with the existing laws, is crucial for maintaining equilibrium. This section lays the foundation for subsequent derivations of constitutive equations, emphasizing the importance of a rigorous approach to ensure thermodynamic consistency, especially when dealing with the novel physics introduced by internal rotation rates within the framework of a non-classical continuum theory.

This section discusses the appropriate measures of stress and moment in the context of a deforming tetrahedron. The authors explore the use of contravariant and covariant bases to define the Cauchy stress tensor and Cauchy moment tensor. They highlight the physical significance of using a contravariant measure for the stress tensor and explain why this approach is most natural. The choice of basis for the Cauchy moment tensor is also discussed, noting its relevance to the presence of moments on the faces of the deformed tetrahedron. This section emphasizes the importance of correctly defining these tensors to accurately capture the physics of the deforming material. Different choices of stress and moment measures are discussed, with a focus on how these choices influence the derivation of the constitutive theories and the subsequent restrictions placed on the material coefficients. The introduction of the Cauchy moment tensor is directly linked to the presence of internal rotation rates and the need for a complete description of the internal moments within the material.

The core of this section focuses on the derivation of constitutive theories for non-classical viscous fluent continua. The methodology relies heavily on the representation theorem (or theory of generators and invariants), coupled with the concept of integrity basis. This ensures that the derived constitutive relationships are complete and account for all possible tensorial contributions. The authors emphasize that this approach inevitably leads to nonlinear constitutive theories, a crucial distinction from the often-used simplified linear models. The constitutive theories are developed for the deviatoric part of the symmetric Cauchy stress tensor, the Cauchy moment tensor (considering cases where the balance of moments of moments is and isn't a balance law), and the heat vector. The representation theorem helps systematically construct these theories from a complete set of invariants formed from the argument tensors, ensuring completeness and avoiding omissions. This section is fundamental in establishing the mathematical framework for the subsequent analysis of material coefficients.

This section details how restrictions on the material coefficients within the derived constitutive theories are established. The key constraint used is the entropy inequality, a fundamental principle in thermodynamics. The authors rigorously apply this inequality to ensure the thermodynamic consistency of the developed models. Satisfying the entropy inequality requires that certain dissipation functions, representing the rate of work, must remain positive for all admissible choices of kinematic variables. This condition necessitates specific restrictions on the material coefficients. The paper emphasizes that examining the positivity of the dissipation function is the primary means of establishing these restrictions. In cases where the dissipation function cannot be guaranteed to be unconditionally positive, some material coefficients may need to be set to zero to ensure thermodynamic consistency. The approach is directly applicable to the nonlinear constitutive theories derived earlier, highlighting their inherent complexity and the need for rigorous thermodynamic analysis.

This section compares the derived non-classical constitutive theories with their classical counterparts. Specifically, it analyzes a simplified linear constitutive theory for the deviatoric Cauchy stress tensor commonly used for compressible viscous fluids. The analysis directly addresses Stokes' hypothesis (or assumption), which posits a relationship between two material coefficients (µ and λ). The paper demonstrates that Stokes' hypothesis lacks a thermodynamic basis; it's not derived from the entropy inequality. Alternative approaches aiming to refine or replace Stokes' hypothesis are also criticized for lacking thermodynamic justification. The authors contrast this with their approach, emphasizing the importance of establishing restrictions on material coefficients strictly through the application of the entropy inequality to ensure thermodynamic consistency. The findings directly challenge widely accepted assumptions in classical fluid mechanics.

This section directly challenges the widely accepted Stokes' hypothesis (or assumption), which states that 3λ + 2µ = 0 for nearly incompressible fluids. The paper argues that this hypothesis, used extensively in fluid mechanics, lacks a thermodynamic basis. The authors demonstrate that Stokes' hypothesis cannot be derived from fundamental thermodynamic principles, specifically the entropy inequality. The analysis focuses on the simplest linear constitutive theory for the deviatoric Cauchy stress tensor in classical continuum mechanics for compressible viscous fluids. In this context, the material coefficients µ and λ are shown to be independent, meaning that no thermodynamic constraint necessitates a relationship between them. The paper, therefore, concludes that Stokes' hypothesis is not thermodynamically justified and should be viewed as incorrect, emphasizing the need for a strictly thermodynamically grounded approach to deriving relationships between material coefficients.

The paper examines alternative proposals attempting to address the limitations of Stokes' hypothesis, such as the suggestion that 3λ + 2µ > 0. However, it critiques these alternatives, demonstrating that they too lack a robust thermodynamic basis. The authors point out that these alternative conclusions are often reached using methods that do not adhere strictly to the constraints imposed by the entropy inequality. One approach criticized is expressing the rate of deformation tensor (D) in terms of the deviatoric Cauchy stress tensor to infer relationships between material coefficients; this method is deemed invalid as it lacks a thermodynamic foundation. The paper emphasizes that for thermodynamically consistent results, restrictions on material coefficients must be directly derived from the conditions resulting from the entropy inequality. The critique of these alternative proposals reinforces the need for a rigorous, thermodynamically sound approach to defining restrictions on material coefficients in constitutive theories.

This section presents the paper's conclusions regarding thermodynamically consistent restrictions on the material coefficients µ and λ in the linear constitutive theory for classical compressible viscous fluids. Applying the entropy inequality directly and correctly to the linear constitutive theory, the authors demonstrate that the only valid constraints are µ > 0 and λ > 0. This conclusion holds for both nonlinear and classical constitutive theories, emphasizing the generality of their approach. The paper concludes that µ and λ are independent material coefficients, a fundamental departure from the assumptions embedded in Stokes' hypothesis. The rigorous application of the entropy inequality provides a pathway to determine thermodynamically consistent restrictions on the material coefficients which are not possible with alternative, less rigorous approaches. The paper concludes by asserting that only a thermodynamically consistent approach to deriving constitutive theories can correctly define the restrictions on material coefficients.

The paper emphasizes the importance of thermodynamically consistent modeling in continuum mechanics, particularly for viscous fluids. The authors highlight that their approach, based on rigorous application of the entropy inequality, provides a more accurate and reliable framework for understanding fluid behavior than previous methods. This is particularly relevant when dealing with nonlinear constitutive theories, which are shown to be unavoidable when incorporating the complete physics of deformation via the representation theorem. The approach is contrasted with methods that rely on assumptions or simplifications not grounded in thermodynamics, leading to potentially inaccurate conclusions about material coefficients and fluid behavior. The emphasis on thermodynamic consistency ensures the resulting models accurately reflect physical reality and avoid spurious results. The non-classical continuum theory presented offers a substantial advancement over classical models, particularly in accurately predicting the behavior of complex fluids.

The key contribution of this research is the development of a thermodynamically consistent framework for modeling non-classical viscous fluids, which explicitly includes the effects of internal rotation rates. This contrasts sharply with classical approaches. The paper successfully derives constitutive theories for the Cauchy stress tensor, Cauchy moment tensor, and heat vector, utilizing the representation theorem and integrity basis. Crucially, the research refutes Stokes' hypothesis, a widely used assumption in fluid mechanics, demonstrating that it lacks a thermodynamic foundation. The correct restrictions on the independent material coefficients (µ and λ) are established through rigorous application of the entropy inequality, highlighting the independence of µ and λ. The findings provide a more accurate and reliable approach to analyzing fluid behavior, particularly in situations where internal rotation rates play a significant role. This is particularly valuable for advanced fluid mechanics and engineering applications.

The conclusion emphasizes the thermodynamic consistency and rigor of the presented approach to deriving constitutive theories and establishing restrictions on material coefficients for both non-classical and classical thermoviscous compressible and incompressible fluent continua. The paper's refutation of Stokes' hypothesis is highlighted as a significant contribution, challenging a widely held assumption in fluid mechanics. The established restrictions on material coefficients for both classical (µ > 0 and λ > 0) and non-classical theories (all coefficients must be greater than zero to ensure positive dissipation functions) are presented as key results. The research strongly advocates for a thermodynamically consistent approach to modeling viscous fluids, offering a refined framework for future studies in fluid mechanics and related engineering disciplines. The work paves the way for further research into more complex scenarios and provides a solid theoretical foundation for advanced applications.