Rotational Inertial Physics in Non-Classical Thermoviscous Fluent Continua Incorporating Internal Rotation Rates

In this paper, we derive non-classical continuum theory for physics of compressible and incompressible thermoviscous non-classical fluent continua using the conservation and balance laws (CBL) by incorporating additional physics of internal rotation rates arising from the velocity gradient tensor as well as their time varying rates and the rotational inertial effects. In this non-classical continuum theory time dependent deformation of fluent continua results in time varying rotation rates i.e., angular velocities and angular accelerations at material points. Resistance offered to these by deforming fluent continua results in additional moments, angular momenta and inertial effects due to rotation rates i.e., angular velocities and angular accelerations at the material points. Currently, this physics due to internal rotation rates and inertial effects is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM). In this paper, we present a derivation of conservation and balance laws in Eulerian description: conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), balance of moment of moments (BMM), first and second laws of thermodynamics (FLT, SLT) that include: (i) Physics of internal rotation rates resulting from the velocity gradient tensor; (ii) New physics resulting due to angular velocities and angular accelerations due to spatially varying and time dependent rotation rates. The balance laws derived here are compared with those that only consider the rotational rates but neglect rotational inertial effects and angular of the conservation and balance laws and constitutive theories presented in this paper has closure. Influence of new physics in the conservation and balance laws on compressible and incompressible thermoviscous fluent continua is demonstrated due to presence of angular velocities and angular accelerations arising from time varying rotation rates when the deforming fluent continua offer rotational inertial resistance. The fluent continua are considered homogeneous and isotropic. Model problem studies are considered in a fol-low-up paper.

material points. Resistance offered to these by deforming fluent continua results in additional moments, angular momenta and inertial effects due to rotation rates i.e., angular velocities and angular accelerations at the material points. Currently, this physics due to internal rotation rates and inertial effects is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM). In this paper, we present a derivation of conservation and balance laws in Eulerian description: conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), balance of moment of moments (BMM), first and second laws of thermodynamics (FLT, SLT) that include: (i) Physics of internal rotation rates resulting from the velocity gradient tensor; (ii) New physics resulting due to angular velocities and angular accelerations due to spatially varying and time dependent rotation rates. The balance laws derived here are compared with those that only consider the rotational rates but neglect rotational inertial effects and angular accelerations to demonstrate the influence of the new physics. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and principle of equipresence when appropriate. Constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity (complete basis). It is shown that the mathematical model consisting

Introduction, Literature Review and Scope of Work
In the spatial or Eulerian mathematical description of deforming continua such as fluent continua [1] [2] [3] [4] the velocities v and the velocity gradient tensor L are fundamental measures of deformation physics. In general v and L vary between material points. Polar decomposition of L at material points into stretch rate tensor (left or right) and rotation rate tensor shows that if L varies between material points so do the stretch rate and rotation rate tensors.
Alternatively, we can additively decompose L at a material point into symmetric ( D ) and skew-symmetric (W ) tensors in which the symmetric tensor is the first convective time derivative of the Green strain tensor as well as the first convected time derivative of the Almansi strain tensor which are shown to be basis independent [4]. The skew-symmetric tensor is a measure of pure rotation rate, referred to as internal rotation rate tensor or angular velocity tensor. In classical continuum mechanics when considering thermoviscous fluent continua, Cauchy stress tensor σ is a rate of work conjugate to D and σ is basis independent when only conjugate to D . In the constitutive theory for Cauchy stress tensor we can also consider higher order convected time derivatives of the strain tensors in which case Cauchy stress tensor is basis dependent i.e., contravariant Cauchy stress tensor ( ) 0 σ or covariant Cauchy stress tensor ( ) 0 σ . In CCM, influence of time varying rotation rates at each material point due to L is not considered. Surana et al. [2] [3] [5] have presented conservation and balance laws for non-classical continuum theory in which additional physics due to time varying rotation rates is incorporated into the conservation and balance laws. Thus, this non-classical continuum theory incorporates L in its entirety in the conservation and balance laws. In subsequent papers, yang et al. [6] and Surana et al. [1] [7] showed that the presence of new physics due to time varying rotation rates requires additional balance law "balance of moment of moments" (BMM). This balance law was originally proposed by Yang et al. [ [1] [7] showed that the derivation of a balance law must be based on rates and presented derivation of BMM balance law for solid continua [7] as well as fluent continua [1].
Ordered rate constitutive theory for thermoviscous fluent continua incorporating internal rotation rates has been presented by Surana et al. [5]. Prior [8].
In a recent paper, Surana et al. [9] presented a comprehensive literature re- can be found in references [10]- [29]. The micropolar theories consider microdeformation of micro-constituents in the continuum and associated homogenization so that the matter at macro scale is isotropic and homogeneous. The theories related to the non-local effects are believed to be originated by Eringen [30] in which a definition of non-local stress tensor is introduced through an integral relationship using the product of macroscopic stress tensor and a distance kernel representing non-local effects. The works by Eringen [18] [19] [20] [21] [22] establish conservation and balance laws, constitutive theories, micromechanics considerations and their use in non-classical theories for fluent continua. Some stability and boundary considerations for non-classical theories are discussed in references [23] [24]. In reference [25] authors present a discussion on a collection of papers related to the macro-micro mechanics' aspects of deformation physics. In reference [26] a micropolar theory is presented for binary media with applications to phase transition of fiber suspensions to show flow during the filling state of injection molding of short fiber reinforced thermoplastics. A similarity solution for boundary problem flow of a polar fluid is given in reference [27]. In references [28] [29] phenomenological theory of ferrofluids and statistical mechanical theory of polar fluids are presented.

Notations
The notations used in this paper conform to reference [4]

Internal Rotation Rates and Their Gradients
The velocities v and the velocity gradients ( Furthermore, using and following a similar procedure we can establish where t R     defined by (4) and (7) is unique. We note that in this approach t R     is a rotation rate transformation matrix, hence does not contain rotation angle rates. Alternatively, we can consider decomposition of L     into symmetric ( D     ) and skew-symmetric ( W     ) tensors.
We define positive rotation rates t i Θ using or ( ) ( ) ( ) We note that where i.e., related to half of the rate of change of 90 degree angle. It is obvious that W is a tensor of rank two, whereas the rotation rates defined in (13) are clearly a tensor of rank one. In other words, rotation rates in (13) constitute a tensor of rank one, but the components of this tensor arranged in the form in which they appear in W     constitute a tensor of rank two. We determine gradients of the rotation rate tensor (13). Let be a vector representation of (13), then the gradient of t i Θ can be defined by The gradient tensor when the velocity gradient tensor varies between the neighboring material points so do the internal rotation rates t i Θ (or W     ), their rates as well as their gradients and their rates. Varying t i Θ and t i Θ J , when resisted by deforming fluent continua, results in moments, angular momenta and angular inertial effects as a consequence. Thus, on the oblique plane of the tetrahedron defining part of ( ) or defining a part of the bounding surface due to cut principle of cauchy, resultant moment can exist.

Stress, Moment and Strain Rate Tensors
Consider a volume of matter V  T whose edges (under finite deformation) are non-orthogonal covariant base vectors i g  . The planes of the tetrahedron formed by the covariant base vectors are flat but obviously non-orthogonal to each other. We assume the tetrahedron to be the small neighborhood of material point o so that the assumption of the oblique plane ABC being flat but still part of V ∂ is valid. When the deformed tetrahedron is isolated from volume V it must be in equilibrium under the action of disturbance on surface ABC from the volume surrounding V and the internal fields that act on the flat faces which equilibrium with the mating faces in volume V when the tetrahedron 1 T is 4 the volume V . ; Columns of J are covariant base vectors i g  that form non-orthogonal covariant basis. Contravariant base vectors of j g  are normal to the faces of the tetrahedron formed by the covariant base vectors The rows of J are contravariant base vectors j g  . These form a non-orthogonal contravariant basis. Covariant and contravariant bases are reciprocal to each other [4].  (25) or We define the contravariant and covariant Cauchy moment tensor in similar fashion and the corresponding Cauchy principle and and are all nonsymmetric tensors of rank two. Thus, we note that the Cauchy stress tensors and the Cauchy moment tensors are basis dependent. It has been shown that [4] for finite strain rates the contravariant measures are meritorious. However, in deriving conservation and balance laws and the constitutive theories either measure yields a covariant mathematical model. We introduce stress measure ( )

Conservation and Balance Laws
In the following we present conservation and balance laws in Eulerian descrip-tion for non-classical fluent continua incorporating internal rotation rates and their spatial and temporal gradients. The fluent continua is assumed homogeneous and isotropic.

Conservation of Mass: CM
The continuity equation resulting from the principle of conservation of mass remains the same in the non-classical continuum theory considered here as in case of classical continuum mechanics. The differential form of the continuity equation in Eulerian description for compressible matter is given by

Balance of Linear Momenta: BLM
For a deforming volume of matter, the rate of change of linear momentum must be equal to the sum of all other forces acting on it. This is Newton's second law applied to a volume of matter. The derivation of the balance laws is exactly same as in case of CCM [4] and we can write the following (Using ( ) 0 σ as Cauchy stress measure) in Eulerian description. (38) in which b F is body force per unit mass.

Balance of Angular Momenta: BAM
The principle of balance of angular momenta for non-classical continuum me- . Then, according to this balance law: We consider each term of (39) and using Divergence Theorem we note the following using (46) in (45) we can write substituting from (41), (43), (47) and (48) in (39) and rearranging terms, we ob- in the second term in (49) is zero due to balance of linear momenta, hence (49) reduces to For isotropic homogeneous matter V is arbitrary hence we can obtain diffe- Remarks 1. If we set the first and the last term in (51) to zero, then we recover balance of angular momenta for classical continuum mechanics in Eulerian description.
2. If we set the first term in (51) to zero but retain second and third order terms, then we have balance of angular momenta for NCCM incorporating internal rotation rates without the rotational inertial physics.
3. Appearance of the first term in (51) is due to consideration of time varying rotation rates and rotational inertia I θ . This is new physics considered in the present work that neither appears in CCM nor NCCM published works. (51) is the final form of balance of angular momenta.

Balance of Moment of Moments: BMM
This is a new balance law originally proposed by Yang et al. [6] for NCCM. This balance law was derived based on static considerations (hence cannot be referred to as a balance law). Later, Surana et al. explained the rationale for this balance law and pointed out that a balance law must be derived using rate considerations. In references [1] [7] [33] they presented derivation of the "balance of moment of moments" balance law for NCCM for fluent and solid continua in the presence of internal rotation rates and internal rotations. In the work presented in this paper, the physics considered is different than in reference [1], hence a rederivation of this balance law is necessary. According to this balance law the rate of change of moment of angular momenta due to rotation rates in a deformed volume V must be equal to the sum of the moment of moments due to the antisymmetric components of the Cauchy stress tensor over the same deformed volume V and the moment of M acting on boundary V substituting from (54) and (57) using balance of angular momenta (51) in (60), we obtain For homogeneous, isotropic continua, V is arbitrary, hence we obtain the following from (61)

First Law of Thermodynamics: FLT
The sum of work and heat added to a volume of matter must result in increase of the energy of the volume. This can be expressed as a rate equation in Eulerian description.
where t E , Q and W are total energy, heat added and work done. Their rates can be written as ( ) where e is specific internal energy, b F are body forces per unit mass and g is heat vector. The second term in the integrand is due to additional rate of work due to rotation rates. We expand integrals in (65)-(67). Following reference [4] we can show ( ) ( ) using Divergence Theorem (66) can be written as using Cauchy principle for P and M we can show that ( ) following reference [4], we can show substituting from (72) and (73) Substituting from (65), (66) and (74) in (64) Using balance of linear momenta (37) in (75) and grouping last two terms in the integrand we obtain (noting that t For isotropic, homogenous continua, V is arbitrary, hence we can set the integrand in (76) to zero. : in which ( ) 0 τ is a vector, containing three components, and noting that using (80) and (81) We can show that This is the final form of the energy equation resulting from the first law of thermodynamics.

Second Law of Thermodynamics: SLT
If η is the entropy density in the volume V , h is the entropy flux between V and the volume of matter surrounding it and s is the source of entropy in V due to non contacting sources (bodies), then the rate of increase of entropy in volume V is at least equal to that applied to V from all contacting and non-contacting sources [4]. Thus For homogeneous. isotropic matter volume V is arbitrary hence we can write the following from (95) Equation (96) is the most fundamental form of the SLT or entropy inequality (Clausius Duhem inequality). We note that entropy inequality is strictly a statement that contains entropy terms, hence contains no information regarding reversible deformation physics. In this form (96) the entropy inequality provides no mechanism(s) for deriving constitutive theories. Only when the mechanical rate of work that results in rate of entropy production is introduced in the entropy inequality, will the entropy inequality contain information regarding conjugate pairs resulting in rate of entropy production. We also note entropy inequality (96) does not provide any information regarding constitutive theory for heat vector q . In the following we derive another form of the entropy inequality using a relationship between ψ and q and relationship between Φ , e and η . Since the energy equation has all possible mechanisms that result in energy storage and dissipation, the form of entropy inequality derived using energy equation is expected to be helpful in the derivation of the constitutive theories. Using where θ is absolute temperature and r is a suitable potential , , , substituting from (98) into (96) and multiplying through by θ ( ) Equation (101) is the final form of the entropy inequality resulting from the second law of thermodynamics.

Complete Mathematical Model Resulting from CBL of NCCM
The system of partial differential equations and algebraic equations resulting from the conservation and balance laws of NCCM incorporating internal rotation rates and their material derivatives and rotational inertial effects are given  3. From entropy inequality we can conclude the following.
(a) From the term θ ⋅ q g , we conclude that , q g is a conjugate pair. 5. In this paper, we consider compressible as well as incompressible thermoviscous fluent continua. 6. In a recent paper Surana et al. [9] presented non-classical continuum theory for thermoelastic solid continua (small deformation, small strain physics for homogeneous and isotropic) incorporating internal rotations with rotational inertial effects. Authors showed the existence of rotational waves similar to translational waves due to BAM when rotational inertial effects are considered. In this derivation the kinetic energy due to i ω was not considered i.e., the term

Constitutive Theories
The conjugate pairs in the entropy inequality (101) expressed in terms of Helmholtz free energy density are instrumental in determining the constitutive variables, their argument tensors as well as derivation of some constitutive theo- The argument tensors of Φ and η at this stage can be chosen using principle of equipresence [4] [34], we remark that principle of equipresence is not used in (110)-(112) as the conjugate pairs in entropy inequality specifically dictate the choice of argument tensors used and additionally 1 ρ and θ .
The argument tensors of ( ) In (117) : : For arbitrary but admissible ( ) From (128)-(130) we can conclude that ρ and θ are the only argument tensors of Φ . From (131) we conclude that η is not a constitutive variable as it is deterministic using θ ∂Φ ∂ . Using (128)-(131) the entropy inequality (127) reduces to We remark that setting coefficient of D in (132) to zero and obtaining ( )

Constitutive Theory for ( )
Since Φ is a function of ρ and θ so is ( )

Constitutive Theory for q
We consider ( ) ,θ = q q g and use representation theorem [35]- [51]. The combined generators of the argument tensors g and θ that are tensors of rank one is just g and the combined invariant is ⋅ g g (or q I ). Thus, the constitutive theory for q in the current configuration can be written as The material coefficients in the constitutive theory for q given by (169)

Significance and Influence of Internal Rotation Rates and Rotational Inertial Effects
In this section, we discuss the influence of internal rotation rates and rotational inertial effects on the deformation physics of thermoviscous incompressible and compressible fluent continua. In thermoviscous fluent continua (both incompressible and compressible) fluid particles experience motion (displacements) but strains are negligible, hence such fluids are considered to have no elasticity. Thus, thermoviscous fluent continua cannot support propagation of waves of deviatoric Cauchy stress tensor in a similar fashion as solid continua does as this physics requires elasticity and mass ( E ρ is the wave speed in solids, E is elastic modulus and ρ is mass density). We discuss details of the deformation physics in the following for incompressible and compressible fluent continua in view of the present work.

Incompressible Thermoviscous Fluent Continua
It is well known that speed of sound in incompressible classical thermoviscous fluent continua (CCM) is infinity. In such fluids equilibrium Cauchy stress is mechanical pressure and/or thermal pressure field (is Lagrange multiplier) that cannot be determined from the deformation but its presence influences the flow physics. Deviatoric Cauchy stress tensor causes distortion of the volume of fluid as well as dissipation (as ( ) 0 σ is conjugate with D ) that results in entropy production which in term influences thermal field.
Surana et al. [2] [3] have shown when CBL of NCCM with internal rotation rate physics (but without rotational internal effects) are employed, the presence of Cauchy moment tensor (symmetric based on BMM balance law) that is conjugate with the symmetric part of the gradients of rotation rate tensor results in added resistance to flow and additional entropy production that alters the thermal field due to the CBL of CCM. In the presence of internal rotations and inertial physics considered in this paper a part of the applied rate of work gets converted into kinetic energy due to angular velocities, thus effecting the rate of production of entropy which influences thermal field. Thus, rate of entropy production differs in the absence and in the presence of rotational inertial effects when using CBL of NNCM based on internal rotation rates. Model problem studies are in progress to compare with the results reported by Surana et al. (in the absence of rotational inertial effects) with those obtained using CBL of NCCM with rotational inertial effects considered in this paper. Thus, in incompressible thermoviscous fluent continua we do not have translational or rotational waves due to Cauchy deviatoric stress tensor and Cauchy moment tensor (as in elastic solid continua [9]), instead the entropy production is affected by the additional rotational inertial physics. The entropy production due to CBL of NCCM is expected to be different depending upon the consideration of absence of rotational inertial effects. Interdependence of the different sources of entropy production and final total entropy productions will be reported in the model problem studies in a follow up paper.

Compressible Thermoviscous Fluent Continua
In compressible thermoviscous fluent continua (CCM) we also decompose sym- , p ρ θ , the equation of state that is known for a given compressible fluid. The physics of change in volume i.e., compressibility, is due to ( ) 0 es σ whereas the change in shape or distortion of the fluid volume and dissipation mechanism resulting in entropy production is due to ( ) 0 ds σ . In this physics ( ) 0 ds σ and D are conjugate pairs. That is ( ) is rate of work that causes change of shape and entropy production.
When a disturbance is applied to a compressible thermoviscous fluent continua, due to compressibility of the fluid, local compression of the medium occurs resulting in localized higher density. In other words, a localized compression wave is generated purely due to ( ) 0 es σ or ( ) , p ρ θ . The deviatoric stress merely causes localized entropy production. If the disturbance is weak, the resulting compression wave or pressure wave is also weak (small pressure disturbance) resulting in insignificant changes in local density. Thus, the compression wave behind the current compression wave, although moving in a slightly compressed medium, will move almost at the same speed as the wave ahead of it due to insignificant changes in density in the weak compression wave. In other words, in this physics progressively generated compression waves propagate at almost the same speed, hence no "piling up" of the compression waves occurs. This physics of compressible thermoviscous medium is generally referred to as sound waves.
Since sound waves are weak compression waves that exist and move only because of compressibility of the medium their weak nature suggests that density changes and the entropy production are almost insignificant. In this physics, CBL of CCM are sufficient and there is hardly any need for CBL of NCCM with or without rotational inertial effects as in this physics entropy production is not significant to consider.
If the disturbance applied to a thermoviscous compressible fluent continua (CCM) is of significant strength such as the two compartments of a shock tube containing compressed gases with higher pressure ratio [52] separated by a diaphragm or high Mach number external flows, then the physics of evolution is quite different from sound waves and may require different considerations. We use this shock tube as an example to illustrate the significance of CBL of NCCM with internal rotation rate physics with or without rotational inertia. Let the shock tube be divided in two compartments of equal length by a diaphragm in the middle. Let ρ and 2 ρ with the same base as the pressure wave. The compression wave behind this wave when it reaches the compression zone will travel at a faster speed, hence will "pile up" on the waves ahead of it. This process of compression waves "piling up" on the waves ahead of them eventually creates a steady wave that no longer changes in time and propagates to the left of the diaphragm. This is a shock wave. In the compressed zone high velocity gradients D and ( ) 0 d σ (in CCM) results in entropy production which stabilizes once the shock wave is fully formed and remains constant during propagation.
Reflection of the shock waves from the impermeable boundaries and the details of the physics can be found in reference [52]. The purpose of describing this problem in detail is to point out that this problem contains high pressure, high temperature physics with large changes in density in which determination of correct entropy production during the entire evolution is extremely important as it allows us to determine if the shocks are sustaining (entropy production remaining constant) or diffusing, indicated by diminishing entropy production.
Secondly, rate of entropy production controls the evolution and formation of the shock wave.
When we consider CBL of NCCM with internal rotation physics we have additional mechanism of rate of entropy production due to ( ( ) The physics of compression waves, hence the shock waves are expected to be influenced the most as the entropy production is most significant in the compression zone. Consideration of rotational inertial physics in conjunction with CBL of NCCM (as presented in this paper) will further influence rate of entropy production. This problem illustrates that in high pressure, high temperature compressible physics in thermoviscous fluent continua such as high Mach number flows, the use of CBL of NCCM with internal rotation rates with and without rotational inertial physics may be more realistic for describing the deformation physics compared to CBL of CCM used currently. Model problem studies in progress will be presented in a follow up paper.

Summary and Conclusions
In this paper conservation and balance laws of non-classical continuum mechanics with internal rotation rate physics [2] [3] and the constitutive theories for thermoviscous fluent continua are rederived by incorporating rotational inertia effects. In the evolution of deforming fluent continua, when the time varying rotation rates (angular velocities) and angular accelerations are resisted by the deforming continua, moments, angular momentum and angular inertial effects are realized. The paper presents complete derivation of CBL and the constitutive theories in the presence of internal rotation rates due to L and the rotational inertial effects. The paper considers homogeneous and isotropic thermoviscous fluent continua. We summarize the work and draw some conclusions in the following.
1. As in most non-classical continuum theories, the Cauchy stress tensor is not symmetric in this work also. 2. In the non-classical continuum theories for fluent continua incorporating internal rotation rates [2] [3], the Cauchy moment tensor is symmetric as a consequence of the balance of moment of moments balance law [1]. In the present work BMM balance law does not establish symmetry of the Cauchy moment tensor, but yields three additional equations in ( ) 0 a m . 3. In the CBL presented here for NCCM with internal rotation rates and rotational inertial effects, BAM balance law is not just a relationship between the gradients of the Cauchy moment tensor and the skew symmetric Cauchy stress tensor, but additional contains rotational inertial effects. 6. Constitutive theory for ( ) 0 e σ , the equilibrium Cauchy stress tensor is derived using Helmholtz free energy density Φ for compressible thermoviscous fluent continua. The constitutive theories for ( ) 0 d σ and ( ) 0 s m are derived using representation theorem. It is shown that the constitutive theory for q based on integrity is cubic in the temperature gradient g . 7. Unlike non-classical solid continua, in fluent continua translational stress waves and rotational moment waves [1] can not exist as the fluent continua has no elasticity (translational or rotational). Thus, in fluent continua only the pressure waves can be realized. 8. It is shown that NCCM with internal rotation rate physics also results in rate of entropy production due to ( ) 0 : t i s s Θ m J  that differs in the absence and presence of rotational inertial effects. We also have rate of entropy production due to ( ) 0 : ds D σ . Both mechanisms of entropy production exist in compressible as well as incompressible fluent continua. In high pressure, high temperature compressible flow physics (with or without shocks) accurate determination of rate of entropy production is important as it controls shock formation, shock structure and shock relations (in general, isolated high gradient physics of dependent variables).
9. The NCCM work proposed here with internal rotation rates and rotational inertial physics may be more realistic approach to describing the flow physics at high pressures and high temperatures that may result in a severe change in state of matter that is critically influenced by the rate of entropy production.