Rate Constitutive Theories of Orders n and 1 n for Internal Polar Non-Classical Thermofluids without Memory

In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to orders n and Cauchy moment tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient tensor is used as rotation rates to determine rate of rotation gradient tensor. This is an approximation to true convected time derivatives of the rotation gradient tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient tensor; (2) first convected derivative of the symmetric part of the rotation gradient tensor as used by Surana et al. is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy moment tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, moment tensor How to cite this paper: Surana, K.S., Long, S.W. and Reddy, J.N. (2016) Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory. Applied Mathematics, 7, 20332077. http://dx.doi.org/10.4236/am.2016.716165 Received: August 10, 2016 Accepted: October 28, 2016 Published: October 31, 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access

and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1 n are conjugate with the moment tensor.Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress tensor, moment tensor and heat vector.Stress tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress tensor and the moment tensor are further decomposed into equilibrium and deviatoric tensors.It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress tensor, moment tensor and heat vector.Density in the current configuration, convected time derivatives of the strain tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1 n, temperature gradient tensor and temperature are considered as argument tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence.The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates.The n th and 1 n th order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1 n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used pre-

Introduction
Conservation and balance laws: conservation of mass, balance of linear momenta, balance of angular momenta, balance of moments of moments (or couples), first law of thermodynamics (energy equation) and second law of thermodynamics (entropy inequality) for internal polar non-classical fluent continua were presented in references [1] [2].A summary of these was also presented in reference [3] in which Surana et al. also presented constitutive theories for internal polar non-classical thermofluids without memory that incorporated convected time derivatives of strain tensor up to order n, density, rate of the symmetric part of the rotation gradient tensor, temperature gradient tensor and temperature as argument tensors of the dependent variables in the constitutive theories at the onset of the derivation.In references [1] [2] [3] comprehensive literature was presented regarding various aspects of non-classical theories that were pertinent in context with internal polar non-classical continuum theory used here for fluent continua.For the sake of brevity these are not repeated here instead interested readers can see references [1] [2] [3].
Another significant discussion in references [1] [2] [3] is the discussion of mathematical description for fluent continua.It was established that in fluent continua one monitors the state of the matter at fixed locations, hence mathematical models describing such processes do not contain information regarding displacements therefore these descriptions can neither be Lagrangian nor Eulerian.Nonetheless since the fixed locations are occupied by different material particles during evolution, the fixed location can be viewed as current positions of some material particle during evolution.This thinking persuades one to believe that the mathematical descriptions used for fluent continua are Eulerian descriptions.This thinking is not contested in this paper, but is rather used as this approach is what is used for mathematical descriptions of fluent continua.
The notations used in this paper have been used by the authors in the current literature, nonetheless some description and their use in deriving conservation and balance laws are presented in the following.Over bar is used on quantities to express quantities in the current configuration in Eulerian description, that is, all quantities with over bars are functions of current coordinates i x and time t.ρ is the density of the fluid in the current configuration and is a function of x and Φ , θ , and η denote the Helmholtz free-energy density, temperature, and entropy density, respectively in the current configuration and are also functions of x . ( ) is the Cauchy stress tensor (in Eulerian description in contravariant basis).The superscript "0" is used to signify that it is rate of order zero and the lowercase parenthesis destinguish it from the second Piola-Kirchhoff stress tensor [ ] 0 σ used in Lagrangian description.Dot on any quantity refers to the material derivative.As explained above undeformed and deformed configurations can be used in the derivatives as long as the final equations from the conservation and balance laws contain i x and t and do not have displacements and strains in them as these are not available for fluent continua.In the following a brief explanation of notations is necessary as some of the notations are new.i x and i x denote the position coordinates of a material point in the reference and current configurations, respectively, in a fixed frame (x-frame) ( ) , , , or ( ) , , , , , dx dx dx dx = and { } [ ] , , dx dx dx dx = are the components of length ds and ds in the reference and current configurations, and if one neglects the infi- nitesimals of orders two and higher in both configurations, then one obtains [ ] In Murnaghan's notation , , , , ; , , , , in which the columns of ( )  are covariant base vectors i g  , whereas the rows of ( ) are Jacobians of deformation in covariant and contravariant bases.Furthermore is Eulerian description.The basis defined by  is reciprocal to the basis defined by . The following relations are useful in the paper: ( ) ( ) where in which D Dt stands for material time derivative, L     is the spatial velocity gradient tensor, and i v are velocity components of a material point i x in the current confi- guration in the x-frame.Over bar on all dependent quantities refers to their Eulerian description, i.e. they are functions of i x and t whereas the quantities without over bar are their Lagrangian description i.e. they are functions of i x and t.Thus ( ) ( ) x are Eulerian and Lagrangian description of a quantity Q in the current configuration.
The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, Cauchy moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1 n are conjugate with the Cauchy moment tensor.

Rotation Gradients, Their Convected Time Derivatives and Conservation and Balance Laws
In reference [1] [2] conservation and balance laws were derived for internal polar (nonclassical) fluent continua.The derivations were presented using contravariant and covariant measures of stress, moment tensors as well as using Jaumann rates.Measures of stress, moment and strain tensors and their convected time derivatives in the respective bases can be considered.Following references [3] [4] for example

Likewise one can let
be the convected time derivatives of Almansi, Green's strain tensors and the Jaumann rates.Where, , symmetric part of the velocity gradient tensor.Let define Cauchy stress tensor, Cauchy moment tensor, and convected time derivatives of the conjugate strain tensor in a chosen basis.Derivations of the constitutive theories is presented using this notation so that the resulting derivations are basis independent.By replacing ), and ( the constitutive theories in contravariant basis, covariant basis and in Jaumann rates can be obtained.In addition to the convected derivatives of the strain tensors one must also consider convected derivatives of the rotation gradient tensor that are also basis dependent.In reference [3] the authors show that Cauchy moment tensor and symmetric part of the gradient of the rate of rotation tensor are conjugate.In reference [3], the authors considered symmetric part of the gradients of rates of rotation obtained using skew symmetric part of the velocity gradient tensor.One notes that the Cauchy moment tensor is basis dependent: J m m m being moment tensors in contravariant basis, covariant basis, and in Jaumann rates.Thus, the convected time derivatives of the symmetric part of the rotation gradient tensor in general must also be basis dependent.Let ( ) and ( ( ) be the convected time derivatives of the rotation gradient tensors in contravariant basis, covariant basis, and Jaumann rates.With these convected time derivatives, the conjugate pairs are ( ( ) ( ) ) and ) in contravariant and covariant bases and in Jaumann rates.Covariant and contravariant bases are important in conservation and balance laws as well as constitutive theories.Jacobian of deformation ( ) are covariant base vectors [4].Thus, quantities derived using ( ) 0 J are in covariant basis and are Lagrangian descriptions.Likewise ( ) is Eulerian description of Jacobian of deformation.Rows of ( ) are contravariant base vectors.Hence, quantities derived using ( ) 0 J are in contravariant basis and are Eulerian descriptions.The convected time derivatives of the rotation gradient tensors in covariant and contravariant bases must be derived using rotation gradient tensor obtained using ( ) 0 J and ( ) 0 J .Details are presented in the following.
In finite deformation, a tetrahedron in the undeformed configuration with its orthogonal edges deforms into one in which the edges are non-orthogonal covariant base vectors and the vectors normal to the faces of the deformed tetrahedron are contravariant non-orthogonal base vectors that are reciprocal to the covariant base vectors.The covariant and contravariant bases are fundamental in the measures of finite deformation, rotations, etc.Consider deformed coordinates x of a material point in the current configuration with undeformed coordinates x in the reference configuration.Then ( ) Let T 0 0 0 0 , , in which Alternatively one can also derive (15) as follows.
ijk  is the permutation tensor.
The sign differences between ( 15) and ( 18) are due to clockwise and counterclockwise internal rotations and will only affect sign of M term in the balance of angular momenta.If one uses (15) as the definition of rotations then the term containing M in the balance of angular momenta must have negative sign.If the rotations in (18) are defined as ( ) ( )

Θ
then the term containing M in the bal- ance of angular momenta must have positive sign.Regardless, the resulting equations and the following derivations are not affected.One notes that decomposition in (11) enables explicit description of stretches and rotations contained in ( ) 0 J due to defor- mation of solid matter.The stretch tensor and the rotation tensor can also be obtained using polar decomposition of ( )

S
or left stretch tensor ( ) ( ) and pure rotation tensor The stretch tensors ( ) ( ) in general is not unique and may not even be possible without some approximation [5] [6] [7] [8].
4) It suffices to note that internal rotations at a material point present in ( ) Θ .Both forms contain mathematical description of same physics, hence either can be used as deemed suitable, but determination of ( ) 5) The internal rotation angles ( ) Θ are present at every material point and are a result of deformation.Between two neighboring material points the variation of ( ) 0 Θ is perhaps small otherwise there may be permanent damage or separation between them.Regardless of the magnitude of ( ) 0 Θ , these are strictly deterministic from Let ( ) 0 Θ J be the internal rotation gradient tensor, a tensor of rank three.Using (20) one can define Alternatively ( 16) can be written as and then In (22) the internal rotations ( ) 0 a J are expressed as a tensor of rank one (i.e. ( ) ( ) as a vector), hence its gradient 23) appears as a tensor of rank two.The representation ( 22) is more appealing for matrix and vector forms given in the following.Let T 0 0 0 0 , , Then, one defines rotation gradient tensor ( ) 0 Θ J and its decomposition into sym- metric and skew-symmetric tensors One can also define the velocity gradients as in which Θ is the rotation rate then its gradients are given by Remarks 1) Symmetric rotation gradient tensor in ( 26) is a covariant measure in Lagrangian description.It describes symmetric part of the gradients in x-frame of rotations about covariant axes expressed about the axes of the x-frame.
2) Since this measure is covariant rotation rate its work conjugate measure will be contravariant.
3) The covariant nature of this measure is intrinsic in its derivation due to ( ) 0 J , hence can not be changed.However, by replacing ( ) be the corresponding second Piola-Kirchhoff covariant rotation gradient tensor acting on the undeformed tetrahedron.Consider the following correspondence rule [4].
Thus, one obtains using ( 36) and (37) in (34) one obtains and Equations ( 40) and (41) are Lagrangian and Eulerian descriptions for second Piola-Kirchhoff covariant rotation gradient tensor.These are useful in deriving covariant convected time derivatives of the rotation gradient tensor , hence for this case (40) and (41) can be modified.(d) Convected time derivatives of the covariant rotation gradient tensor: compressible matter In this section derivation of convected time derivative of the covariant rotation gradient tensor One intentionally chooses Eulerian description for Cauchy and second Piola-Kirchhoff tensor as this is what is needed in the case of mathematical model for fluent continua.Consider material derivative of [ ] in (43), factoring and regrouping, one can write then one obtains the following from (46 is the first convected time derivative of the covariant rotation gradient tensor ( ) for compressible matter.To obtain the second convected time derivative of the covariant rotation gradient tensor one can take material derivative of (49), and following the same steps as in case of where In general one can write the following recursive relations that can be used to obtain the convected time derivative of any desired order k of the covariant rotation gradient tensor For incompressible case in (52).

Contravariant Basis: Internal Rotations, Rotation Matrix, Rotation Gradient Tensor and Their Convected Time Derivatives
(a) Internal rotations and rotation matrix Following the derivations for covariant measures, one can derive the following if one considers Jacobian of deformation ( ) 0 J in contravariant basis.Rows of ( ) 0 J are contravariant base vectors.Consider decomposition of ( ) 0

J
into symmetric and skewsymmetric tensors.
be the components of the rotations about ox axes of the x-frame, then one can write Alternatively one can also derive (57) as follows. (58) ( ) The reason for the sign difference in (57) and ( 60) is exactly same as for covariant measures.One notes that decomposition (53) enables explicit description of stretches (elongation per unit length and change in angles between the pair of orthogonal material lines in the undeformed configuration) and rotation tensor contained in ( ) The stretch tensors and the rotation tensor can also be obtained using polar decomposition of ( ) into right stretch tensor The stretch tensors on the other hand contains rotation angles due to deformation about the axes of the x-frame due to rotations about contravariant axes.
3) One notes that determination of ( ) is not necessary.Two different mathematical forms of rotation physics is sufficient in derivation of the conservation and balance laws.However, this process of obtaining ( ) in general is not unique and may not even be possible without some approximation [5] [6] [7] [8].
4) It suffices to note that internal rotations at a material point present in ( ) Both forms contain mathematical description of same physics, hence either can be used as deemed suitable, but determination of are present at every material point and are a result of deformation.Between two neighboring material points the variation of ( ) 0 Θ is perhaps small otherwise there may be permanent damage or separation between them.Regardless of the magnitude of ( ) 0 Θ , these are strictly deterministic from be the internal rotation gradient tensor, a tensor of rank three.Using (62) one can define Alternatively (62) can be written as and then In (64) the internal rotations are expressed as a tensor of rank one (i.e. ( ) Θ as a vector), hence its gradient in (65) appears as a tensor of rank two.The representation (64) is more appealing for matrix and vector representations given in the following.Let Then the rotation gradient tensor and its decomposition into symmetric and skew-symmetric tensors are defined as: Remarks 1) Symmetric rotation gradient tensor in (67) is a contravariant measure in Eulerian description.It describes symmetric part of the gradients of rotations about contravariant axes expressed about the axes of the x-frame.
2) Since this measure is contravariant its work conjugate moment measure is expected to be covariant (see derivation of first law of thermodynamics).
3) Contravariant nature of this measure is intrinsic in its derivation, hence can not be changed.However by replacing ( ) { Let the contravariant Cauchy rotation gradient tensor be acting on the deformed tetrahedron and let be the corresponding second Piola-Kirchhoff covariant rotation gradient tensor acting on the faces of the undeformed tetrahedron derived from using correspondence rule (70).Then one can write Substituting ( 71) and ( 72) in (70) Hence, one obtains and Also J in (77) can be replaced by 76) and (77) define contravariant second Piola-Kirchhoff rotation gradient tensor in Lagrangian and Eulerian descriptions.For incompressible fluent continua [ ] ( and regrouping the terms one obtains then one can write Here ( ) is the first convected time derivative of the contravariant Cauchy rotation gradient tensor for compressible matter.To obtain the second convected time derivative of the contravariant Cauchy rotation gradient tensor , one takes material derivative of (84) and follows the same steps as in case of ( ) , then one obtains the following: where In general one can write the following recursive relation that can be used to obtain the convected time derivative up to any desired order k of the tensor For incompressible case 1 J = and ( ) , hence the expressions for the convected time derivatives can be modified for this case.
It is advantageous to introduce basis independent notations so that the derivations of conservation and balance laws could be carried out independent of the basis.These can then be made basis dependent by simply replacing the basis independent quantities.Similar to Cauchy stress tensor and Cauchy moment tensor, introduce as basis independent convected time derivatives of the rotation gradient tensor.By choosing one can obtain convected time derivatives of the rotation gradient tensor in contravariant basis, covariant basis and in Jaumann rates.

Polar Decomposition of Velocity Gradient Tensor and Consideration of Local Rotation Rates
Polar decomposition of the velocity gradient tensor is helpful in decomposing deformation into stretch rate tensor and rotation rate tensor.Whether one uses left stretch rate tensor or right stretch rate tensor, the rotation rate tensor is unique.Thus, at each location with infinitesimal volume surrounding it, the velocity gradient tensor L     can be decomposed into pure rates of rotation t R     and right or left stretch rate tensors     are symmetric and positive definite.The rotation rate tensor can equivalently be obtained due to rotation rates t Θ at each location in the flow domain.Thus, at each location in the flow do- main the rotation rate t R     matrix can be viewed as being due to t Θ .If varying ro- tation rates at varying locations in the flow domain are resisted by the constitution of the fluent continua then this must result in additional dissipation that requires existence of energy conjugate moments M in the deforming matter.Thus, at the onset t Θ and its conjugate M are considered in the derivation of the polar continuum theory for the fluent continua.Details of polar decomposition of L     and rotation rates t Θ are given in the following.Let Then (89) holds, hence t r S     can be defined using (90).t R     can now be deter- mined using (88) and following a similar procedure one can establish the following 91) or (94) is unique.The rate of rotation matrix t R     can equivalently be obtained due to rotation rates t Θ at each location.Thus, at each location t R     can be viewed as be- ing due to rates of rotations t Θ .Rate of energy dissipation due to t Θ requires coex- istence of moments M (per unit area) on the oblique surface of the tetrahedron in the deforming matter.Thus where ( ) Explicit forms of t Θ i.e. m , then there is basis dependency in t Θ .In the energy equation and likewise in entropy inequality their derivations are continued with t Θ until at a later stage when gradients of t Θ are needed, convected time derivatives of the rotation gradient tensor in the appropriate basis are introduced.

Conservation and Balance Laws
In reference [1] [2] conservation and balance laws were derived for internal polar fluent continua.These derivations were presented using ( ) 0 σ , ( ) 0 m and ( ) 0 q as basis in- dependent constitutive tensors which were then given appropriate definitions of ( ) q depending on whether the basis of choice is contravariant, covariant or Jaumann rates.Additionally were used as argument tensors that are basis independent convected time derivatives of strain tensor that could be as conjugate pairs in the derivation.Since ( ) 0 m is basis dependent, it's conjugate pair(s) must also be basis dependent.Choice of the rotation rate gradient resulting from the velocity gradient tensor as conjugate to ( ) 0 m limits the applicability of the resulting theory to small strain rates and small rotation rates and its gradients.
Furthermore, this choice can not be extended to higher order time derivatives of the rotation gradient tensor as it is not the convected time derivative of rotation gradient tensor in Eulerian description.The work presented in this paper proposes and replaces D θ     with the true convected time derivatives of the rotation gradient tensor that can be considered in a chosen basis.Use of correct convected time derivatives of the rotation gradient tensor requires the rederivation of energy equation and entropy inequality using correct measures.Using ( ) At this stage the Cauchy stress tensor ( ) 0 σ is nonsymmetric but the Cauchy mo- ment tensor ( ) 0 m is symmetric (due to moment of moments Equation (100)).In (98) -(100) basis independent ( ) 0 σ and ( ) 0 m have been used as opposed to their contravariant measures, but the details of the derivations remain similar.Since the energy equation and the entropy inequality require rate of work due to ( ) 0 m in addition to rate of work due to ( ) 0 σ , their derivations in reference [3] does not hold here as the conjugate pair to ( ) 0 m is no longer gradient of rotation rate tensor resulting from the velocity gradient tensor, instead it is convected time derivative of the rotation gradient tensor in appropriate basis depending upon the choice of basis for the Cauchy moment tensor.

First Law of Thermodynamics: Energy Equation
The sum of work and heat added to a deforming volume of matter must result in the increase in energy of the system.Expressing this as a rate statement one can write [4] where e is specific internal energy, b F is body force per unit mass, u are dis- placement, and 0 q is rate of heat.Note the additional term tributes additional rate of work due to rates of rotation in (104).Expand each of the integrals in (102)-(104).Following reference [4], it is straight forward to show that: Using basis independent Cauchy stress tensor ( ) 0 σ , Cauchy principle, and following the details in reference [4] one can write Likewise using basis independent moment tensor (per unit area) ( ) 0 m , Cauchy principle, and following the details similar to these used in deriving (108), one can write The first convected time derivative of the rotation gradient tensor, ( ) ( ) that is conjugate to the Cauchy moment tensor ( )  0 m has been used in (109).Using (105)-( 109) in (101) Transferring all terms to left of equality and regrouping Using (98) (balance of linear momenta) and (99) balance of angular momenta, (110) reduces to : 0 Equation ( 113) is the final form of the energy equation in which ( ) 0 σ is a nonsym- metric Cauchy stress tensor and ( ) 0 m is a symmetric Cauchy moment tensor.Thus in (113) one can use In (114) the following decomposition of ( ) ( ) Θ γ into symmetric and antisymmetric tensors has been used. ( ) By appropriate choices of ( , and s Θ m σ γ , the explicit form of the energy equ- ation in any desired basis can be obtained.

Second Law of Thermodynamics: Entropy Inequality
If η is the entropy density in volume ( ) V t , h is the entropy flux between ( ) and the volume of matter surrounding it and s is the source of entropy in V due to non-contacting bodies, then the rate of increase in entropy in volume ( ) V t is at least equal to that supplied to ( ) V t from all contacting and non-contacting sources [4].
Using (117) in (116) One recalls that [4] ( ) ( ) and Substituting from ( 119) and ( 120) in ( 118) and transferring all terms to the left of inequality Since volume ( ) Equation ( 122) is called the Clausius-Duhem inequality and is the most fundamental form resulting from the second law of thermodynamics.A different form of (122) can be derived if one assumes where θ is absolute temperature, ( ) 0 q is heat vector, and r is a suitable potential. Using ; Ψ from (124) into (122) and multiplying by θ .
From energy Equation (113) (after inserting r ρ term) in basis independent form one can write the following. ( Let Φ be Helmholtz free energy density (specific Helmholtz free energy) defined by Substituting from (130) into (128) ( ) 0 m is symmetric but ( ) 0 σ is not symmetric.Since ( ) 0 m is symmetric, one can use the following in (132).
The entropy inequality (132) in contravariant basis, covariant bases and in Jaumann rates can be obtained by replacing ( )  0 σ , ( ) 0 m with corresponding quantities in appropriate basis.
3) From ( 145) and (146) one can conclude that , s σ γ and ( ) , s Θ m γ are con- jugate pairs, hence are responsible for conversion of mechanical energy into heat or entropy.The conjugate pairs are instrumental in deciding the dependent variables in the constitutive theories and some of their argument tensors.These conjugate pairs suggest that ( ) 0 s σ can be expressed as a function of ( ) 1 γ and ( ) 0 m as a function of One notes that ( ) 0 q and g are also conjugate, thus ( ) 0 q can be expressed as a func- tion of g .These details of the constitutive theories are presented in the following sections.4) This mathematical model has closure once the constitutive theories for

Dependent Variables in the Constitutive Theories
The choice of dependent variables in the constitutive theories must be consistent with the axiom of casualty [4] [9] [10].The self observable quantities and those that can be derived from them by simple differentiation and/or integration can not be considered as dependent variables in the constitutive theories.Thus velocities, temperatures, tem-perature gradients, etc. are ruled out as choices of dependent variables in the constitutive theories.From the entropy inequality one notes that are possible choices of dependent variables in the constitutive theories.The choice of ( ) 0 s σ , ( ) 0 m , and q as dependent variables in the constitutive theories is also supported by balance of linear momenta, balance of angular momenta, and the energy equation.is a matter of preference as these are related through Φ .In the present work ,η Φ are chosen, hence e need not be considered as a dependent variable in the constitutive theories.Thus, and q are the possible dependent variables in the constitutive theories.At a later stage of the derivation, some of these may be ruled out as dependent variables in the constitutive theories if so warranted by some other considerations.
Possible choices of argument tensors of dependent variables are considered, keeping in mind the principle of equipresence [4] [9] [10], i.e. at the onset all dependent variables in the constitutive theories possibly must contain the same argument tensors.
For compressible fluent media, density ρ is certainly an argument tensor.θ is a natural choice as an argument tensor.The choice of g as an argument tensor is ne- cessitated due to the dependent variable q in the constitutive theory and the physics of heat conduction.The choice of ( ) as argument tensors is also clear as these are conjugate to One notes that ( ) is the first convected time derivative of the strain tensor (Almansi tensor or Green's tensor or Jaumann rates) and is a fundamental kinematic tensor.In addition to ( )  , are also fundamental kinematic tensors up to order n that are convected time derivatives of orders 2,3, , n  of strain , , , ; 1, 2, , , , From the entropy inequality one notes that , has no dependence on ( ) . Thus one can modify the argument tensors of ( )

Entropy Inequality and Constitutive Theories
Consider the entropy inequality (146) with the arguments of Φ defined in (149).Now D Dt Φ = Φ  can be obtained which is needed in the entropy inequality.
2) Equation ( 158) implies that Φ is not a function of ( ) ; 1, 2, , 3) Equation ( 159) implies that Φ is not a function of 4) Based on (160), η is not a dependent variable in the constitutive theories as 5) The last inequality is essential in the form it is stated.For example the following (or any other separation of terms) are inappropriate due to the fact that these imply that ( ) as Φ is not a function of these.This is contrary to (149).In- equality (161) in this form is unable to provide one with further details regarding the derivation of the constitutive theories.
In view of these remarks the arguments of the dependent variables in the constitutive theories in (149) can be modified.One can use ρ instead of , 0, 0, , 0 , ; One notes that there are no mechanisms or conditions that permit eliminating from the argument list of ( ) 0 q , hence one must keep them as in (163).Based on (160) and remark (4), η is no longer a dependent variable in the constitutive theories.With (161) and ( 163) one has no further mechanisms to proceed with the derivation of the constitutive theories.

Decomposition of Stress Tensor ( )
in which one considers the following ) and g are zero.Substituting (164) into ( Inequality ( 170) is satisfied if ) Inequalities (171) imply that the rate of work due to ( Constitutive theories for ( ) ( ) 0 m , and 0 q must satisfy (171) and (172).

Constitutive Theory for Equilibrium Stress
The incompressibility condition must be enforced.Based on (174) one can add to (167).( ) Using 0 ρ ∂Φ = ∂ in (176) and regrouping terms. ( In the case of incompressible internal polar thermofluids ( ) ( ) 0 e s σ is a function of θ only, hence from (177) one obtains ( ) p θ is called mechanical pressure.Since ( ) p θ is an arbitrary Lagrange multi- plier, ( ) p θ is not deterministic from the deformation field.In view of (178), (177) reduces to Inequality (179) will hold if ) Conditions ( 180) and ( 181) are the same for the compressible case i.e. the rate of work due to Constitutive theories for  2) Use of stress decomposition (164) in the conditions resulting from the entropy inequality permits determination of the constitutive theory for equilibrium stress tensor for compressible as well as incompressible internal polar thermofluids in terms of thermodynamic pressure and mechanical pressure.
3) The inequalities (170) or (179) require the rate of work due to 4) The inequality (172) or (181) can be used (shown later) to derive a simple constitutive theory for ( ) 0 q (Fourier heat conduction law), but better constitutive theories are possible for ( ) 0 q (shown in subsequent sections) 5) The equilibrium stress ( ) ( ) 0 e s σ is independent of the basis for compressible as well as incompressible polar thermofluids due to the fact that [ ] I is basis independent.This implies that 6) The rate constitutive theories for deviatoric Cauchy stress tensor, Cauchy moment tensor and heat vector are derived using theories of generators and invariants [4] [9] [12]- [27].

Symmetric Cauchy Stress Tensor
The coefficients To determine material coefficients from (186), one considers Taylor series expansion of each One notes that 1, 2, , , ; 0,1, , Collecting coefficients (quantities defined in Ω ) of the terms in (188) that are de- fined in the current configuration and also grouping those terms that are completely defined in the known configuration Ω .Let , Using (189), one can write (188) as follows , , , .This constitutive theory is based on integrity, hence it is complete.

Rate Constitutive Theories of up to Order n and 1 n for Heat Vector
( ) q 0 : Compressible Consider (from (173)) , , , Let { } ; 1, 2, , be the combined generators of the argument tensors 1, 2, , , and g that are tensors of rank one.
Let ; 1, 2, , be the combined invariants of the same argument tensors.Then, one can express ( ) { } 0 q as a linear combination of { } ; 1, 2, , The absence of unit vector in (192) is due to the fact uniform temperature field does not contribute to ( ) { } 0 q .The negative sign in (192) is because a positive ( ) { } 0 q in the direction of the exterior unit normal to the surface of the volume of matter results in heat removal from the volume of matter.The coefficients ; 1, 2, , in the current configuration.To determine the material coefficients from ; 1, 2, , (that are defined in the current configuration) in (192), one considers Taylor series expansion of each ; 1, 2, , about a known configuration Ω in θ and ; 1, 2, , and retains only up to linear terms in θ and the invariants and then substitutes these back in (192).If one defines the following in the final expression for ( ) Then, using (193) the resulting form of (192) can be written as 194) q i b , q ij c , and q i d are material coefficients defined in known configuration Ω .

Constitutive Theory for Cauchy Moment Tensor
be the combined generators of the argument tensors of ( ) 0 m that are symmetric tensors of rank two and ; 1, 2, , be the combined invariants of the same argument tensors of ( ) 0 m .One can express ( ) and identity tensor [ ] The coefficients are functions of , ρ θ and ; 1, 2, , , To determine the material coefficients from (197), one considers Taylor series expansion of each ; 0,1, , about a known configuration Ω and retains only up to linear terms in θ and the invariants (for sim- plicity) and then substitutes these back in (196).Define the following ( Then using (198) in (196) can be written as .This constitutive theory is based on integrity, hence is complete.

Remarks
1) The constitutive theories for 2) The configuration Ω can be chosen to be reference configuration (undeformed configuration before the commencement of the evolution) in which case the material coefficients will be independent of the deformation.If one chooses Ω to be a known deformed configuration, then the deformation dependent material coefficients are possible in the constitutive theories.Dependence of the material coefficients on the invariants of the argument tensor in a known configuration Ω permits complex de- scription of material coefficients.
3) An important point to note is that the material coefficients in the final forms of the constitutive theories are defined in a known configuration Ω , whereas the consti- tutive equations hold in the current configuration for which the deformation field is yet to be determined.This of course is a consequence of the Taylor series expansion of the coefficients in the linear combination (using combined generators) about a known configuration.In the currently used constitutive models in the published works [28] for variable material coefficients, the coefficients are expressed as a function of the unknown deformation field in the current configuration.This is obviously not supported by the derivations presented in Sections 5.2-5.4.
4) Using the derivations presented in Sections 5.2-5.4rate constitutive theories of various orders in desired basis can be derived by choosing values of n and 1 n, the orders of the rate theory.As the orders of the rate theory increase, the number of material constants increases significantly.Thus, the higher order rate theories necessitate elaborate experiments to calibrate them.5) In the following rate theories of orders one ( In this case the combined generators of ( ) and { } g that are symmetric ten- sors of rank two are ( and the combined invariants of ( ) tr , tr , tr , , Thus, one can write Following the general derivations in Section 5.2 for N generators and M invariants, for this specific case one can write The definitions of material coefficients , , , and tm α as well as 0 σ Ω remain the same as defined in (189).This constitutive theory requires 46 material coefficients, still too many to determine experimentally.

Simplified Rate Constitutive Theory of Order One
In this case there are only two generators ( 2 N = ) and three invariants ( and the following constitutive theory (using (190) for 2 N = and 3 M = ) is obtained.
This constitutive theory requires 14 material coefficients and contains up to fifth degree terms in the components of ( ) Begin with (204) and neglect those terms on the right side of (204) that are of degree higher than two in the components of ( ) This constitutive theory requires 8 material coefficients.
If one further neglects the product terms in ( ) θ θ Ω − (last two terms on the right side of (208)) in (208) one obtains This constitutive theory requires only six material coefficients.The dependence of the material coefficients on the invariants in (209) can be modified based on the assumptions used here or can be maintained as originally defined in (189).
Material coefficients η Ω and κ Ω can be functions of ρ Ω , θ Ω and invariants of ( ) in the known configuration Ω .The constitutive theory (211) is the simplest possible constitutive theory for deviatoric symmetric Cauchy stress tensor.2) Some specific remarks can be made for the simplified rate theory of order one given by (211).When one compares (211) with the similar theory for  That is, the linear constitutive theory of order one in (214) for deviatoric Cauchy stress tensor is basis independent.

Remarks on Constitutive Theories for
3) Since the material coefficients η Ω and κ Ω are functions of ρ Ω , θ Ω and inva- riants of D     in the known configuration Ω , they can be defined using power law, Carreau-Yasuda model, Sutherland law, etc. similar to classical thermofluids (see reference [28]).In case of incompressible fluid tr 0 D   =   in (214).

Simplified Constitutive Theories for
This constitutive theory requires only six material coefficients.The dependence of the material coefficients on the invariants in (220) can be modified based on the assumptions used here or can be maintained as originally defined in (198)., then one can write ( The material coefficients m κ Ω and m η Ω can be functions of ρ Ω , θ Ω and in- variants of ( ) ( ) in the known configuration Ω .The constitutive theory (222) is the simplest possible constitutive theory for Cauchy moment tensor ( )  0 m but permits deformation dependent material coefficients.Thus, here also one can use concepts similar to power law, Carreau-Yasuda model, Sutherland law etc. that are used for the material coefficients in the constitutive theory for

Remarks
In reference [3] ( ) as ( ) This is an approximation as D Θ     is an approximation to the symmetric part of the convected time derivative of the rotation gradient tensor.Use of D Θ     is justified only when the rates of rotation gradients are very small in which case ( ) ( ) Recall that or can be considered as measures of Cauchy stress and Cauchy moment tensors in contravariant and covariant bases and corresponding to Jaumann rates.

2 . 1 .J
Covariant Basis: Internal Rotations, Rotation Matrix, Rotation Gradient Tensor and Their Convected Time Derivatives (a) Internal rotations and rotation matrix Consider decomposition of the Jacobian of deformation ( ) 0 into symmetric and skew-symmetric tensors.

be the components of the rotations ex- pressed as rotations about 1 ox , 2 ox , and 3 ox
axes of the x-frame, then one can write

RRΘ 3 )
positive-definite and the rotation tensor ( )0 is orthogonal.Since ( ) 0 R in (19) and ( ) 0 Θ in (15) are both obtained from the same deformation in ( ) 0 J , these contain details of the same internal rotation physics but in different forms.One may make the following remarks.1)( ) 0 is rotation matrix, hence relates undeformed orthogonal frame to a new or- thogonal rotated frame (due to deformation).2) ( ) 0 on the other hand contains rotation angles due to deformation about the axes of the x-frame.Determination of ( ) different mathematical forms of rotation physics in ( ) , this process of obtaining ( )

a
polar decomposition.(b) Internal rotation gradient tensor and its rates using ( ) J is a tensor of rank two, hence alterna- tively one can write

R
are both obtained from the same deformation in ( ) 0 J , these contain details of the same internal rotation physics but in different forms.The following remarks parallel to those for covariant measures can be made.1) ( ) is rotation matrix due to deformation, hence relates two orthogonal frames.2) ( ) 0 Θ

J
the polar decomposition.(b) Internal rotation gradient tensor using ( ) 0 The contravariant internal rotation tensor ( ) 0 a J is a tensor of rank two, hence alternatively one can define

J
, these measures will become Lagran- gian descriptions.4) Convected time derivatives of ( ) 0 s Θ J of orders up to 1 n in contravariant basis are defined as ( ) Second Piola-Kirchhoff contravariant rotation gradient tensor Consider isotropic, homogeneous compressible matter.Consider oblique planes of the deformed and the undeformed tetrahedra with scalar areas n dA and n dA and vector areas { } dA and { } dA .Let the resultant rotation gradient vector acting on these areas be { } and (77) can be modified for this case.(d) Convected time derivatives of the contravariant rotation gradient tensor: compressible matter Consider the material derivative of [ ] second Piola-Kirchhoff rotation gradient tensor (Equation (77)) derived using contravariant Cauchy rotation gradient tensor.

2 t Θ , and 3 t
Θ are de- fined in terms of velocity gradients.These rotation rates on the oblique plane of the tetrahedron are conjugate with the resultant moment tensor M , hence result in rate of work.At this stage these are not basis dependent.When M is converted to moment tensor contravariant or covariant basis or Jaumann rates.The constitutive theory for thermofluids without memory in reference[3] utilizes ( ) ( ) Cauchy stress tensor, Cauchy moment tensor, and heat vector, convected time derivatives of the rotation gradient tensor (all will eventually be basis dependent) one can write the following for conservation of mass, balance of linear momenta, balance of angular momenta and balance of moments of moments (or couples).
total of 15 equations) are added to the already existing eight equa- tions for the conservation and balance laws (conservation of mass, balance of linear momenta, balance of angular momenta and energy equation) giving rise to a total of 23 equations in 23 variables,

( ) 0 a
σ can not be chosen as dependent variables in the constitutive theories as these are deterministic from the balance of angular momenta.Choice of , e η or ,η Φ

s σ 0
In order to remedy the situation discussed in remark(5), one considers decomposition of symmetric Cauchy stress tensor into equilibrium Cauchy stress tensor

(
is called thermodynamic pressure for compressible internal polar thermoflu- ids and is generally referred to as an equation of state[4] [9] in which p is expressed as a function of ρ and θ or 1 v ρ = and θ , where v is specific volume.If one assumes the compressive pressure to be positive, then ( ) , p ρ θ in (168) can be re-

ψ
must be positive.In view of (168) one can write the following for com- pressible internal polar thermofluids.
incompressibility condition given by (174) must be incorpo- rated in the entropy inequality.

(
be positive and the constitutive theory for ( ) 0 q must satisfy (181).In view of (178) one can write the following for incompressible internal polar thermofluids.

(
σ and ( ) be positive but provide no mechanisms for deriving constitutive theories for

Let ; 1
g that are symmetric tensors of rank two [

Ω
and retains only up to linear terms in θ and the invariants (for simplicity).


are functions of the same quantities but in the current configuration (186).When (187) is substituted in (185), one obtains the final expression for the most general rate constitutive theory of up to order n for compressible internal polar thermofluids.The final expression defines the material coefficients in the known configuration Ω .Details are given in the following.First, substitute (187) in (185),

(
defined in known configurations Ω .This constitutive theory requires ( )( ) ( ) 1 M N M N N + + + + material coefficients.The material coefficients defined in (190) are functions of ρ Ω , θ Ω and ( ) ; 1, 2, , tm m α are material coefficients defined in known configura- tions Ω .This constitutive theory requires material coefficients defined in (198) are functions of ρ Ω , θ Ω and ( ) and their further simplifications are considered to present rather simple theories that could be used in model problem studies.

5. 6 . 3 .
Simplified Rate Constitutive Theory of Order One (If one neglects quadratic terms in ( )1 γ     in (209), then one obtains a constitutive theory for σ that is linear in ( ) same as those of ( )0 d σ ,Cauchy stress tensor for classical thermofluids[4].Thus the constitutive theories for σ for internal polar thermofluids are the same as those for ( ) 0 d σ for classical thermofluids.The fundamental difference is that even though the constitutive theories are the same, they are for different stress measures.

( ) 0 d
σ is the deviatoric part of the total Cauchy stress tensor, whereas σ is the deviatoric part of the symmetric part of the Cauchy stress tensor.
one notes that η and κ are similar to first and second viscosities and tm α is thermal modulus.Since most general constitutive theory for Cauchy moment tensor ( ) 0 m has been presented in Section 5.4.Unfortunately this constitutive theory for( )  0 m requires forty seven material coefficients.In this section simplified constitutive theories are presented that are derived using the general constitutive theory presented in Section 5.4.

5. 8 . 3 .
Constitutive Theory for( ) part of the convected time derivatives of the rotation gradient tensor, only D i.e. it must be the convected time derivative of the rotation gradient tensor i.e. one must replace it with ( ) is consistent.Authors in[3] use the following.
is the gradients of the rotation rates, not the convected time derivatives of the rotation gradient tensor.