Conservation laws for the Cattaneo heat propagation
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov () . SUPPLEMENTARY BALANCE LAWS FOR CATTANEO HEATPROPAGATION.
SERGE PRESTON
Abstract.
In this work we determine for the Cattaneo heat propagation sys-tem all the supplementary balance laws (conservation laws ) of the same or-der (zero) as the system itself and extract the constitutive relations (expres-sion for the internal energy) dictated by the Entropy Principle. The spaceof all supplementary balance laws (having the functional dimension 8) con-tains four original balance laws and their deformations depending on 4 func-tions of temperature ( λ ( ϑ ) , K A ( ϑ ) , A = 1 , , K A = 0 , A = 1 , ,
3) and to further restriction to the form of internal en-ergy. In its final formulation, entropy balance represent the deformation of theenergy balance law by the functional parameter λ ( ϑ ). October 9, 2018 1.
Introduction.
In this work we determine the form of all supplementary balance laws for theCattaneo heat propagation system (CHP-system) (2.1) bellow. We will solve theLL-equations [5, 8] directly, and get the constitutive relation on the internal energyas the function of temperature θ and heat flux q . If this condition is fulfilled, thetotal space of SBL (modulo trivial balance laws) is 8-dimensional, if this conditiondoes not hold, there are no new SBL. Then we show that the positivity conditionfor the production in the new balance laws place additional restriction to the formof internal energy and determine the unique SBL having nonnegative production -entropy balance law.2. Supplementary balance laws of a balance system.
Let one has a system of balance equations for the fields y i ( t, x A , A = 1 , , ∂ t F i + ∂ X A F Ai = Π i , i = 1 , . . . , m with the densities F i , fluxes F Ai , A = 1 , , i being functions of space-time point ( t, x A , A = 1 , , y i and their derivatives (by t, x A ) up to theorder k ≧
0. Number k is called the order of balance system (1.1). In continuumthermodynamics people mostly work with the balance system of order 0 (case ofRational Extended Thermodynamics) and 1.A balance law of order r (in the same sense as the system (1.1) is of order k ).(2.2) ∂ t K + X A =1 ∂ x A K A = Q is called a supplementary balance law for the system (SBL) (1.1) if anysolution of the system satisfies to the balance equation (1.2) . Examples of supplementary balance laws are: entropy balance, provided theEntropy Principle is admitted for system (1.1), see [10, 5, 11], Noether symmetriesin the sense of works [7, 8] and some linear combinations of the balance equationsof system (1.1) satisfying to some condition (gauge symmetries of system (1.1), see[7, 8] ).As a rule, in classical physics one looks for entropy balance laws of the sameorder as the original balance systems. Higher order SBL are also of an interest forstudying the system (1.1) - for example, study of integrable systems leads to thehierarchy of conservation laws (often having form of conservation laws themselves)of higher order.For a balance systems (1.10 of order 0 (case of Rational extended Thermody-namics, [5]), density and flux of a SBL (1.2) satisfy to the system of equations(2.3) λ i F µi,y j = K µ,y j , where summation by repeated indices is taken. Functions λ i ( y j ) (main fields interminology of [5]) are to be found from the conditions of solvability of this system.We call this system - the LL-system refereing to the Liu method of using La-grange method for formulating dissipative inequality for a system (1.1), see [4, 5].Source/production in the system (1.2) is then found as Q = P i λ i Π i .3. Cattaneo Heat propagation balance system.
Consider the heat propagation model containing the temperature ϑ and heatflux q as the independent dynamical fields. y = ϑ, y A = q A , A = 1 , , ( ∂ t ( ρǫ ) + div ( q ) = 0 ,∂ t ( τ q ) + ∇ Λ( ϑ ) = − q. Second equation can be rewritten in the conventional form ∂ t ( τ q ) + λ · ∇ ϑ = − q, where λ = ∂ Λ ∂ϑ . If coefficient λ may depend on the density ρ , equation is morecomplex.Constitutive relation specify dependence of the internal energy ǫ on ϑ, q and pos-sible dependence of coefficients τ, Λ on the temperature (including the requirementΛ ,ϑ = 0). Simplest case is the linear relation ǫ = kϑ , but for our purposes it is toorestrictive, see [3], Sec.2.1.Since ρ is not considered here as a dynamical variable, we merge it with thefield ǫ and from now on and till the end it will be omitted. On the other hand, inthis model the the energy ǫ depends on temperature ϑ and on the heat flux q (see[3],Sec.2.1.2) or, by change of variables, temperature ϑ = ϑ ( ǫ, q ) will be consideredas the function of dynamical variables.Cattaneo equation q + τ ∂ t ( q ) = − λ · ∇ ϑ has the form of the vectorial balancelaw and, as a result there is no need for the constitutive relations to depend on thederivatives of the basic fields. No derivatives appears in the constitutive relation,therefore, this is the RET model. In the second equation there is a nonzero pro-duction Π A = − q A . Model is homogeneous, there is no explicit dependence of anyfunctions on t, x A . UPPLEMENTARY BALANCE LAWS 3 LL-system for supplementary balance laws of CHP-system
To study the LL-system for the supplementary balance laws we start with the i × µ matrix of density/flux components F µi = ǫ τ q τ q τ q q Λ( ϑ ) 0 0 q ϑ ) 0 q ϑ ) . Assuming that coefficients τ and the function Λ are independent on the heat fluxvariables q A we get the ”vertical (i. e by fields ϑ, q A ) differentials” of densities andflux components F µi d v F µi = ǫ ϑ dϑ + ǫ q A dq A τ ϑ q dϑ + τ dq τ ϑ q dϑ + τ dq τ ϑ q dϑ + τ dq dq Λ ϑ dϑ dq ϑ dϑ dq ϑ dϑ . Let now(4.1) ∂ t K ( x, y ) + ∂ x A K A ( x, y ) = Q ( x, y )be a supplementary balance law for the Cattaneo balance system (2.1). It is easyto see that the LL-system has the form As a result, LL subsystem for µ = 0 takesthe formFor µ = 0,(4.2) ( λ ǫ ϑ + τ ϑ λ A q A = K ,ϑ ,λ ǫ q A + λ A τ = K ,q A , A = 1 , , . For A = 1 , ,
3, using cyclic notations, we have LL-equations(4.3) Λ ϑ λ A = K A,ϑ ,λ = K A,q A , K Aq A +1 , K Aq A +2 , A = 1 , , . Looking at systems (4.2-4.3) we see that if we make the change of variables:˜ ϑ = Λ( ϑ ) then the system of equations (4.2-3)takes the form ( wherever is thederivative by ϑ we multiply this equation by Λ ,ϑ )(4.4) ( λ ǫ ˜ ϑ + τ ˜ ϑ λ A q A = K , ˜ ϑ ,λ ǫ q A + λ A τ = K q A ; ( K A, ˜ ϑ = λ A ,K Aq B = λ δ AB , A, B = 1 , , . Second subsystem is equivalent to the relation d v K A = λ A d ˜ ϑ + λ dq A . These integrability conditions imply the expression K A = K A ( x µ , ˜ ϑ, q A ) and K Aq A = λ , A = 1 , , ⇒ λ = λ ( ˜ ϑ ) . SERGE PRESTON
Integrating equation K Aq A = λ ( ˜ ϑ ) by q Q we get(4.5) K A = λ ( ˜ ϑ ) q A + ˜ K A ( ˜ ϑ )with some functions ˜ K A ( ˜ ϑ ) . First equation of each system now takes the form(4.6) λ A = K A ˜ ϑ = λ ϑ q A + ˜ K A, ˜ ϑ ( ˜ ϑ ) . Substituting these expressions for λ A into the 0-th system ( λ ǫ ˜ ϑ + τ ˜ ϑ λ A q A = K , ˜ ϑ ,λ ǫ q A + λ A τ = K q A , A = 1 , , , we get(4.7) ( K , ˜ ϑ = λ ǫ ˜ ϑ + τ ˜ ϑ ( λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A ) ,K q A = λ ǫ q A + τ ( λ ϑ q A + ˜ K A, ˜ ϑ ( ˜ ϑ )) , A = 1 , , , where k q k = P A q A .Integrating A -th equation by q A and comparing results for different A we obtainthe following representation(4.8) K = λ ǫ + τ ( ˜ ϑ )[ 12 λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A ] + f ( ˜ ϑ )for some function f ( ˜ ϑ, x µ ).Calculate derivative by ˜ ϑ in the last formula for K and subtract the first formulaof the previous system. We get(4.9) 0 = λ , ˜ ϑ ǫ + τ ( ˜ ϑ )[( 12 λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A )] , ˜ ϑ − τ , ˜ ϑ λ , ˜ ϑ k q k + f , ˜ ϑ ( ˜ ϑ ) . This is the compatibility condition for the system (4.2) for K . As such, it isrealization of the general compatibility system (4.4).Take q A = 0 in the last equation, i.e. consider the case where there are no heatflux . Then the internal energy reduces to its equilibrium value ǫ eq ( ˜ ϑ ) and we get f , ˜ ϑ ( ˜ ϑ ) = − λ , ˜ ϑ ǫ eq . Integrating here we find(4.10) f ( ˜ ϑ ) = f ( x µ ) − Z ˜ ϑ λ , ˜ ϑ ( s ) ǫ eq ( s ) ds. Substituting this value for f into the previous formula and we get expressions for K µ :(4.11) ( K = λ ǫ − R ˜ ϑ λ , ˜ ϑ ǫ eq ds + τ ( ˜ ϑ )[ λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A ] + f ,K A = λ ( ˜ ϑ ) q A + ˜ K A ( ˜ ϑ ) , A = 1 , , . In addition to this, from (4.9) and obtained expression for f ( ˜ ϑ ), we get the expres-sion for internal energy (4.12) ǫ = ǫ eq ( ˜ ϑ ) + 12 τ , ˜ ϑ k q k − τ ( ˜ ϑ ) λ ϑ ( ˜ ϑ ) (cid:20) λ , ˜ ϑ ˜ ϑ k q k + ˜ K A, ˜ ϑ ˜ ϑ ( ˜ ϑ ) q A (cid:21) . This form for internal energy present the restriction to the constitutiverelations in Cattaneo model placed on it by the entropy principle.
UPPLEMENTARY BALANCE LAWS 5
Zero-th main field λ is an arbitrary function of ˜ ϑ while λ A are given by therelations (4.11):(4.13) λ A = ( λ ϑ q A + ˜ K A, ˜ ϑ ( ˜ ϑ )) . Using this we find the source/production term for the SBL (4.1)(4.14) Q = λ A Π A = − λ A q A = − ( λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A ) . Now we combine obtained expressions for components of a secondary balance law.We have to take into account that the LL-system defines K µ only mod C ∞ ( X ). Thismeans first of all that all the functions may depend explicitly on x µ . For energy ǫ , field Λ( ϑ ) and the coefficient τ this dependence is determined by constitutiverelations and is, therefore, fixed. Looking at (4.12) we see that the coefficients ofterms linear and quadratic by q A are also defined by the constitutive relation, i.e.in the representation(4.15) ǫ = ǫ eq ( ˜ ϑ )+ µ ( ˜ ϑ ) k q k + M A ( ˜ ϑ ) q A = ǫ eq ( ˜ ϑ )+ 12 τ , ˜ ϑ k q k − τ ( ˜ ϑ ) λ ϑ ( ˜ ϑ ) (cid:20) λ , ˜ ϑ ˜ ϑ k q k + ˜ K A, ˜ ϑ ˜ ϑ ( ˜ ϑ ) q A (cid:21) , coefficients(4.16) µ ( ˜ ϑ, x ) = 12 τ , ˜ ϑ − τ ( ˜ ϑ ) λ ϑ ( ˜ ϑ ) λ ϑ ˜ ϑ , M A = − τ ( ˜ ϑ ) λ ϑ ( ˜ ϑ ) ˜ K A, ˜ ϑ ˜ ϑ ( ˜ ϑ )are defined by the CR - by expression of internal energy as the quadratic functionof the heat flux.More then this, quantities λ ϑ ˜ ϑ λ ϑ and ˜ K A, ˜ ϑ ˜ ϑ ( ˜ ϑ ) λ ϑ are also defined by the constitutiverelations.Rewriting the first relation (14.15) we get(4.17) (cid:0) ln ( λ ϑ ) (cid:1) , ˜ ϑ = ln ( τ ) , ˜ ϑ − µ ( ˜ ϑ ) τ ( ˜ ϑ ) ⇒ ln ( λ ϑ ) = ln ( τ ) + b − Z ˜ ϑ µτ ( s ) ds ⇒⇒ λ ϑ = ατ e − R ˜ ϑ µτ ( s ) ds , α = e b > . From this relation we find(4.18) λ ( ˜ ϑ, x ) = a + α ˆ λ = a + α Z ˜ ϑ [ τ e − R u µ ( s ) τ ( s ) ds ] du Here a and α are constants (or, maybe, functions of x µ (?).Using obtained expression for λ ( ˜ ϑ, x ) in the second formula (3.16) we get theexpression for coefficients ˜ K A and, integrating twice by ˜ ϑ , for the functions K A ( ˜ ϑ )(4.19) ˜ K A, ˜ ϑ ˜ ϑ = − M A · λ ϑ ( ˜ ϑ ) τ ( ˜ ϑ ) = − M A αe − R ˜ ϑ µτ ( s ) ds ⇒⇒ ˜ K A = k A ˜ ϑ + m A + α · ˆ K A ( ˜ ϑ ) = k A ˜ ϑ + m A − α Z ˜ ϑ dw Z w [ M A ( u ) e − R u µτ ( s ) ds ] du. Functions ˆ K A ( ˜ ϑ ) are defined by the second formula in the second line.Thus, functions λ ϑ , ˜ K A,ϑϑ are defined by the constitutive relations while coeffi-cients α > , a , k A , m A are arbitrary functions of x µ . SERGE PRESTON Supplementary balance laws for CHP-system.
Combine obtained results, returning to the variable ϑ (and using repeatedly therelation f , ˜ ϑ = ϑ , ˜ ϑ f ,ϑ = ( ˜ ϑ ,ϑ ) − f ,ϑ = Λ − ,ϑ f ,ϑ ) we get the general expressions foradmissible densities/fluxes of the supplementary balance laws(5.1) K = λ ǫ − R ˜ ϑ λ , ˜ ϑ ǫ eq ds + τ ( ˜ ϑ )[ λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A ] + f == ( a + α ˆ λ ) ǫ − α R ϑ ˆ λ ,ϑ ǫ eq ds + τ ( ϑ )Λ ,ϑ [ α ˆ λ ϑ k q k + (Λ ,ϑ k A + α ˆ K A,ϑ ( ϑ )) q A ] + f ,K A = λ ( ˜ ϑ ) q A + ˜ K A ( ˜ ϑ ) = ( a + α ˆ λ ( ϑ )) q A + k A Λ( ϑ ) + m A + α ˆ K A ( ϑ ) , A = 1 , , .Q = − λ A q A = − ( λ ϑ k q k + ˜ K A, ˜ ϑ ( ˜ ϑ ) q A ) = − Λ − ,ϑ ( λ ϑ k q k + Λ ,ϑ k A q A + α ˆ K A,ϑ ( ϑ ) q A ) = − Λ − ,ϑ ( α ˆ λ ϑ k q k + Λ ,ϑ k A q A + α ˆ K A,ϑ ( ϑ ) q A ) . Collecting previous results together we present obtained expressions for sec-ondary balance laws first in short form and then - in the form where original balancelaws and the trivial balance laws are separated from the general form of SBL(5.2) K K K K Q = λ ǫ − R ϑ λ ,ϑ ǫ eq ds + τ ( ϑ )Λ − ϑ [ λ ϑ k q k + α ˜ K A,ϑ ( ϑ ) q A ] + f λ ( ϑ ) q + ˜ K ( ϑ ) λ ( ϑ ) q + ˜ K ( ϑ ) λ ( ϑ ) q + ˜ K ( ϑ ) − Λ − ,ϑ ( λ ,ϑ k q k + ˜ K A,ϑ ( ϑ ) q A ) == a ǫq q q + X A k A τ ( ϑ ) q A δ A Λ( ϑ ) δ A Λ( ϑ ) δ A Λ( ϑ ) − q A + ατ Λ( ϑ ) − ˆ K A,ϑ ( ϑ ) q A ˆ K ( ϑ )ˆ K ( ϑ )+ ˆ K ( ϑ ) − Λ − ,ϑ ˆ K A,ϑ ( ϑ ) q A + α ˆ λ ǫ − R ϑ ˆ λ ,ϑ ǫ eq ds + τ ( ϑ )Λ − ϑ [ ˆ λ ,ϑ k q k ]ˆ λ ( ϑ ) q ˆ λ ( ϑ ) q ˆ λ ( ϑ ) q − Λ − ,ϑ ˆ λ ,ϑ k q k + f m m m . To get the second presentation of the SBL we use the decompositions (4.19) λ = α ˆ λ + a and (4.18) ˜ K A ( ˜ ϑ ) = k A ˜ ϑ + m A − ˆ K A . Remark 1.
Notice the duality between the tensor structure of the basic fields ofCattaneo system - one scalar field (temperature ϑ ) and one vector field (heat flux q A , A = 1 , ,
3) and the structure of space
SBL ( C ) of supplementary balance laws- elements of SBL ( C ) depend on one scalar function of temperature λ ( ϑ ) and onecovector function of temperature ˆ K A . Remark 2.
It is easy to see that none of new SBL can be written as a linear com-bination of original balance equations with variable coefficients (Noether balancelaws generated by vertical symmetries v = v k ( y i ) ∂ y k , see [7, 8] ). Easiest way toprove this is to compare the source terms of different balance equations.Returning to the variable ϑ in the expression (3.12) and using the relation ∂ ˜ ϑ = ϑ ) ,ϑ ∂ ϑ we get the expression for the internal energy(5.3) ǫ = ǫ eq ( ϑ ) + τ ,ϑ ,ϑ k q k − τ ( ϑ ) λ ,ϑ λ ,ϑ Λ ,ϑ ! ,ϑ k q k + ˜ K A,ϑ Λ ,ϑ ! ,ϑ q A == Λ ,ϑ = κ − const ǫ eq ( ϑ ) + τ ,ϑ κ k q k − τ ( ϑ ) κλ ,ϑ (cid:20) λ ,ϑϑ k q k + ˜ K A,ϑϑ q A (cid:21) . UPPLEMENTARY BALANCE LAWS 7
Notice that for λ = 0, balance law given by the 4th column in (12.24) vanish. Thesame is true for deformations of the Cattaneo equation defined in the third columnwhen ˜ K A ( ϑ ) = 0.First and second balance laws in the system (12.24) are the balance laws of theoriginal Cattaneo system. Last one is the trivial balance law. Third and forthcolumns give the balance law(5.4) ∂ t " ˆ λ ǫ − Z ϑ λ ,ϑ ǫ eq ds + τ ( ϑ )Λ − ϑ [ 12 λ ϑ k q k + ˆ K A,ϑ ( ϑ ) q A ] + ∂ x A h ˆ λ ( ϑ ) q A + ˆ K A ( ϑ ) i == − Λ − ,ϑ (ˆ λ ϑ k q k + ˆ K A,ϑ ( ϑ ) q A ) . Source/production term in this equation has the form(5.5) − Λ − ,ϑ (ˆ λ ϑ k q k + ˆ K A,ϑ ( ϑ ) q A ) = − Λ − ,ϑ ˆ λ ϑ ( k q k + ˆ K A,ϑ ( ϑ )ˆ λ ϑ q A ) == − Λ − ,ϑ ˆ λ ϑ X A ( q A + ˆ K A,ϑ ( ϑ )2ˆ λ ϑ ) − X A ˆ K A,ϑ ( ϑ )2ˆ λ ϑ ! By physical reasons, Λ ,ϑ > . As (3.18) shows, λ ,ϑ may have any sign. We assumethat this sign does not depend on ϑ .For a fixed ϑ expression (4.5) for the production in the balance law (4.4) mayhave constant sign for all values of q A if and only if ˆ K A,ϑ ( ϑ ) = 0 , A = 1 , , ǫ = ǫ eq ( ϑ )+ τ ,ϑ ,ϑ − τ ( ϑ )2ˆ λ ,ϑ ˆ λ ,ϑ Λ ,ϑ ! ,ϑ k q k = τ − const, Λ ,ϑ − const ǫ eq ( ϑ ) − τ ( ϑ )2 k ˆ λ ,ϑ ˆ λ ,ϑϑ k q k with some function ˆ λ ( ϑ ). This being so, Cattaneo system has the supplementarybalance law(5.7) ∂ t " ˆ λ ǫ − Z ϑ ˆ λ ,ϑ ǫ eq ds + 12 τ ( ϑ )Λ − ϑ ˆ λ ,ϑ k q k + ∂ x A h ˆ λ ( , ϑ ) q A i = − Λ − ,ϑ ˆ λ ,ϑ k q k with the production term that may have constant sign - nonnegative,provided (we use the fact that ˆ λ ,ϑ = λ ,ϑ ) (5.8) Λ − ,ϑ λ ,ϑ ≦ . This inequality (which is equivalent, if Λ ,ϑ ≧ , to the inequality λ ,ϑ ≦ )is the II law of thermodynamics for Cattaneo heat propagation model. If we take q = 0 in the entropy balance (4.7) we have to get the value of entropyat the equilibrium s el :(5.9) s eq = ˆ λ ǫ eq − Z ϑ λ ,ϑ ǫ eq ds = Z ϑ ˆ λ ǫ eq,ϑ dϑ. SERGE PRESTON
From this it follows that at a homogeneous state ds eq = ˆ λ dǫ eq . Comparing thiswith the Gibbs relation dǫ eq = ϑds eq we conclude that(5.10) ˆ λ = 1 ϑ . Using (3.13) we also conclude that(5.11) λ A = − q A ϑ , A = 1 , , . It follows from this that the condition (4.8) (II law) takes the form well known fromthermodynamics (see [3, ? , 5]:(5.12) Λ ,ϑ ≧ . Substituting (4.10) into (4.6) and calculating − τ ( ϑ )2ˆ λ ,ϑ (cid:18) ˆ λ ,ϑ Λ ,ϑ (cid:19) ,ϑ = τ ( ϑ ) ϑ (cid:16) − ϑ Λ ,ϑ (cid:17) ,ϑ = − τ ( ϑ ) ϑ − (2 ϑ Λ ,ϑ + ϑ Λ ,ϑϑ ) ϑ Λ ,ϑ = τ ( ϑ ) ϑ Λ ,ϑ + τ ( ϑ )Λ ,ϑϑ ,ϑ ) we get the expression for internal energyin the form(5.13) ǫ = ǫ eq ( ϑ ) + (cid:20) τ ,ϑ ,ϑ + τϑ Λ ϑ + τ Λ ,ϑϑ ,ϑ ) (cid:21) k q k = τ − const, Λ ,ϑ − const ǫ eq ( ϑ ) + τϑ Λ ,ϑ k q k . For the entropy density we have(5.14) s = s eq + ˆ λ ( ǫ − ǫ eq ) + 12 τ ( ϑ )Λ − ϑ ˆ λ ,ϑ k q k == s eq + 1 ϑ (cid:20) τ ,ϑ ,ϑ + τϑ Λ ϑ + τ Λ ,ϑϑ ,ϑ ) (cid:21) k q k − τ ( ϑ )2 ϑ Λ ϑ k q k == s eq + 1 ϑ (cid:20) τ ,ϑ ,ϑ + τ ϑ Λ ϑ + τ Λ ,ϑϑ ,ϑ ) (cid:21) k q k = s eq + τ ϑ Λ ,ϑ (cid:20) τ ,ϑ τ + 1 ϑ + Λ ,ϑϑ Λ ,ϑ (cid:21) k q k == τ − const, Λ ,ϑ − const s eq + τ ϑ Λ ,ϑ k q k . Correspondingly, the entropy balance law takes the form(5.15) ∂ t (cid:18) s eq + τ ϑ Λ ,ϑ (cid:20) τ ,ϑ τ + 1 ϑ + Λ ,ϑϑ Λ ,ϑ (cid:21) k q k (cid:19) + ∂ x A ( q A ϑ ) = 1Λ ,ϑ k q ϑ k . Remark 3.
If in the absence of the heat flow ( q = 0) the ”equilibrium state” isnot homogeneous, more general constitutive relations with λ different from (5.10)and more general form of energy and entropy entropy balances satisfying to the IIlaw of Thermodynamics, are possible.We collect obtained results in the following Theorem 1. (1)
For the Cattaneo heat propagation balance system (2.1) com-patible with the entropy principle and having a nontrivial supplementarybalance law that is not a linear combination of the original balance lawswith constant coefficients, the internal energy has the form (4.3). If (4.3)holds, all supplementary balance laws for Cattaneo balance system (in-cluding original equations and the trivial ones) are listed in (4.2). New
UPPLEMENTARY BALANCE LAWS 9 supplementary balance laws depend on the 4 functions of temperature - ˆ λ ( ϑ ) , ˜ K A ( ϑ ) , A = 1 , , . Corresponding main fields λ µ , µ = 0 , , , havethe form (4.10-11). (2) Additional balance law (4.4) given by the sum of third and forth columns in(4.2) has the nonnegative production term if and only if the internal energy ǫ has the form (4.12) and, in addition, the condition (4.11) holds. Cattaneosystems satisfying to these conditions depend on one arbitrary function oftime ǫ eq ( ϑ ) . (3) Supplementary balance law having nonnegative production term (entropy)is unique modulo linear combination of original balance laws and the trivialbalance laws. Conclusion.
Description of the supplementary balance laws for Cattaneo heat propagationsystem given in this paper can probably be carried over for other systems of balanceequations for the couples of fields: scalar + vector field.One observes a kind of duality between the tensorial structure of dynamical fields(here θ, q ) and the list of free functions of temperature λ ( ϑ ) , ˜ K A ( ϑ ) , A = 1 , , References [1] H. Callen,
Thermodynamics , Whiley, 2nd ed. 1985.[2] P.Glensdorf, I.Prigogine,
Thermodynamical Theory of Structure, Stability and fluctuations ,Wiley, Brussels, 1971.[3] D.Jou, J.Casas-Vasquez, G.Lebon,
Extended Irreversible Thermodynamics , 3rd ed., Springer,2001.[4] I-Shish Liu. Method of Lagrange multipliers for exploatation of the entropy principle, Arch.Rational Mech. Anal., v.46, 1972, pp.131-148.[5] I. Muller, T. Ruggeri,
Rational Extended Thermodynamics , 2nd ed., Springer, 1998.[6] I.Muller,
Thermodynamics , Pitman Adv. Publ., Boston, 1985. co.,1985.[7] S. Preston,
Geometrical Theory of Balance Systems and the Entropy Principle , Proceedingsof GCM7, Lancaster, UK, Journal of Physics: Conference Series, vol.62, pp.102-154, 2007.[8] S.Preston, ”Variational theory of balance systems”, Intern. J. of Geom. Methods of ModernPhys., v7, N5 (August) 2010.[9] S.Preston, Supplementary balance laws for the navier-Stokes-fourier Fluid, Manuscript, un-published.[10] T. Ruggeri,
Galilean Invariance and Entropy Principle For Systems of Balance Laws , Cont.Mech.Thermodyn. 1 (1989).[11] T. Ruggeri,
The Entropy Principle: from Continuum Mechanics to Hyperbolic Systems ofBalance Laws , Entropy, v.10, pp.319-333, 2008.[12] S. Pennisi, T. Ruggeri,
A new method to exploit the entropy Principle and galilean invariancein the macroscopic approach to Extended Thermodynamics , Ricerche di Matematica, 55,2006, pp. 319-339.[13] D. Serre,
Systems of Conservation Laws I
CUP, Cambridge, 1999.[14] C. Truesdell, W. Noll,
The Non-Linear Field Theories of Mechanics , 2nd ed., Springer, 1992.
Department of Mathematics and Statistics, Portland State University, Portland,OR, U.S.
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