On Entropy Flux Heat Flux Relation in Thermodynamics with Lagrange Multipliers

Transcription

1 Continuum Mech. Thermodyn. (1996) 8: On Entropy Flux Heat Flux Relation in Thermodynamics with Lagrange Multipliers I-Shih Liu Instituto de Matemática Universidade do rio de Janeiro, Caixa Postal 68530, CEP , Rio de Janeiro, Brazil A typical problem often encountered in thermodynamics with Lagrange multipliers is to deduce the relation between the entropy flux and the heat flux from conditions imposed by the entropy principle employing the general entropy inequality. This problem is usually trivial if linear constitutive relations are assumed, and otherwise it can be quite difficult even though such a relation is expected. In this paper, we shall try to formulate such a problem in general and prove as theorems. Some previous thermodynamic theories with Lagrange multipliers are mentioned as typical examples. 1 Introduction Statement. Let Φ and q be isotropic vector functions of an arbitrary number of vector and tensor variables. Then Φ = Λ q, (1) where Λ is a scalar function, if for every one of the vector variable v the following relation in components holds, + Φ ( j qi = Λ + q ) j. (2) v j v i v j v i This statement is a typical problem arises in thermodynamics with Lagrange multipliers [1], for various classes of isotropic materials, in its exploitation of the 1

2 entropy principle based on the general entropy inequality for supply-free bodies, ρ η + div Φ 0, (3) first proposed by Müller ([2, 3]). In most of such theories, relations of the forms (2) were encountered and the entropy flux - heat flux relation (1) is asserted. The scalar function Λ is the Lagrange multiplier to be identified as the reciprocal of the absolute temperature in the theory. In the classes of isotropic materials involved only one or two constitutive vector variables [4, 5], the assertion of the statement is a straight forward proof using the well-known representation theorem for isotropic vector functions. In more general case, the assertion is either a nearly trivial consequence from the assumed linear constitutive representation for the entropy flux Φ and the heat flux q on the variable v [6], or a result from a formidable task based on the polynomial isotropic representations and series expansions [7, 8]. In some cases [9], the proportionality relation (1) has been adopted without a proof and the validity of the statement is merely speculated. The difficulty in proving the the proportionality between the entropy flux and the heat flux in more sophisticated material classes from condition (1), greatly restrains the furtherance and the sympathy of the rational thermodynamics with Lagrange multipliers based on the general entropy inequality (3). On the other hand, it also raises the question of whether such a statement is valid in general. In this paper, we shall see indeed it is not true as stated, and we shall prove the entropy flux - heat flux relation (1) under some additional conditions. Those conditions are usually encountered alongside the condition (2) in most of the thermodynamic theories with Lagrange multipliers, but have seldom been reckoned as necessary for the assertion of the entropy flux - heat flux relation (1). 2 A Counter Example First we shall recall the definition of isotropic functions. Let L(V ) be the space of second order tensors on a three-dimensional vector space V, and O(V ) be the group of orthogonal tensors on V. We say that φ, h, and S are isotropic scalar, vector, and tensor functions respectively, if for any v V, A L(V ), we have φ(qaq T, Qv) = φ(a, v), h(qaq T, Qv) = Q h(a, v), S(QAQ T, Qv) = Q S(A, v) Q T, for any Q O(V ). The isotropic functions of arbitrary number of vectors and tensors are similarly defined. General solutions of isotropic functions are usually given in the so-called representation theorems available in the literature (see [10, 11]). 2

3 Let Φ and q be vector functions and Λ be a scalar function of three vector variables (u, v, w) given by Φ = γ u v, q = γ 1 u v, Λ = γ 2, (4) where γ = u v w is assumed not to vanish. One can easily verify that they are isotropic functions, 1 since for any orthogonal tensor Q, Qu Qv = (det Q) Q(u v), Qu Qv Qw = (det Q) (u v w), and they satisfy the following relations: + Φ ( j qi = Λ + q ) j, u j u i u j u i + Φ ( j qi = Λ + q ) j, (5) v j v i v j v i = Λ q i. w j w j However, from (4) it follows that Φ Λ q = 2γ u v. This is a counter example to the statement given in the introduction. Other counter examples can be constructed by giving Λ(γ) as an even function of γ and defining Φ = φ(γ) u v, q = κ(γ) u v, with φ(γ) and κ(γ) given by κ = α ( dλ ) 1, φ = Λ κ + α γ, dγ and α being a constant. They satisfy the relations (5) and Φ Λq = αγ u v. 3 Preliminary Lemmas For the proof of the main theorem, we shall first establish two simple lemmas. 1 Note that the expression Φ = (u v w) (u v) can also be written as Φ = ((w v)(v u) (w u)(v v)) u+((w u)(v u) (w v)(u u)) v+((u u)(v v) (v u) 2 )) w, confirmed to the usual representation theorem. 3

4 Lemma 1. Let F(Q) be a function (scalar-, vector-, or tensor-valued) defined on L(V ) and suppose that F(Q) = 0 for any Q O(V ). Then the gradient of F at the identity tensor is symmetric, i.e., for any skew symmetric W L(V ), Q F(1 )[W ] = 0. Proof. For W = W T and 0 < ε 1, since (1 +εw )(1 +εw ) T = 1 ε 2 W 2, the tensor 1 +εw is orthogonal to within second order terms o(2) in ε. In other words, there exist an Q W O(V ) such that 1 + εw = Q W + o(2). By the assumption we have F(Q W ) = 0 and F(1 ) = 0. Consequently, by the definition of the gradient, Q F is a linear transformation on L(V ) and F(1 + εw ) F(1 ) = Q F(1 )[εw ] + o(2), = F(Q W + o(2)) F(Q W ) = Q F(Q W )[o(2)] + o(2) = o(2), which implies that the first order term ε Q F(1 )[W ] must vanish, and the lemma is proved. Lemma 2. Let h(a, v) be an isotropic vector function. Then the function h satisfies the following relation, ( hi (δ ij h k δ ik h j ) = v k h ) ( i hi v j + A kl + h i A lk h i A jl h ) i A lj. v j v k A jl A lj A kl A lk Proof. If we define F(Q) = h(qaq T, Qv) Q h(a, v). Then it follows that Q F(1 )[W ] = A h[w A + AW T ] + v h[w v] W h. Since h(a, v) is an isotropic function, we have F(Q) = 0 for any Q O(V ), and hence by the Lemma 1 we obtain, in the terms of components, ( hi A kl + h i A lk + h ) i v k δ ij h k W jk = 0, A jl A lj v j for any skew symmetric tensor W. Therefore the lemma is proved. One can easily extend the above lemma to the case of an isotropic vector function of an arbitrary number of vectors and tensor variables, (δ ij h k δ ik h j ) = v ( hi v j v k h i v k v j ) + A ( hi A jl A kl + h i A lj A lk h i A kl A jl h i A lk A lj ), where the summations are taken over all the vector and the tensor variables respectively. 4

5 4 The Proportionality Theorem Now we shall restate the main problem in the following theorems, but first let us introduce the following abbreviations: k i = Φ i Λq i, H v ij = v j Λ q i v j, H A ijk = A jk Λ q i A jk. (6) Theorem 1. Let Φ and q be isotropic vector functions, and Λ be an isotropic scalar function, of an arbitrary number of vector and tensor variables. Assume that i) for N vector variables v a, a = 1,, N, ( Φi v a j + Φ ) ( j qi vi a Λ vj a ii) for every other vector variable u, + q ) j vi a = 0, (7) u j Λ q i u j = 0, (8) iii) for every tensor variable A, A jk Λ q i A jk = 0. (9) Then Λ is a constant and Φ = Λ q, holds, for N = 1 and for N = 2 with the assumption that q and v 1 v 2 be functionally independent 2 Proof. With the abbreviations (6), the assumption (i) implies that H va ij is skew symmetric, H va ij + H va ji = 0, (10) and the assumptions (ii) and (iii) give H u ij = 0, H A ikj = 0. 2 By functional independence of a function h from a set {v, u, } we mean the function h can not be expressed as a linear combination of {v, u, } with scalar functions depending on {v, u, } as coefficients not all vanishing simultaneously. See the concept of functional independence in the representation theorems [10]. 5

6 Applying Lemma 2 to the vectors Φ and q, we have (δ ij Φ k δ ik Φ j ) = v (δ ij q k δ ik q j ) = v ( Φi v j v k v k v j ) + A ( qi v j v k q i v k v j ) + A ( Φi A jl A kl + A lj A lk A kl A jl A lk A lj ), ( qi A jl A kl + q i A lj A lk q i A kl A jl q i A lk A lj ), where the summations are taken over all vector variables (including the vectors v a and all other vectors u) and all tensor variables respectively. Multiplying the second equation with Λ and subtracting it from the first one, we obtain by the use of the abbreviations (6) and the assumptions (ii) and (iii), (δ ij k k δ ik k j ) = (H a ij vk a H a ik vj a ), (11) in which for simplicity, we have written H a ij for Hva ij. This implies that k 1 = H a 12 v2, a k 2 = H a 23 v3, a k 3 = H a 31 v1, a (12) and the following system of six equations: (H a 12 v2 a + H a 31 v3) a = 0, (H a 23 v3 a + H a 12 v1) a = 0, (H a 31 v1 a + H a 23 v2) a = 0, (H a 12 v3 a + H a 31 v2) a = 0, (H a 23 v1 a + H a 12 v3) a = 0, (13) (H a 31 v2 a + H a 23 v1) a = 0, by the use of the relation (10). The three equations on the right of (13) reduce to H a 12 v3 a = 0, H a 23 v1 a = 0, H a 31 v2 a = 0. (14) We shall now proceed to prove the theorem for N = 1 and for N = 2 separately. 6

7 For N = 1, since v 1 0 in general, from (14) we obtain H 1 12 = 0, H 1 23 = 0, H 1 31 = 0. Therefore from (12) it follows that k must vanish, in other words, the relation Φ = Λ q holds. By the substitution of the relation Φ = Λ q into the conditions (7), (8), and (9), it follows immediately that the partial derivatives of Λ with respect to all the vector and the tensor variables must vanish, since in general the vector function q need not vanish, and the theorem is proved for N = 1. For N = 2, from (13) and (14) we have the following linear system of six equations for six variables (H 1 23, H 1 31, H 1 12, H 2 23, H 2 31, H 2 12), H 1 12 v H 1 31 v H 2 12 v H 2 31 v 2 3 = 0, H 1 23 v H 1 12 v H 2 23 v H 2 12 v 2 1 = 0, H 1 31 v H 1 23 v H 2 31 v H 2 23 v 2 2 = 0, (15) H 1 12 v H 2 12 v 2 3 = 0, H 1 23 v H 2 23 v 2 1 = 0, H 1 31 v H 2 31 v 2 2 = 0. The coefficient matrix of this system is of rank equal to 5, and the system admits a one-parameter solution given by H 1 23 v 2 1 = H1 31 v 2 2 = H1 12 v 2 3 = H2 23 v 1 1 = H2 31 v 1 2 = H2 12 v 1 3 = γ, which imply from (12) that k = Φ Λ q = γ (v 1 v 2 ), (16) where γ is a scalar function of the vector and the tensor variables. By the relation (16), the assumptions (ii) and (iii) lead to Λ X q i + γ X (v1 v 2 ) i = 0, where X stands for the components of of any vector variable u and any tensor variable A. Since q and v 1 v 2 are functionally independent by assumption, the above relations are possible only if both Λ and γ are independent of u and A. Therefore, Λ and γ are functions of v 1 and v 2 only, Λ = Λ(v 1, v 2 ), γ = γ(v 1, v 2 ). Consequently, we also have k = k(v 1, v 2 ). 7

8 On the other hand, since k = Φ Λ q is an isotropic vector function, it can be represented by, k = k 1 v 1 + k 2 v 2, or we have γ v 1 v 2 = k 1 v 1 + k 2 v 2, where k 1 and k 2 are isotropic scalar functions of (v 1, v 2 ). Taking inner product of this relation with v 1 v 2 we obtain γ (v 1 v 2 ) (v 1 v 2 ) = 0, which implies that γ must vanish. Therefore k = 0 and the relation Φ = Λ q holds. Finally, by the substitution of Φ = Λ q into (7), it follows that the partial derivatives of Λ with respect to v 1 and v 2 must vanish. Therefore Λ is independent of any vector and tensor variables. This completes the proof for N = 2. The assumption that q must be functionally independent from v 1 v 2 is physically reasonable, since one of the vectors v a is the temperature gradient, and usually the component of the heat flux vector q in the direction of the temperature gradient does not vanish. Corollary 1. Theorem 1 remains valid, i) if for any symmetric tensor variable A, the condition (9) is replaced by ( Φi + Φ ) ( j qi Λ + q ) j = 0; (17) A kj A ki A kj A ki ii) by if for any skew symmetric tensor variable W, the condition (9) is replaced ( Φi Φ ) ( j qi Λ q ) j = 0. (18) W kj W ki W kj W ki Proof. With the abbreviation (6), the assumptions becomes H A ikj = H A jki, H W ikj = H W jki, and since A is symmetric and W is skew symmetric, we also have H A ikj = H A ijk, H W ikj = H W ijk. 8

9 Combining these conditions, we can obtain H A ikj = H A jki = H A jik = H A kij = H A kji = H A ijk = H A ikj, H W ikj = H W jki = H W jik = H W kij = H W kji = H W ijk = H W ikj. Therefore, H A ikj and HW ikj must vanish. In other words, both conditions (17) and (18) imply the condition (9) when the tensor variable A is symmetric and skew symmetric respectively. For the case N > 2 and conditions more general than the ones given in the previous theorem, sometimes encountered in a more complicated constitutive class, we can prove the following theorem. Theorem 2. Let Φ and q be isotropic vector functions, and Λ be an isotropic scalar function, of an arbitrary number of vector and tensor variables. Assume that i) for vector variables v a, a = 1,, N, ( Φi vj a + Φ ) ( j qi vi a Λ vj a + q ) j vi a = 0, (19) ii) for every other vector variables u a, a = 1,, M, u a j Λ q i u a j = µ a δ ij, (20) iii) for every tensor variables W a, a = 1,, L W a jk Λ q i W a jk = λ a ε ijk, (21) in particular, if W is symmetric then it reduces to the condition (9). Then Φ i Λ q i = a,b=1 a<b γ ab (v a v b ) i + M µ a u a i + λ a ε ijk Wjk, a (22) where γ ab as well as µ a, λ a are scalar functions. Proof. By the use of Lemma 2 and the abbreviations (6), the present assumptions lead to (δ ij k k δ ik k j ) M = (H a ij vk a H a ik vj a )+ µ a (δ ij u a k δ ik u a j )+ 2λ a (ε ijl Wkl a ε ikl Wjl). a 9

10 Note that in the last term, the summation is taken over all skew symmetric tensor variables only. It implies that M k 1 = H a 12 v2 a + µ a u a 1 + M k 2 = H a 23 v3 a + µ a u a 2 + M k 3 = H a 31 v1 a + µ a u a 3 + 2λ a W23, a 2λ a W31, a 2λ a W12, a (23) and a system of six equations identical to the system (13). Therefore, the case for N 2 is already proved in Theorem 1. Now we shall prove the general case. Since H a ij is skew symmetric, it can be associated with an axial vector ha k by H a ij = ε ijk h a k, and the system (13) become (h a 3 v2 a + h a 2 v3) a = 0, (h a 1 v3 a + h a 3 v1) a = 0, (h a 2 v1 a + h a 1 v2) a = 0, (h a 3 v3 a + h a 2 v2) a = 0, (h a 3 v3 a + h a 1 v1) a = 0, (h a 2 v2 a + h a 1 v1) a = 0. This system of equations for h a k can be solved and it yields the following solutions, with the scalar functions γ ab = γ ba as parameters, h a k = γ ab vk, a b=1 Substitution of this relation into (23) gives immediately the result (22) and the theorem is proved. Corollary 2. Theorem 2 with the additional assumption that γ ab and λ a be constants implies Φ i Λ q i = M µ a u a i. (24) 10

11 Proof. Since k = Φ Λ q is an isotropic vector function of (v a, u a, W a ) it satisfies k(qv a, Qu a, QW a Q T ) = Q k(v a, u a, W a ), for any orthogonal tensor Q, and from (20) µ a are isotropic scalar functions. With Qv a Qv b = (det Q) Q (v a v b ) and ε ijk Q ip Q jq Q kr = (det Q) ε pqr, we obtain from (22), a,b=1 a<b γ ab v a v b + 2λ a w = (det Q) ( a,b=1 a<b γ ab v a v b + 2λ a w), where w is the axial vector associated with the skew symmetric tensor W. By taking Q = 1, it follows that a,b=1 a<b γ ab v a v b + 2λ a w = 0, and the corollary is proved. We have not given any conditions sufficient to guarantee the constancy of those scalar functions in the last theorem as we have done in Theorem 1, since such conditions are assumptions on constitutive functions and they can be made more realistic and understandable in the specific problem. 5 Remarks on Some Previous Theories In the exploitation of the entropy inequality with Lagrange multipliers, conditions of the form (7), namely only the symmetric parts vanish, are encountered for vector variables which are the gradients of some scalar functions, for example the temperature gradient. For other vector variables usually conditions of the form (8) are obtained. On the other hand, for tensor variables the conditions (9) (or the condition (17) for symmetric tensor variables) are also encountered usually in the process. Therefore, although we have proved the theorem with more assumptions than the one in the original statement put forth in the introduction, Theorem 1 would be applicable to many material classes whose constitutive functions depend on not more than two vector variables for the conditions of the form (7) (N 2). In the following, we shall comment on some of the previous theories with the application of the present theorem to the assertion of the proportionality between the entropy flux and the heat flux. 11

12 5.1 Theories with one or two vector variables For the theory of rigid heat conductors [5] with only one constitutive vector variable, the temperature gradient, and for the theory of a non-simple fluid [4] with two constitutive vector variables, also the density gradient, the entropy flux - heat flux relation (1) can be proved in a straightforward manner using the general isotropic representation theorems. Theorem 1 is also adequate. The first works which raised this typical problem of entropy flux - heat flux relation are the two papers by Müller [2, 3], they involved temperature gradient and a symmetric tensor variable, the rate of strain tensor for viscous fluids and the Cauchy Green tensor for isotropic elastic solids. The proof of the proportionality was not adequate in the original papers. 3 Both cases satisfy the assumptions of Theorem 1, and hence it provides a proof of their assertions. The proof of entropy flux - heat flux relation in a thermodynamic theory with linear constitutive equations is usually trivial and straightforward. The viscous heat-conducting fluid considered in [6] is characterized by two vector and a symmetric tensor variables, namely, the density gradient, the energy gradient, and the rate of strain tensor. In this case, Theorem 1 would be adequate to deduce the entropy flux - heat flux relation in general were it necessary, instead of restricting to linear constitutive equations only. 5.2 Theories of materials in electromagnetic fields The thermodynamic theory of fluids in electromagnetic fields formulated in [8] involved three vector variables: the temperature gradient g i = (grad θ) i, the electromotive intensity E i, and the magnetic flux density B i. The vector B i is an axial vector quantity, which can be represented as a skew symmetric tensor B with components given by B ij = ε ijk B k. Among the conditions obtained from the exploitation of the entropy principle, there are 4 Φ (i Λ Q (i = 0, g j) g j) Φ (i Λ Q (i = 0, (25) E j) E j) Λ Q i = λ ε ijk. B jk B jk It has been proved in [7] that the relations (25) imply that Φ i = Λ Q i as well as λ = 0, and Λ does not depend on (g, E, B). The proof is based on the polynomial isotropic representations and series expansions and is very long and tedious. The following is a proof by the use of Theorem 2. 3 In [2] and [3] the seemingly trivial arguments were later found to be insufficient. It has been correctly restated in p. 256 of [12] but the details of the proof were not given. 4 The parentheses around the two indices indicate the symmetrization, e.g., X (ij) = 1 2 (X ij+ X ji ). 12

13 Proof. The conditions (25) satisfy the assumptions of Theorem 2 with N = 2, M = 0, and L = 1, therefore we can conclude immediately that k i = Φ i Λ Q i = γ (g E) i + 2λ B i. (26) From (25) 3 and (26), we have Λ Q i + 2 λ B i + γ (g E) i = 0, B jk B jk B jk which imply Λ B jk = 0, λ B jk = 0, γ B jk = 0, if we assume that {Q, B, g E} is functionally independent, which essentially means that the constitutive vector quantity Q can not be parallel to B or to g E. Moreover, with arguments similar to that used in the proof of Theorem 1, we can conclude that Λ, λ, and γ are independent of (g, E, B). Furthermore, by Corollary 2, since λ is a constant, it follows that Φ i = Λ Q i and λ = 0. Generalizations of this theory have been considered in [9]. In that theory two more symmetric tensor variables were involved, namely, the Cauchy Green and the rate of Cauchy Green tensors, and the relation Φ i = Λ Q i was taken as an assumption. Since the additional tensor variables satisfy the condition (17), the above analysis is sufficient for the proof of the assumption. 5.3 Theories of mixtures Theories of mixtures are usually characterized by a large set of constitutive variables including the temperature gradient and the relative velocities of the constituents, and more generally also density gradients of each constituent and other tensor variables. In [13], the theory of a simple mixture of ν constituents involved ν vector variables, namely, the temperature gradient g and relative velocities u a for a = 1,, ν 1. Conditions of the following form are encountered, Φ (i g j) Λ q (i g j) = 0, u a j Λ q i u a j = µ a δ ij, which satisfy the assumptions of Theorem 2 for N = 1, M = ν 1, and L = 0. Therefore, we obtain immediately that ν 1 Φ i = Λ q i + µ a u a i, 13

14 which is proved in [13] under the assumption that the constitutive functions are linear in the vector variables. With Theorem 2 we do not need such an assumption. To prove that Λ and µ a are independent of (g, u 1,, u ν 1 ), it is sufficient to assume that q and {u 1,, u ν 1 } are functionally independent. References [1] Liu, I-Shih: Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Rational Mech. Anal., 46, (1972) [2] Müller, I.: Die Kältefunktion, eine universelle Funktion in der Thermodynamik viskoser wärmeleitender Flüssigkeiten, Arch. Rational Mech. Anal., 40, 1-36 (1971) [3] Müller, I.: The coldness, a universal function in thermoelastic bodies, Arch. Rational Mech. Anal., 41, (1971) [4] Liu, I-Shih: A non-simple heat-conducting fluid, Arch. Rational Mech. Anal., 50, (1973) [5] Batra, R. C.: A thermodynamic theory of rigid heat conductors, Arch. Rational Mech. Anal., 53, (1974) [6] Liu, I-Shih: On Fourier s law of heat conduction, Continuum Mech. Thermodyn., 2, (1990) [7] Liu, I-Shih: On irreversible thermodynamics, Dissertation, The Johns Hopkins University, Baltimore (1972) [8] Liu, I-Shih, & Müller, I.: On the thermodynamics and thermostatics of fluids in electromagnetic fields, Arch. Rational Mech. Anal., 46, (1972) [9] Hutter, K.: On thermodynamics and thermostatics of viscous thermoelastic solids in electromagnetic fields, Arch. Rational Mech. Anal., 58, (1975) [10] Wang, C.-C.: A new representation theorem for isotropic functions, Part I and II, Arch. Rational Mech. Anal., 36, (1970), Corrigendum, 43, (1971) [11] Smith, G. F.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors, Int. J. Engng Sci., 9, (1971) [12] Müller, I.: Thermodynamics, Pitman Publishing, London (1985) [13] Müller, I.: A new approach to thermodynamics of simple mixtures, Zeitschrift für Naturforschung, 28, (1973) 14