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Oldroyd fluids with hyperbolic heat conduction
Accepted Manuscript Title: Oldroyd Fluids with Hyperbolic Heat Conduction Author: Martin Ostoja-Starzewski Roger Khayat PII: DOI: Reference:
S0093-6413(17)30205-7 http://dx.doi.org/doi:10.1016/j.mechrescom.2017.07.012 MRC 3197
To appear in: Received date: Revised date: Accepted date:
21-4-2017 24-7-2017 26-7-2017
Please cite this article as: Martin Ostoja-Starzewski, Roger Khayat, Oldroyd Fluids with Hyperbolic Heat Conduction, (2017), http://dx.doi.org/10.1016/j.mechrescom.2017.07.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Oldroyd Fluids with Hyperbolic Heat Conduction
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Martin Ostoja-Starzewski Department of Mechanical Science & Engineering also Institute for Condensed Matter Theory and Beckman Institute University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.
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Roger Khayat Department of Mechanical and Materials Engineering The University of Western Ontario London, Ontario, Canada N6A 5B9
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July 24, 2017
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Keywords: Oldroyd fluids; hyperbolic heat conduction; thermal relaxation times; primitive thermodynamics; thermodynamic orthogonality
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Abstract Development of constitutive equations of Oldroyd fluids with hyperbolic heat conduction, directly from the free energy and dissipation functions, is the focus of this work. Two models from hyperbolic thermoelasticity of solids − theory with one thermal relaxation time and theory with two thermal relaxation times − provide a stepping-stone for this task. The setting for these models is offered, respectively, by the primitive thermodynamics of Edelen and the thermodynamic orthogonality of Ziegler, both cases involving an internal parameter: the inelastic strain. The identified free energy and dissipation functions show that, in the first case, the hyperbolic (telegraph-like) heat conduction is governed by the Maxwell-Cattaneo model involving one thermal relaxation time, whereas, in the second case, by the Fourier law and two thermal relaxation times in the constitutive laws for stress and entropy. In both cases, the thermal fields vanish from the mechanical equations as the Oldroyd fluid becomes incompressible, whereas the mechanical fields are present in the telegraph-type equations for temperature.
1
Motivation
The Oldroyd fluid is specified by the (mechanical) constitutive equation [1] σ + λ1 σ˙ = 2η0 d + 2η0 λ2 d˙
σij + λ1 σ˙ ij = 2η0 dij + 2η0 λ2 d˙ij ,
or
(1)
1
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5
σ=
∂σ T + v · ∇σ − L·σ − (L·σ) , ∂t
cr
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where λ1 is the relaxation time, λ2 (= λ1 ηs /η0 ) the retardation time, and η0 (= ηs + ηp ) the total fluid viscosity composed of a solvent’s viscosity (ηs ) and polymer’s viscosity (ηp ). We are focusing on incompressible fluids, so that σ (or σij in index notation) is the deviatoric part of the Cauchy stress, while d is the deviatoric part of the deformation rate (v(i ,j) ); the spherical part of the Cauchy stress is the negative of pressure. The overdot on σ and d denotes the upper-convected time derivative, e.g.
(2)
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d
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an
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which indicates our focus on the Oldroyd-B fluid; here L = ∇v (i.e., Lij = vj ,i ) is the velocity gradient, e.g. [2]. The overdot on a scalar quantity signifies a material time derivative. The somewhat more general picture of the overdot standing for a convex combination of the upper-convected and lower-convected derivatives (eventually leading to the Oldroyd-A fluid) is not being considered here, but can be treated by the same approach as outlined in this paper. As is well known, various liquids (e.g. bloods, paints) are best modeled as Oldroyd-B. Our ongoing experiments indicate that porcine blood transfers heat in a hyperbolic way with thermal relaxation time on the same order as the mechanical times λ1 and λ2 . This is an example of challenge where a generalization of (1) to a thermo-fluid model is needed. Simply attaching to (1) the thermal conductivity model of either a Fourier or Maxwell-Cattaneo type is an ad hoc way to postulate the heat transfer to be either parabolic or hyperbolic. However, as is well known, the parabolic heat conduction suffers from the paradox of infinitely fast propagation of heat and this motivates development of hyperbolic models. In this study we derive such models from an energetic-thermodynamic standpoint. We proceed through the thermomechanics with internal variables (TIV) which, fundamentally, bases the entire constitutive behavior on two functionals: free energy and dissipation function. In essence, the challenge is to find a pair of two such functionals which will allow a systematic derivation of simplest thermal-Oldroyd models with restriction to linear phenomena. The guidance is provided by two well established models in thermoelasticity: the Lord-Shulman model [3] (also called a theory with one relaxation time) and that of Green-Lindsay [4] (also called a theory with two relaxation times); see [5] for a comprehensive picture. The Lord-Shulman model involves these constitutive equations σij = Cijkl kl + Mij ϑ, θ0 s = −θ0 Mij ij + CE ϑ,
(3)
qi + t0 q˙i = −kij ϑ,j , where t0 > 0 is the relaxation time and ϑ := θ − θ0 is the temperature difference from the reference temperature θ0 . Also, σij is the whole (not just deviatoric) 2
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Cauchy stress, Cijkl the stiffness tensor, kl the infinitesimal strain, Mij the stress-temperature tensor, s the entropy density, CE the specific heat at zero strain, and kij the thermal conductivity tensor, respectively. These quantities satisfy the following symmetry, positivity, and positive-definiteness relations
CE > 0,
aij Cijkl akl > 0, kij = kji ,
∀a 6= 0,
bi kij bj > 0,
∀b 6= 0.
(4)
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Mij = Mji ,
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Cijkl = Cjikl = Cijlk = Cklij ,
Let ui denote the displacement. The resulting system of coupled displacementtemperature field equations is
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(Cijkl uk ,l ),j + (Mij ϑ) ,j = ρ¨ ui , (kij ϑ,j ) ,i +θ0 Mij (u˙ i ,j +t0 u ¨i ,j ) = CE ϑ˙ + t0 ϑ¨ ,
(5)
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which shows that the heat conduction is governed by a telegraph-type equation. The Green and Lindsay model involves the constitutive relations ˙ σij = Cijkl kl + Mij (ϑ + t1 ϑ) (6)
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˙ θ0 s = −θ0 Mij ij + CE (ϑ + t0 ϑ)
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qi = −kij ϑ,j ,
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where t0 and t1 are two relaxation times, satisfying the inequalities t1 ≥ t0 > 0. The relaxation time t0 in (6) is, in general, different from that in (3). The remaining symbols appearing in (6) have the same meaning as in the LordShulman model and are subject to the restrictions (4). It is important to note that, even though the Fourier law holds, the heat conduction process itself is hyperbolic (not parabolic!) [5]. This is seen from the second (telegraphtype) equation in the resulting system of coupled displacement-temperature field equations is ˙ j = ρ¨ (Cijkl uk ,l ),j +[Mij (ϑ + t1 ϑ)], ui , ¨ (kij ϑ,j ),i +θ0 Mij u˙ i ,j = CE (ϑ˙ + t0 ϑ).
(7)
In light of the above, for a thermal-Oldroyd fluid there will likely be two types of relaxation times: mechanical (λ1 , λ2 ) and thermal (t0 , t1 ); this notation shall be kept throughout this paper. Also note that both thermoelasticity models can be reformulated within the TIV framework in terms of the free energy and the dissipation function rather then only from the free energy as done in the original papers [3, 4]. Such a reformulation allows one to arrive at, say, viscothermoelastic solids with hyperbolic heat conduction [6]. That reformulation was motivated by the fact that the Fourier-type heat conduction is a dissipative 3
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Oldroyd fluids with one thermal relaxation time
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process, and thus should be described by the dissipation function rather than by the free energy function as in [4]. In the present study, we want to introduce thermal effects in the Oldroyd (i.e. viscoelastic) fluid by proceeding from TIV, thereby extending the range of viscous fluids that can thus be derived [7, 8] and a more general TIV-type theory [9].
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In order to derive the constitutive equations of a viscothermoelastic fluid with one relaxation time, we begin in spatial (Eulerian) description with the specific (per unit volume) internal energy u as a function of the entropy s, the heat flux qi , the strain ij , and the internal variable αij : u = u(ij , αij , s, qi ).
(8)
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Since we focus on incompressible fluids, σij will stand for deviatoric Cauchy stress, just as introduced in the Motivation. The specific power of deformation is l = σij ˙ij and the classical relation ψ = u − θs
(9)
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holds, whereby, in view of (8) and, by a partial Legendre transformation switching the argument s to the absolute temperature θ, we recognize that and
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ψ = ψ(ij , αij , θ, qi )
s = s(ij , αij , θ, qi , ).
(10)
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The first fundamental law (energy balance) has the form in spatial (Eulerian) description ρu˙ = σij ˙ij − qi ,i , (11) R R qi while the second fundamental law (starting from sdV ˙ ≥ − θ ni dA) s˙ ≥ −
q i
θ
,i =
qi θ,i qi ,i − 2 θ θ
(12)
is written in terms of the reversible (s∗(r) ) and irreversible (s∗(i) ) parts of entropy production rate (s) ˙ q i s˙ = s∗(r) + s∗(i) with s∗(r) = − ,i and s∗(i) ≥ 0. (13) θ The material time derivative of the free energy is found on the one hand from (9) as ˙ ψ˙ = u˙ − θs˙ − θs, (14) and, on the other hand from (10)1 , as ∂ψ ∂ψ ˙ ∂ψ ∂ψ ψ˙ = ˙ij + α˙ ij + θ+ q˙i . ∂εij ∂αij ∂θ ∂qi
(15)
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(q)
The latter, on account of the standard definitions of external (σij ) and internal (q)
(βij ) quasi-conservative stresses and entropy (s)
becomes (q)
(q)
ρψ˙ = σij ˙ij + βij α˙ ij − sθ˙ + ρ
∂ψ q˙i , ∂qi
leading to
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∂ψ ∂ψ ∂ψ (q) , βij = ρ , s=− , ∂ij ∂αij ∂θ
(16)
(17)
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(q)
σij = ρ
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∂ψ (q) (q) ρψ˙ + ρsθ˙ = ρu˙ − ρθs˙ = σij ˙ij + βij α˙ ij + ρ q˙i . ∂qi
(18)
With the external and internal dissipative stresses defined by (d)
(q)
(d)
(q)
the second equality in (18) becomes (d)
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σij = σij − σij , βij = −βij ,
(d)
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ρu˙ − ρθs˙ = σij ˙ij − σij ˙ij − βij α˙ ij + ρ
∂ψ q˙i . ∂qi
(19)
(20)
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On account of the first fundamental law (ρu˙ = σij ˙ij − qi ,i ), this is equivalent to ∂ψ (d) (d) ρθs˙ = σij ˙ij + βij α˙ ij − qi ,i −ρ q˙i (21) ∂qi or 1 (d) qi θ,i qi ∂ψ 1 (d) ,i −ρ ρs˙ = σij ˙ij + βij α˙ ij − 2 − q˙i (22) θ θ θ θ ∂qi
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By inserting the above into (12), we obtain the Clausius-Duhem inequality in the form ∂ψ qi θ,i (d) (d) −ρ q˙i = ρθs∗(i) ≥ 0, (23) σij ˙ij + βij α˙ ij − θ ∂qi
In any fluid having an isotropic response, the free energy function ψ can, in general, depend on all the basic invariants of ij and αij . Since the elastic spring-back is involved in the response of the Oldroyd fluid [2], ψ is taken as a functional of the difference ij − αij . Since the constitutive equation involving stress (and its rate) and deformation rate (and its rate) is linear, ψ is a quadratic functional of (ij − αij ). Thus, ρψ per unit volume (i.e. ψ being the free energy per unit mass) is taken as ρψ(ij , αij , θ, qi ) = ρψ0 − ρs0 (θ − θ0 ) + µ(ij − αij )2 C t0 2 + Mij ij (θ − θ0 ) − (θ − θ0 ) + qi qi 2θ0 2θ0 κ
(24)
Here ψ0 and s0 are, respectively, the free energy and entropy in the reference state, µ is the elastic shear modulus, Mij is the stress-temperature tensor (effectively accounting for the thermal expansion), C is the specific heat at zero 5
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strain, t0 is the thermal relaxation time, and κ is the Fourier-type thermal conductivity. By hyperelasticity we obtain ∂ψ = 2µ(ij − αij ) + Mij (θ − θ0 ) ∂ij ∂ψ (q) βij = ρ = 2µ(αij − ij ) ∂αij C ∂ψ = ρs0 − Mij ij + (θ − θ0 ) ρs = −ρ ∂θ θ0 ∂ψ t0 −ρ =− qi . ∂qi θ0 κ
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(q)
σij = ρ
(q)
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(25)
(q)
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Thus, (25)1 and (25)2 give σij and βij , respectively, while (25)3 is immediately identified as the constitutive equation for entropy. The significance of (25)3 (q) will be clarified further below. The stress σij models the spring-back effect in the viscoelastic model [2]. The spring-back strain is very small, so that (q) (q) σij ˙ij ' σij dij , which justifies our replacing ˙ij by the deformation rate dij below and the use of µ in (24). Also, just like σij is the deviatoric part of the (q) (q) Cauchy stress, σij and βij are the deviatoric parts of the external and internal quasi-conservative stresses. The irreversible part of entropy production rate is a function of the heat flux, its rate q˙i , and the strain rate ε˙ij , and the internal parameter rate α˙ ij : ρθs∗(i) = ρθs∗(i) (dij , α˙ ij , qi , q˙i ).
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(26)
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Clearly, the inequality (23) may be written as a scalar product Y · V ≥ 0,
(27)
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whereby we have defined
Y ≡ (y1 , y2 , y3 , y4 ) :=
∂ψ θ,i (d) (d) σij , βij , − , − θ ∂qi
(28)
as the vector of dissipative thermodynamic forces. Thus, we have defined the four sub-vectors of Y, while its conjugate flux is V ≡ (V1 , V2 , V3 , V4 ) := (dij , α˙ ij , qi , q˙i ) ,
(29)
in which we have defined four sub-vectors vi of v, the vector of its conjugate thermodynamic velocities. A general procedure based on the representation theory in the so-called primitive thermodynamics due to Edelen [9] allows a derivation of the most general form of the constitutive relation either for V as a function of Y or for Y as a function of V, subject to (27). Thus, we first introduce a functional φE (Edelen’s dissipation) which depends on dij , α˙ ij , qi , q˙i according to (26), and take Y(V) as the gradient in the velocity space plus an extra vector U: Y = ∇V φE + U,
or
Yi =
∂φE + Ui . ∂Vi
(30)
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Here the vector U = (u1 , u2 , u3 , u4 ) does not contribute to the entropy production according to the condition U · V = 0,
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(31)
∂ [Yi (τ V) − Ui ] ∂ [Yj (τ V) − Uj ] = ∂vj ∂vi
cr
and hence R 1 is called a powerless dissipative force, while the dissipation function is φE = 0 V · Y(τ V)dτ . Also, the symmetry relations (32)
iff
U = 0.
(33)
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∂Yj (τ V) ∂Yi (τ V) = ∂vj ∂vi
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must hold, and these reduce to the classical Onsager reciprocity conditions
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In order to derive a model with one thermal relaxation time (i.e. MaxwellCattaneo type), we generalize the formulation of [6] by taking the dissipative vector U = (U1 , U2 , U3 , U4 ) (corresponding to Y and V above) as, in components, t0 t0 U1 = 0 U2 = 0 U3i = q˙i U4i = − qi . (34) θκ θκ The dissipation function φE of an isotropic fluid can, in general, depend on all the basic and mixed invariants of dij , α˙ ij , qi , and q˙i . To have a fluid model linear in heat conduction as well as mechanically linear, φE must simply be a quadratic form in dij , α˙ ij , qi , and q˙i , so that, in view of (19), ρφE (V) ≡ ρφE (dij , α˙ ij , qi , q˙i ) =
H F 1 G dij dij + α˙ ij α˙ ij + qi qi + q˙i q˙i . (35) 2 2 2θκ 2
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Note that φE cannot depend on qi q˙i , qi dij qj , qi dij djk qk , q˙i dij q˙j , but such invariants might appear in a nonlinear model. Now, on account of (30), it follows that ∂φE + u1ij = Hdij ∂dij ∂φE (d) + u2ij = F α˙ ij βij ≡ Y2ij = ρ ∂ α˙ ij θ,i ∂φE 1 t0 − ≡ Y3i = ρ + u3i = qi + q˙i θ ∂qi θκ θκ t0 ∂ψ ∂φE t0 − qi = −ρ ≡ Y4i = ρ + u4i = Gq˙i − qi . θ0 κ ∂qi ∂ q˙i θκ (d)
σij ≡ Y1ij = ρ
(36)
First, by (19)1 , the total stress is σij = 2µ(ij − αij ) + Mij ϑ + Hdij .
(37)
F α˙ ij = −2µ(ij − αij ).
(38)
while by (19)2
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Taking the second-order time derivative of (37) − effectively being the upperconvected derivative of σij and dij (recall the statement following (2)) and the material time derivative of ϑ − we obtain σ˙ ij = 2µ(˙ij − α˙ ij ) + Mij ϑ˙ + H d˙ij .
(39)
σij +
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Here we recall that, one needs to interpret both time derivatives in (1) not as a conventional material derivative but an upper-convected one. Thus, the overdot in (39) needs to be interpreted as the Lie (i.e., the Oldroyd upper-convected) ¨ In the case of a rigid derivative [8]. An analogous argument applies to ϑ˙ and ϑ. conductor, the overdot reduces to the conventional material time derivative as shown by the requirement of Galilean invariance [10, 11, 12]. Now, eliminating the internal variable αij from (37), (38), and (39), we arrive at the constitutive equation linking stress (and its rate) with strain (and ˙ its rate) and with temperature difference ϑ := θ − θ0 and its rate ϑ: FH ˙ F F σ˙ ij = (F + H) dij + dij + Mij ϑ + Mij ϑ˙ 2µ 2µ 2µ
or, in symbolic notation,
F FH ˙ F ˙ σ˙ = (F + H) d + d + Mϑ + M ϑ. 2µ 2µ 2µ
M
σ+
(40)
(41)
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This is the Oldroyd viscoelastic fluid model directly coupled with the temperature field and its rate. With reference to (1), we now find the energetic-thermodynamic interpretation of the conventional constitutive coefficients in the Oldroyd model: λ1 =
F , 2µ
2η0 = F + H,
2η0 λ2 =
FH . 2µ
(42)
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Note that only the deviatoric part of Mij enters the constitutive equation (40). Thus, when Mij is taken as an isotropic tensor Mij = M δij with M = 0, the coupling between mechanical and thermal fields vanishes and we recover the ˙ classical Oldroyd fluid equation relating σ, σ, ˙ d, and d. The formulation is completed by setting G = 0 and θ ' θ0 , thus assuring the identity in (36)4 . The approximation of small temperature changes is the same as in the Lord-Shulman thermoelasticity model (3); see also equations (R.1.1)-(R.1.11) in [5]. According to (36)3 , the heat flux and temperature are related through the Maxwell-Cattaneo model qi + t0 q˙i = −κϑ,i . (43) indicating that the temperature field is governed by a telegraph equation for temperature with extra nonlinear mechanical terms which have a very complicated expression; we do not reproduce that equation here. It is simple to show that, in the special case of F = 0 and t0 = 0, that equation reduces to the ˙ which is the form of energy equaparabolic equation κϑ,ii +Hdij dij = ρC ϑ, tion used for Newtonian-Fourier fluids; C is the specific heat under constant strains and internal parameters. 8
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Oldroyd fluids with two thermal relaxation times
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The internal energy u is now taken as a function of the strain ij , the internal variable αij , and the entropy s: (44)
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u = u(ij , αij , s). The specific power of deformation is
As before, switching from u to ψ = u − θs, we find and
s = s(ij , αij , θ).
(46)
an
ψ = ψ(ij , αij , θ)
(45)
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l = σij ˙ij /ρ.
(47)
qi θ,i = ρθs∗(i) ≥ 0. θ
(48)
The relations (9), (11), (12), (13), (16), and (19) hold as before, so that (d)
(d)
ρu˙ − ρθs˙ = σij ˙ij − σij ˙ij − βij α˙ ij ,
(d)
(d)
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while the Clausius-Duhem inequality takes the form σij ˙ij + βij α˙ ij −
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te
d
Similar to the model with one relaxation time, in any fluid having an isotropic response, the free energy function ψ can, in general, depend on all the basic invariants of ij and αij . Since the spring-back is involved in the response of the Oldroyd fluid, ψ is taken as a functional of the difference ij − αij . Since the constitutive equation involving stress (and its rate) and deformation rate (and its rate) is linear, ψ is a quadratic functional of (ij − αij ). Thus, ρψ per unit volume (i.e. ψ being the free energy per unit mass) is taken as ρψ(ij , αij , θ) = ρψ0 − ρs0 (θ − θ0 ) + µ(ij − αij )2 C C 2 ˙ (49) + Mij ij (θ − θ0 ) − (θ − θ0 ) − t0 (θ − θ0 ) θ. 2θ0 θ0
Here ψ0 and s0 are the free energy and entropy in the reference state, µ is the shear elastic modulus, Mij is the stress-temperature (effectively accounting for the thermal expansion), C is the specific heat at zero strain, and t0 is the thermal relaxation time. Again invoking the hyperelasticity property, from the free energy, we find ∂ψ = 2µ(ij − αij ) + Mij (θ − θ0 ) ∂ij ∂ψ (q) βij = ρ = 2µ(αij − ij ) ∂αij ∂ψ C C ˙ ρs = −ρ = ρs0 − Mij ij + (θ − θ0 ) + t0 θ. ∂θ θ0 θ0 (q)
σij = ρ
(50)
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cr
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The relation (50)3 is immediately identified as the constitutive equation for (q) (q) entropy, exactly the same as (6)2 . As in the previous section, here σij and βij are the deviatoric parts of the external and internal quasi-conservative stresses, the corresponding spherical parts being zero in view of the incompressibility of fluid. Also, the same argument as in the paragraph following eq. (25) allows a replacement of the strain rate ˙ij by dij . Clearly, the inequality (48) may be written as a scalar product1 Y · V ≥ 0, whereby we have defined Y ≡ (y1 , y2 , y3 ) :=
(d)
(d)
us
(51)
σij , βij , −
θ,i θ
(52)
an
as the vector of dissipative thermodynamic forces. Corresponding to the three sub-vectors of Y there are three conjugate velocity sub-vectors V ≡ (V1 , V2 , V3 ) := (dij , α˙ ij , qi ) .
M
(53)
The dissipation function ϕZ in the sense of Ziegler [7] (as opposed to φE of Edelen) is taken equal to the irreversible entropy production rate and, since we are aiming at linear processes, is adopted as quadratic form
d
ρθs∗(i) ≡ ρϕZ (V) ≡ ρϕZ (dij , α˙ ij , qi )
te
1 = Hdij dij + 2t1 Mij dij ϑ˙ + F α˙ ij α˙ ij + qi qi , (54) θκ
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Next, recall the thermodynamic orthogonality (or the maximum entropy production principle) [7, 8] which gives Y as a gradient in the space of velocities V (without any powerless dissipative force): Y=
1 ∇V ϕZ , 2
or
Yi =
with the factor 1/2 determined by the condition result, we obtain (d)
1 ∂ϕZ , 2 ∂Vi 1 2
(55) −1
= ϕZ (Vi ∂ϕZ /∂Vi )
1 ∂ϕZ ρ = Hdij + t1 Mij ϑ˙ 2 ∂dij 1 ∂ϕZ (d) βij ≡ Y2ij = ρ = F α˙ ij 2 ∂ α˙ ij θ,i 1 ∂ϕZ 1 − ≡ Y3i = ρ = qi . θ 2 ∂qi θκ
. As a
σij ≡ Y1ij =
(56)
The final constitutive relations follow by noting from (19)2 that 2µ(ij − αij ) = F α˙ ij ,
(57)
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while (50)1 and (56)1 yield the total (deviatoric) stress ˙ σij = 2µ(ij − αij ) + Mij ϑ + Hdij + t1 Mij ϑ.
ip t
(58)
Taking the time derivative of (58) with respect to time (as before, recall the statement following (2)) we find
(59)
cr
¨ σ˙ ij = 2µ(dij − α˙ ij ) + Mij ϑ˙ + H d˙ij + t1 Mij ϑ,
FH ˙ F σ˙ ij = (F + H) dij + dij 2µ 2µ
an
σij +
us
where − by applying the same argument as in (39) − the overdots on secondrank tensors signify the upper-convected derivatives. Eliminating the internal variable and its rate from (57)-(59), we arrive at the constitutive equation linking stress (and its rate) with strain (and its rate) with temperature difference ϑ := θ − θ0 (and its first and second rates)
or, in symbolic notation, σ+
F FH ˙ σ˙ = (F + H) d + d 2µ 2µ
M
F F ˙ ϑ + t1 Mij ϑ¨ (60) + Mij ϑ + t1 Mij + Mij 2µ 2µ
d
F F ¨ + Mϑ + t1 M + M ϑ˙ + t1 M ϑ. (61) 2µ 2µ
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te
Since the mechanical terms in the above are the same as in (40)-(41), we find the energetic-thermodynamic interpretation (42) of the conventional constitutive coefficients of the Oldroyd model to also apply here. However, the presence of ϑ¨ distinguishes this model from that with one relaxation time obtained in Section 2. Also, when Mij is taken as an isotropic tensor with M = 0, the coupling between mechanical and thermal fields in (60) vanishes and we recover ˙ the classical Oldroyd fluid equation relating σ, σ, ˙ d, and d. Turning to the third relation in (56), we see that it immediately gives the Fourier law for the isotropic medium, just like (6)3 , −κθ,i = qi .
(62)
In order to derive the equation governing the temperature, assume Mij to be an (d) isotropic tensor and start from (50)1 . Next, note that σij dij = Hdij dij , while from (19), (56)2 , and (57) we find (q)
(d)
F α˙ ij = 2µ ( − α) = σij = σij − σij = σij − Hdij
(63)
(d)
which implies βij α˙ ij = F α˙ ij α˙ ij = (σij − Hdij ) (σij − Hdij ) F −1 . Hence, on account of (62) and the energy balance (11), κϑ,ii = −Hdij dij − (σij − Hdij ) (σij − Hdij ) F −1 + ρC ϑ˙ + t0 ϑ¨ , (64) 11
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Conclusions
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which is a hyperbolic equation for the temperature difference ϑ (and, equivalently, for θ), even though the constitutive law of heat conduction is Fourier. The above equation is analogous to (7)2 in that it is coupled with the mechanical field. In the special case of F = 0 and t0 = 0, on account of (63), the hyperbolic ˙ again a equation (64) reduces to the parabolic equation κϑ,ii +Hdij dij = ρC ϑ, form of energy equation for Newtonian-Fourier fluids − the same as mentioned at the end of Section 2.
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Adopting the thermomechanics with internal variables, we have derived the Oldroyd fluid equation with hyperbolic heat conduction from the standpoint of second law of thermodynamics. In particular, we have explored two basic ways for an Oldroyd fluid to possess linear damped hyperbolic heat conduction:
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1. The dissipation function φE following Edelen’s primitive thermodynamics approach, which does not equal ρθs∗(i) but (i) leads to the MaxwellCattaneo relation between heat flux and temperature gradient and (ii) results in a thermomechanical model with one thermal relaxation time. This is a thermo-fluid analogue of the “thermoelasticity with one relaxation time” [3].
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2. The dissipation function ϕZ following Ziegler’s thermodynamic orthogonality, which equals ρθs∗(i) but (i) leads to a Fourier model of heat conduction with temperature being governed by a telegraph equation (3.21) with extra nonlinear mechanical terms, and (ii) results in a thermomechanical model with two thermal relaxation times. It may appear puzzling at first sight to have a Fourier model with a hyperbolic field equation, but a completely analogous picture occurs in the “thermoelasticity with two relaxation times” [4].
The presence of mechanical terms in either of the temperature equations is unavoidable because it is, basically, an energy equation, whereas the temperature terms do not enter the mechanical equations of either model because of the incompressibility. Indeed, given that Mij is typically isotropic (i.e., temperature is uncoupled from shear stresses), the Oldroyd mechanical equation is practically uncoupled from the thermal equation. Overall, both models derived here fall within the class of hyperelastic and hyperdissipative media [13, 14]. Various simpler thermo-fluid models are obtained as special cases of both models given here. For example, upon setting F = 0, we recover, in both models, the Newtonian fluid with independent Fourier heat conduction. Overall, while many different material behaviors may be obtained from Ziegler’s maximum entropy production principle (MEPP) (e.g. [15], [16]), some
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systems may actually require a more general primitive thermodynamics (of Edelen). Expanding on [7] and [8], the key role is played by two functionals (of free energy and dissipation), whereby one can also relate the MEPP continuum approach to molecular dynamics of nonlinear fluids [17] and even account for to spontaneous violations of the second law of thermodynamics [18].
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Acknowledgment We have benefitted from constructive comments of J.D. Goddard (UCSD). This material is based upon work partially supported by the NSF under grant CMMI-1462749.
References
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[1] J.C. Oldroyd (1950). On the formulation of rheological equations of state. Proc. Roy. Soc. A 200, 523-541.
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[2] R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager (1989), Dynamics of Polymeric Liquids - 1 and II (2nd Ed). John Wiley and Sons. [3] H.W. Lord, Y. Shulman (1967). A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299-309.
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[4] A.E. Green, K.A. Lindsay (1972). Thermoelasticity. J. Elast. 2(1), 1-7.
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[5] J. Ignaczak, M. Ostoja-Starzewski (2009). Thermoelasticity with Finite Wave Speeds. Oxford University Press.
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[6] M. Ostoja-Starzewski (2014), Viscothermoelasticity with finite wave speeds: Thermomechanical laws. Acta Mech. 225(4-5), 1277-1285. [7] H. Ziegler (1983) An Introduction to Thermomechanics. North-Holland. [8] G.A. Maugin (1999), The Thermomechanics of Nonlinear Irreversible Behaviours, World Scientific Pub. Co. [9] D.G.B. Edelen (1974). Primitive thermodynamics: A new look at the Clausius-Duhem inequality. Int. J. Eng. Sci. 12, 121-141.
[10] C.I. Christov, P.M. Jordan (2005). Heat conduction paradox involving second-sound propagation in moving media. Phys. Rev. Lett. 94, 1543011-4.
[11] C.I. Christov (2005). On frame-indifferent formulation of the MaxwellCattaneo model of finite-speed conduction. Mech. Res. Comm. 36, 481-486. [12] R.E. Khayat, M. Ostoja-Starzewski (2011). On the objective rate of change of heat and stress fluxes, Connection with micro/nano-scale convection. Discr. Cont. Dyn. Syst. - Ser. B (DCDS-B) 15(4), 991-998.
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[13] J.D. Goddard (2014). Edelen’s dissipation potentials and the viscoplasticity of particulate media. Acta Mech. 225, 2239-2259.
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[14] J.D. Goddard (2015). Dissipation potentials for reaction-diffusion systems. Ind. Eng. Chem. Res. 54 (16), 4078–4083.
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[15] L.M. Martyushev, V.D. Seleznev (2006). Maximum entropy producton principle in physics, chemistry and biology. Phys. Reports 426, 1-45.
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[16] Ivan C. Christov, C.I. Christov (2016). Stress retardation versus stress relaxation in linear viscoelasticity. Mech. Res. Comm. 72, 59-63. [17] B. Raghavan, M. Ostoja-Starzewski (2017). Shear-thinning characteristics of molecular fluids un-dergoing planar Couette flow. Phys. Fluids 29, 023103-1-7.
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[18] M. Ostoja-Starzewski, A. Malyarenko (2014). Continuum mechanics beyond the second law of thermodynamics. Proc. R. Soc. A 470, 20140531.
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Highlights
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1. Thermodynamics-based derivation of Oldroyd-B fluids with hyperbolic heat conduction. 2. Model with one thermal relaxation time is obtained via primitive thermodynamics of Edelen. 3. Model with two thermal relaxation times is obtained via thermodynamic orthogonality of Zeigler.
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S0093-6413(17)30205-7 http://dx.doi.org/doi:10.1016/j.mechrescom.2017.07.012 MRC 3197
To appear in: Received date: Revised date: Accepted date:
21-4-2017 24-7-2017 26-7-2017
Please cite this article as: Martin Ostoja-Starzewski, Roger Khayat, Oldroyd Fluids with Hyperbolic Heat Conduction, (2017), http://dx.doi.org/10.1016/j.mechrescom.2017.07.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Oldroyd Fluids with Hyperbolic Heat Conduction
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Martin Ostoja-Starzewski Department of Mechanical Science & Engineering also Institute for Condensed Matter Theory and Beckman Institute University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.
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Roger Khayat Department of Mechanical and Materials Engineering The University of Western Ontario London, Ontario, Canada N6A 5B9
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July 24, 2017
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Keywords: Oldroyd fluids; hyperbolic heat conduction; thermal relaxation times; primitive thermodynamics; thermodynamic orthogonality
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Abstract Development of constitutive equations of Oldroyd fluids with hyperbolic heat conduction, directly from the free energy and dissipation functions, is the focus of this work. Two models from hyperbolic thermoelasticity of solids − theory with one thermal relaxation time and theory with two thermal relaxation times − provide a stepping-stone for this task. The setting for these models is offered, respectively, by the primitive thermodynamics of Edelen and the thermodynamic orthogonality of Ziegler, both cases involving an internal parameter: the inelastic strain. The identified free energy and dissipation functions show that, in the first case, the hyperbolic (telegraph-like) heat conduction is governed by the Maxwell-Cattaneo model involving one thermal relaxation time, whereas, in the second case, by the Fourier law and two thermal relaxation times in the constitutive laws for stress and entropy. In both cases, the thermal fields vanish from the mechanical equations as the Oldroyd fluid becomes incompressible, whereas the mechanical fields are present in the telegraph-type equations for temperature.
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Motivation
The Oldroyd fluid is specified by the (mechanical) constitutive equation [1] σ + λ1 σ˙ = 2η0 d + 2η0 λ2 d˙
σij + λ1 σ˙ ij = 2η0 dij + 2η0 λ2 d˙ij ,
or
(1)
1
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5
σ=
∂σ T + v · ∇σ − L·σ − (L·σ) , ∂t
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where λ1 is the relaxation time, λ2 (= λ1 ηs /η0 ) the retardation time, and η0 (= ηs + ηp ) the total fluid viscosity composed of a solvent’s viscosity (ηs ) and polymer’s viscosity (ηp ). We are focusing on incompressible fluids, so that σ (or σij in index notation) is the deviatoric part of the Cauchy stress, while d is the deviatoric part of the deformation rate (v(i ,j) ); the spherical part of the Cauchy stress is the negative of pressure. The overdot on σ and d denotes the upper-convected time derivative, e.g.
(2)
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which indicates our focus on the Oldroyd-B fluid; here L = ∇v (i.e., Lij = vj ,i ) is the velocity gradient, e.g. [2]. The overdot on a scalar quantity signifies a material time derivative. The somewhat more general picture of the overdot standing for a convex combination of the upper-convected and lower-convected derivatives (eventually leading to the Oldroyd-A fluid) is not being considered here, but can be treated by the same approach as outlined in this paper. As is well known, various liquids (e.g. bloods, paints) are best modeled as Oldroyd-B. Our ongoing experiments indicate that porcine blood transfers heat in a hyperbolic way with thermal relaxation time on the same order as the mechanical times λ1 and λ2 . This is an example of challenge where a generalization of (1) to a thermo-fluid model is needed. Simply attaching to (1) the thermal conductivity model of either a Fourier or Maxwell-Cattaneo type is an ad hoc way to postulate the heat transfer to be either parabolic or hyperbolic. However, as is well known, the parabolic heat conduction suffers from the paradox of infinitely fast propagation of heat and this motivates development of hyperbolic models. In this study we derive such models from an energetic-thermodynamic standpoint. We proceed through the thermomechanics with internal variables (TIV) which, fundamentally, bases the entire constitutive behavior on two functionals: free energy and dissipation function. In essence, the challenge is to find a pair of two such functionals which will allow a systematic derivation of simplest thermal-Oldroyd models with restriction to linear phenomena. The guidance is provided by two well established models in thermoelasticity: the Lord-Shulman model [3] (also called a theory with one relaxation time) and that of Green-Lindsay [4] (also called a theory with two relaxation times); see [5] for a comprehensive picture. The Lord-Shulman model involves these constitutive equations σij = Cijkl kl + Mij ϑ, θ0 s = −θ0 Mij ij + CE ϑ,
(3)
qi + t0 q˙i = −kij ϑ,j , where t0 > 0 is the relaxation time and ϑ := θ − θ0 is the temperature difference from the reference temperature θ0 . Also, σij is the whole (not just deviatoric) 2
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Cauchy stress, Cijkl the stiffness tensor, kl the infinitesimal strain, Mij the stress-temperature tensor, s the entropy density, CE the specific heat at zero strain, and kij the thermal conductivity tensor, respectively. These quantities satisfy the following symmetry, positivity, and positive-definiteness relations
CE > 0,
aij Cijkl akl > 0, kij = kji ,
∀a 6= 0,
bi kij bj > 0,
∀b 6= 0.
(4)
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Mij = Mji ,
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Cijkl = Cjikl = Cijlk = Cklij ,
Let ui denote the displacement. The resulting system of coupled displacementtemperature field equations is
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(Cijkl uk ,l ),j + (Mij ϑ) ,j = ρ¨ ui , (kij ϑ,j ) ,i +θ0 Mij (u˙ i ,j +t0 u ¨i ,j ) = CE ϑ˙ + t0 ϑ¨ ,
(5)
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which shows that the heat conduction is governed by a telegraph-type equation. The Green and Lindsay model involves the constitutive relations ˙ σij = Cijkl kl + Mij (ϑ + t1 ϑ) (6)
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˙ θ0 s = −θ0 Mij ij + CE (ϑ + t0 ϑ)
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qi = −kij ϑ,j ,
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where t0 and t1 are two relaxation times, satisfying the inequalities t1 ≥ t0 > 0. The relaxation time t0 in (6) is, in general, different from that in (3). The remaining symbols appearing in (6) have the same meaning as in the LordShulman model and are subject to the restrictions (4). It is important to note that, even though the Fourier law holds, the heat conduction process itself is hyperbolic (not parabolic!) [5]. This is seen from the second (telegraphtype) equation in the resulting system of coupled displacement-temperature field equations is ˙ j = ρ¨ (Cijkl uk ,l ),j +[Mij (ϑ + t1 ϑ)], ui , ¨ (kij ϑ,j ),i +θ0 Mij u˙ i ,j = CE (ϑ˙ + t0 ϑ).
(7)
In light of the above, for a thermal-Oldroyd fluid there will likely be two types of relaxation times: mechanical (λ1 , λ2 ) and thermal (t0 , t1 ); this notation shall be kept throughout this paper. Also note that both thermoelasticity models can be reformulated within the TIV framework in terms of the free energy and the dissipation function rather then only from the free energy as done in the original papers [3, 4]. Such a reformulation allows one to arrive at, say, viscothermoelastic solids with hyperbolic heat conduction [6]. That reformulation was motivated by the fact that the Fourier-type heat conduction is a dissipative 3
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Oldroyd fluids with one thermal relaxation time
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process, and thus should be described by the dissipation function rather than by the free energy function as in [4]. In the present study, we want to introduce thermal effects in the Oldroyd (i.e. viscoelastic) fluid by proceeding from TIV, thereby extending the range of viscous fluids that can thus be derived [7, 8] and a more general TIV-type theory [9].
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In order to derive the constitutive equations of a viscothermoelastic fluid with one relaxation time, we begin in spatial (Eulerian) description with the specific (per unit volume) internal energy u as a function of the entropy s, the heat flux qi , the strain ij , and the internal variable αij : u = u(ij , αij , s, qi ).
(8)
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Since we focus on incompressible fluids, σij will stand for deviatoric Cauchy stress, just as introduced in the Motivation. The specific power of deformation is l = σij ˙ij and the classical relation ψ = u − θs
(9)
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holds, whereby, in view of (8) and, by a partial Legendre transformation switching the argument s to the absolute temperature θ, we recognize that and
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ψ = ψ(ij , αij , θ, qi )
s = s(ij , αij , θ, qi , ).
(10)
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The first fundamental law (energy balance) has the form in spatial (Eulerian) description ρu˙ = σij ˙ij − qi ,i , (11) R R qi while the second fundamental law (starting from sdV ˙ ≥ − θ ni dA) s˙ ≥ −
q i
θ
,i =
qi θ,i qi ,i − 2 θ θ
(12)
is written in terms of the reversible (s∗(r) ) and irreversible (s∗(i) ) parts of entropy production rate (s) ˙ q i s˙ = s∗(r) + s∗(i) with s∗(r) = − ,i and s∗(i) ≥ 0. (13) θ The material time derivative of the free energy is found on the one hand from (9) as ˙ ψ˙ = u˙ − θs˙ − θs, (14) and, on the other hand from (10)1 , as ∂ψ ∂ψ ˙ ∂ψ ∂ψ ψ˙ = ˙ij + α˙ ij + θ+ q˙i . ∂εij ∂αij ∂θ ∂qi
(15)
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(q)
The latter, on account of the standard definitions of external (σij ) and internal (q)
(βij ) quasi-conservative stresses and entropy (s)
becomes (q)
(q)
ρψ˙ = σij ˙ij + βij α˙ ij − sθ˙ + ρ
∂ψ q˙i , ∂qi
leading to
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∂ψ ∂ψ ∂ψ (q) , βij = ρ , s=− , ∂ij ∂αij ∂θ
(16)
(17)
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(q)
σij = ρ
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∂ψ (q) (q) ρψ˙ + ρsθ˙ = ρu˙ − ρθs˙ = σij ˙ij + βij α˙ ij + ρ q˙i . ∂qi
(18)
With the external and internal dissipative stresses defined by (d)
(q)
(d)
(q)
the second equality in (18) becomes (d)
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σij = σij − σij , βij = −βij ,
(d)
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ρu˙ − ρθs˙ = σij ˙ij − σij ˙ij − βij α˙ ij + ρ
∂ψ q˙i . ∂qi
(19)
(20)
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On account of the first fundamental law (ρu˙ = σij ˙ij − qi ,i ), this is equivalent to ∂ψ (d) (d) ρθs˙ = σij ˙ij + βij α˙ ij − qi ,i −ρ q˙i (21) ∂qi or 1 (d) qi θ,i qi ∂ψ 1 (d) ,i −ρ ρs˙ = σij ˙ij + βij α˙ ij − 2 − q˙i (22) θ θ θ θ ∂qi
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By inserting the above into (12), we obtain the Clausius-Duhem inequality in the form ∂ψ qi θ,i (d) (d) −ρ q˙i = ρθs∗(i) ≥ 0, (23) σij ˙ij + βij α˙ ij − θ ∂qi
In any fluid having an isotropic response, the free energy function ψ can, in general, depend on all the basic invariants of ij and αij . Since the elastic spring-back is involved in the response of the Oldroyd fluid [2], ψ is taken as a functional of the difference ij − αij . Since the constitutive equation involving stress (and its rate) and deformation rate (and its rate) is linear, ψ is a quadratic functional of (ij − αij ). Thus, ρψ per unit volume (i.e. ψ being the free energy per unit mass) is taken as ρψ(ij , αij , θ, qi ) = ρψ0 − ρs0 (θ − θ0 ) + µ(ij − αij )2 C t0 2 + Mij ij (θ − θ0 ) − (θ − θ0 ) + qi qi 2θ0 2θ0 κ
(24)
Here ψ0 and s0 are, respectively, the free energy and entropy in the reference state, µ is the elastic shear modulus, Mij is the stress-temperature tensor (effectively accounting for the thermal expansion), C is the specific heat at zero 5
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strain, t0 is the thermal relaxation time, and κ is the Fourier-type thermal conductivity. By hyperelasticity we obtain ∂ψ = 2µ(ij − αij ) + Mij (θ − θ0 ) ∂ij ∂ψ (q) βij = ρ = 2µ(αij − ij ) ∂αij C ∂ψ = ρs0 − Mij ij + (θ − θ0 ) ρs = −ρ ∂θ θ0 ∂ψ t0 −ρ =− qi . ∂qi θ0 κ
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(q)
σij = ρ
(q)
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(25)
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Thus, (25)1 and (25)2 give σij and βij , respectively, while (25)3 is immediately identified as the constitutive equation for entropy. The significance of (25)3 (q) will be clarified further below. The stress σij models the spring-back effect in the viscoelastic model [2]. The spring-back strain is very small, so that (q) (q) σij ˙ij ' σij dij , which justifies our replacing ˙ij by the deformation rate dij below and the use of µ in (24). Also, just like σij is the deviatoric part of the (q) (q) Cauchy stress, σij and βij are the deviatoric parts of the external and internal quasi-conservative stresses. The irreversible part of entropy production rate is a function of the heat flux, its rate q˙i , and the strain rate ε˙ij , and the internal parameter rate α˙ ij : ρθs∗(i) = ρθs∗(i) (dij , α˙ ij , qi , q˙i ).
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(26)
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Clearly, the inequality (23) may be written as a scalar product Y · V ≥ 0,
(27)
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whereby we have defined
Y ≡ (y1 , y2 , y3 , y4 ) :=
∂ψ θ,i (d) (d) σij , βij , − , − θ ∂qi
(28)
as the vector of dissipative thermodynamic forces. Thus, we have defined the four sub-vectors of Y, while its conjugate flux is V ≡ (V1 , V2 , V3 , V4 ) := (dij , α˙ ij , qi , q˙i ) ,
(29)
in which we have defined four sub-vectors vi of v, the vector of its conjugate thermodynamic velocities. A general procedure based on the representation theory in the so-called primitive thermodynamics due to Edelen [9] allows a derivation of the most general form of the constitutive relation either for V as a function of Y or for Y as a function of V, subject to (27). Thus, we first introduce a functional φE (Edelen’s dissipation) which depends on dij , α˙ ij , qi , q˙i according to (26), and take Y(V) as the gradient in the velocity space plus an extra vector U: Y = ∇V φE + U,
or
Yi =
∂φE + Ui . ∂Vi
(30)
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Here the vector U = (u1 , u2 , u3 , u4 ) does not contribute to the entropy production according to the condition U · V = 0,
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(31)
∂ [Yi (τ V) − Ui ] ∂ [Yj (τ V) − Uj ] = ∂vj ∂vi
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and hence R 1 is called a powerless dissipative force, while the dissipation function is φE = 0 V · Y(τ V)dτ . Also, the symmetry relations (32)
iff
U = 0.
(33)
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∂Yj (τ V) ∂Yi (τ V) = ∂vj ∂vi
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must hold, and these reduce to the classical Onsager reciprocity conditions
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In order to derive a model with one thermal relaxation time (i.e. MaxwellCattaneo type), we generalize the formulation of [6] by taking the dissipative vector U = (U1 , U2 , U3 , U4 ) (corresponding to Y and V above) as, in components, t0 t0 U1 = 0 U2 = 0 U3i = q˙i U4i = − qi . (34) θκ θκ The dissipation function φE of an isotropic fluid can, in general, depend on all the basic and mixed invariants of dij , α˙ ij , qi , and q˙i . To have a fluid model linear in heat conduction as well as mechanically linear, φE must simply be a quadratic form in dij , α˙ ij , qi , and q˙i , so that, in view of (19), ρφE (V) ≡ ρφE (dij , α˙ ij , qi , q˙i ) =
H F 1 G dij dij + α˙ ij α˙ ij + qi qi + q˙i q˙i . (35) 2 2 2θκ 2
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Note that φE cannot depend on qi q˙i , qi dij qj , qi dij djk qk , q˙i dij q˙j , but such invariants might appear in a nonlinear model. Now, on account of (30), it follows that ∂φE + u1ij = Hdij ∂dij ∂φE (d) + u2ij = F α˙ ij βij ≡ Y2ij = ρ ∂ α˙ ij θ,i ∂φE 1 t0 − ≡ Y3i = ρ + u3i = qi + q˙i θ ∂qi θκ θκ t0 ∂ψ ∂φE t0 − qi = −ρ ≡ Y4i = ρ + u4i = Gq˙i − qi . θ0 κ ∂qi ∂ q˙i θκ (d)
σij ≡ Y1ij = ρ
(36)
First, by (19)1 , the total stress is σij = 2µ(ij − αij ) + Mij ϑ + Hdij .
(37)
F α˙ ij = −2µ(ij − αij ).
(38)
while by (19)2
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Taking the second-order time derivative of (37) − effectively being the upperconvected derivative of σij and dij (recall the statement following (2)) and the material time derivative of ϑ − we obtain σ˙ ij = 2µ(˙ij − α˙ ij ) + Mij ϑ˙ + H d˙ij .
(39)
σij +
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Here we recall that, one needs to interpret both time derivatives in (1) not as a conventional material derivative but an upper-convected one. Thus, the overdot in (39) needs to be interpreted as the Lie (i.e., the Oldroyd upper-convected) ¨ In the case of a rigid derivative [8]. An analogous argument applies to ϑ˙ and ϑ. conductor, the overdot reduces to the conventional material time derivative as shown by the requirement of Galilean invariance [10, 11, 12]. Now, eliminating the internal variable αij from (37), (38), and (39), we arrive at the constitutive equation linking stress (and its rate) with strain (and ˙ its rate) and with temperature difference ϑ := θ − θ0 and its rate ϑ: FH ˙ F F σ˙ ij = (F + H) dij + dij + Mij ϑ + Mij ϑ˙ 2µ 2µ 2µ
or, in symbolic notation,
F FH ˙ F ˙ σ˙ = (F + H) d + d + Mϑ + M ϑ. 2µ 2µ 2µ
M
σ+
(40)
(41)
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This is the Oldroyd viscoelastic fluid model directly coupled with the temperature field and its rate. With reference to (1), we now find the energetic-thermodynamic interpretation of the conventional constitutive coefficients in the Oldroyd model: λ1 =
F , 2µ
2η0 = F + H,
2η0 λ2 =
FH . 2µ
(42)
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Note that only the deviatoric part of Mij enters the constitutive equation (40). Thus, when Mij is taken as an isotropic tensor Mij = M δij with M = 0, the coupling between mechanical and thermal fields vanishes and we recover the ˙ classical Oldroyd fluid equation relating σ, σ, ˙ d, and d. The formulation is completed by setting G = 0 and θ ' θ0 , thus assuring the identity in (36)4 . The approximation of small temperature changes is the same as in the Lord-Shulman thermoelasticity model (3); see also equations (R.1.1)-(R.1.11) in [5]. According to (36)3 , the heat flux and temperature are related through the Maxwell-Cattaneo model qi + t0 q˙i = −κϑ,i . (43) indicating that the temperature field is governed by a telegraph equation for temperature with extra nonlinear mechanical terms which have a very complicated expression; we do not reproduce that equation here. It is simple to show that, in the special case of F = 0 and t0 = 0, that equation reduces to the ˙ which is the form of energy equaparabolic equation κϑ,ii +Hdij dij = ρC ϑ, tion used for Newtonian-Fourier fluids; C is the specific heat under constant strains and internal parameters. 8
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Oldroyd fluids with two thermal relaxation times
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The internal energy u is now taken as a function of the strain ij , the internal variable αij , and the entropy s: (44)
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u = u(ij , αij , s). The specific power of deformation is
As before, switching from u to ψ = u − θs, we find and
s = s(ij , αij , θ).
(46)
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ψ = ψ(ij , αij , θ)
(45)
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l = σij ˙ij /ρ.
(47)
qi θ,i = ρθs∗(i) ≥ 0. θ
(48)
The relations (9), (11), (12), (13), (16), and (19) hold as before, so that (d)
(d)
ρu˙ − ρθs˙ = σij ˙ij − σij ˙ij − βij α˙ ij ,
(d)
(d)
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while the Clausius-Duhem inequality takes the form σij ˙ij + βij α˙ ij −
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Similar to the model with one relaxation time, in any fluid having an isotropic response, the free energy function ψ can, in general, depend on all the basic invariants of ij and αij . Since the spring-back is involved in the response of the Oldroyd fluid, ψ is taken as a functional of the difference ij − αij . Since the constitutive equation involving stress (and its rate) and deformation rate (and its rate) is linear, ψ is a quadratic functional of (ij − αij ). Thus, ρψ per unit volume (i.e. ψ being the free energy per unit mass) is taken as ρψ(ij , αij , θ) = ρψ0 − ρs0 (θ − θ0 ) + µ(ij − αij )2 C C 2 ˙ (49) + Mij ij (θ − θ0 ) − (θ − θ0 ) − t0 (θ − θ0 ) θ. 2θ0 θ0
Here ψ0 and s0 are the free energy and entropy in the reference state, µ is the shear elastic modulus, Mij is the stress-temperature (effectively accounting for the thermal expansion), C is the specific heat at zero strain, and t0 is the thermal relaxation time. Again invoking the hyperelasticity property, from the free energy, we find ∂ψ = 2µ(ij − αij ) + Mij (θ − θ0 ) ∂ij ∂ψ (q) βij = ρ = 2µ(αij − ij ) ∂αij ∂ψ C C ˙ ρs = −ρ = ρs0 − Mij ij + (θ − θ0 ) + t0 θ. ∂θ θ0 θ0 (q)
σij = ρ
(50)
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The relation (50)3 is immediately identified as the constitutive equation for (q) (q) entropy, exactly the same as (6)2 . As in the previous section, here σij and βij are the deviatoric parts of the external and internal quasi-conservative stresses, the corresponding spherical parts being zero in view of the incompressibility of fluid. Also, the same argument as in the paragraph following eq. (25) allows a replacement of the strain rate ˙ij by dij . Clearly, the inequality (48) may be written as a scalar product1 Y · V ≥ 0, whereby we have defined Y ≡ (y1 , y2 , y3 ) :=
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(51)
σij , βij , −
θ,i θ
(52)
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as the vector of dissipative thermodynamic forces. Corresponding to the three sub-vectors of Y there are three conjugate velocity sub-vectors V ≡ (V1 , V2 , V3 ) := (dij , α˙ ij , qi ) .
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(53)
The dissipation function ϕZ in the sense of Ziegler [7] (as opposed to φE of Edelen) is taken equal to the irreversible entropy production rate and, since we are aiming at linear processes, is adopted as quadratic form
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ρθs∗(i) ≡ ρϕZ (V) ≡ ρϕZ (dij , α˙ ij , qi )
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1 = Hdij dij + 2t1 Mij dij ϑ˙ + F α˙ ij α˙ ij + qi qi , (54) θκ
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Next, recall the thermodynamic orthogonality (or the maximum entropy production principle) [7, 8] which gives Y as a gradient in the space of velocities V (without any powerless dissipative force): Y=
1 ∇V ϕZ , 2
or
Yi =
with the factor 1/2 determined by the condition result, we obtain (d)
1 ∂ϕZ , 2 ∂Vi 1 2
(55) −1
= ϕZ (Vi ∂ϕZ /∂Vi )
1 ∂ϕZ ρ = Hdij + t1 Mij ϑ˙ 2 ∂dij 1 ∂ϕZ (d) βij ≡ Y2ij = ρ = F α˙ ij 2 ∂ α˙ ij θ,i 1 ∂ϕZ 1 − ≡ Y3i = ρ = qi . θ 2 ∂qi θκ
. As a
σij ≡ Y1ij =
(56)
The final constitutive relations follow by noting from (19)2 that 2µ(ij − αij ) = F α˙ ij ,
(57)
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while (50)1 and (56)1 yield the total (deviatoric) stress ˙ σij = 2µ(ij − αij ) + Mij ϑ + Hdij + t1 Mij ϑ.
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(58)
Taking the time derivative of (58) with respect to time (as before, recall the statement following (2)) we find
(59)
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¨ σ˙ ij = 2µ(dij − α˙ ij ) + Mij ϑ˙ + H d˙ij + t1 Mij ϑ,
FH ˙ F σ˙ ij = (F + H) dij + dij 2µ 2µ
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σij +
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where − by applying the same argument as in (39) − the overdots on secondrank tensors signify the upper-convected derivatives. Eliminating the internal variable and its rate from (57)-(59), we arrive at the constitutive equation linking stress (and its rate) with strain (and its rate) with temperature difference ϑ := θ − θ0 (and its first and second rates)
or, in symbolic notation, σ+
F FH ˙ σ˙ = (F + H) d + d 2µ 2µ
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F F ˙ ϑ + t1 Mij ϑ¨ (60) + Mij ϑ + t1 Mij + Mij 2µ 2µ
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F F ¨ + Mϑ + t1 M + M ϑ˙ + t1 M ϑ. (61) 2µ 2µ
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Since the mechanical terms in the above are the same as in (40)-(41), we find the energetic-thermodynamic interpretation (42) of the conventional constitutive coefficients of the Oldroyd model to also apply here. However, the presence of ϑ¨ distinguishes this model from that with one relaxation time obtained in Section 2. Also, when Mij is taken as an isotropic tensor with M = 0, the coupling between mechanical and thermal fields in (60) vanishes and we recover ˙ the classical Oldroyd fluid equation relating σ, σ, ˙ d, and d. Turning to the third relation in (56), we see that it immediately gives the Fourier law for the isotropic medium, just like (6)3 , −κθ,i = qi .
(62)
In order to derive the equation governing the temperature, assume Mij to be an (d) isotropic tensor and start from (50)1 . Next, note that σij dij = Hdij dij , while from (19), (56)2 , and (57) we find (q)
(d)
F α˙ ij = 2µ ( − α) = σij = σij − σij = σij − Hdij
(63)
(d)
which implies βij α˙ ij = F α˙ ij α˙ ij = (σij − Hdij ) (σij − Hdij ) F −1 . Hence, on account of (62) and the energy balance (11), κϑ,ii = −Hdij dij − (σij − Hdij ) (σij − Hdij ) F −1 + ρC ϑ˙ + t0 ϑ¨ , (64) 11
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Conclusions
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which is a hyperbolic equation for the temperature difference ϑ (and, equivalently, for θ), even though the constitutive law of heat conduction is Fourier. The above equation is analogous to (7)2 in that it is coupled with the mechanical field. In the special case of F = 0 and t0 = 0, on account of (63), the hyperbolic ˙ again a equation (64) reduces to the parabolic equation κϑ,ii +Hdij dij = ρC ϑ, form of energy equation for Newtonian-Fourier fluids − the same as mentioned at the end of Section 2.
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Adopting the thermomechanics with internal variables, we have derived the Oldroyd fluid equation with hyperbolic heat conduction from the standpoint of second law of thermodynamics. In particular, we have explored two basic ways for an Oldroyd fluid to possess linear damped hyperbolic heat conduction:
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1. The dissipation function φE following Edelen’s primitive thermodynamics approach, which does not equal ρθs∗(i) but (i) leads to the MaxwellCattaneo relation between heat flux and temperature gradient and (ii) results in a thermomechanical model with one thermal relaxation time. This is a thermo-fluid analogue of the “thermoelasticity with one relaxation time” [3].
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2. The dissipation function ϕZ following Ziegler’s thermodynamic orthogonality, which equals ρθs∗(i) but (i) leads to a Fourier model of heat conduction with temperature being governed by a telegraph equation (3.21) with extra nonlinear mechanical terms, and (ii) results in a thermomechanical model with two thermal relaxation times. It may appear puzzling at first sight to have a Fourier model with a hyperbolic field equation, but a completely analogous picture occurs in the “thermoelasticity with two relaxation times” [4].
The presence of mechanical terms in either of the temperature equations is unavoidable because it is, basically, an energy equation, whereas the temperature terms do not enter the mechanical equations of either model because of the incompressibility. Indeed, given that Mij is typically isotropic (i.e., temperature is uncoupled from shear stresses), the Oldroyd mechanical equation is practically uncoupled from the thermal equation. Overall, both models derived here fall within the class of hyperelastic and hyperdissipative media [13, 14]. Various simpler thermo-fluid models are obtained as special cases of both models given here. For example, upon setting F = 0, we recover, in both models, the Newtonian fluid with independent Fourier heat conduction. Overall, while many different material behaviors may be obtained from Ziegler’s maximum entropy production principle (MEPP) (e.g. [15], [16]), some
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systems may actually require a more general primitive thermodynamics (of Edelen). Expanding on [7] and [8], the key role is played by two functionals (of free energy and dissipation), whereby one can also relate the MEPP continuum approach to molecular dynamics of nonlinear fluids [17] and even account for to spontaneous violations of the second law of thermodynamics [18].
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Acknowledgment We have benefitted from constructive comments of J.D. Goddard (UCSD). This material is based upon work partially supported by the NSF under grant CMMI-1462749.
References
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[1] J.C. Oldroyd (1950). On the formulation of rheological equations of state. Proc. Roy. Soc. A 200, 523-541.
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[2] R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager (1989), Dynamics of Polymeric Liquids - 1 and II (2nd Ed). John Wiley and Sons. [3] H.W. Lord, Y. Shulman (1967). A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299-309.
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[4] A.E. Green, K.A. Lindsay (1972). Thermoelasticity. J. Elast. 2(1), 1-7.
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[5] J. Ignaczak, M. Ostoja-Starzewski (2009). Thermoelasticity with Finite Wave Speeds. Oxford University Press.
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[6] M. Ostoja-Starzewski (2014), Viscothermoelasticity with finite wave speeds: Thermomechanical laws. Acta Mech. 225(4-5), 1277-1285. [7] H. Ziegler (1983) An Introduction to Thermomechanics. North-Holland. [8] G.A. Maugin (1999), The Thermomechanics of Nonlinear Irreversible Behaviours, World Scientific Pub. Co. [9] D.G.B. Edelen (1974). Primitive thermodynamics: A new look at the Clausius-Duhem inequality. Int. J. Eng. Sci. 12, 121-141.
[10] C.I. Christov, P.M. Jordan (2005). Heat conduction paradox involving second-sound propagation in moving media. Phys. Rev. Lett. 94, 1543011-4.
[11] C.I. Christov (2005). On frame-indifferent formulation of the MaxwellCattaneo model of finite-speed conduction. Mech. Res. Comm. 36, 481-486. [12] R.E. Khayat, M. Ostoja-Starzewski (2011). On the objective rate of change of heat and stress fluxes, Connection with micro/nano-scale convection. Discr. Cont. Dyn. Syst. - Ser. B (DCDS-B) 15(4), 991-998.
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[13] J.D. Goddard (2014). Edelen’s dissipation potentials and the viscoplasticity of particulate media. Acta Mech. 225, 2239-2259.
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[14] J.D. Goddard (2015). Dissipation potentials for reaction-diffusion systems. Ind. Eng. Chem. Res. 54 (16), 4078–4083.
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[15] L.M. Martyushev, V.D. Seleznev (2006). Maximum entropy producton principle in physics, chemistry and biology. Phys. Reports 426, 1-45.
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[16] Ivan C. Christov, C.I. Christov (2016). Stress retardation versus stress relaxation in linear viscoelasticity. Mech. Res. Comm. 72, 59-63. [17] B. Raghavan, M. Ostoja-Starzewski (2017). Shear-thinning characteristics of molecular fluids un-dergoing planar Couette flow. Phys. Fluids 29, 023103-1-7.
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[18] M. Ostoja-Starzewski, A. Malyarenko (2014). Continuum mechanics beyond the second law of thermodynamics. Proc. R. Soc. A 470, 20140531.
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Highlights
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1. Thermodynamics-based derivation of Oldroyd-B fluids with hyperbolic heat conduction. 2. Model with one thermal relaxation time is obtained via primitive thermodynamics of Edelen. 3. Model with two thermal relaxation times is obtained via thermodynamic orthogonality of Zeigler.
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