# A Variational Principle for the Steady-State Heat Transfer Process in a Rigid Continuous Mixture

Authors: Rogério M. Saldanha da Gama
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The subject of this paper is the steady-state heat transfer process in a rigid mixture withN continuous constituents, each of them representing a given continuous body. A continuous mixture consists of a convenient representation for bodies composed by several different materials or phases, in which the actual interfaces do not allow an adequate Classical Continuum Mechanics approach, once that the boundary conditions make the mathematical description of the problem unfeasible (as for instance in reinforced concrete, polymer strengthened concrete, and porous media). The phenomenon is mathematically described by a set of N partial differential equations coupled by temperature-dependent terms that play the role of internal energy sources. These internal energy sources arise because, at each spatial point, there are different temperatures, each one associated with one constituent of the mixture. The coupling among the partial differential equations arises from the thermal interchange among continua in a thermal nonequilibrium context (different temperature levels). In this work, it is presented a functional whose minimization is equivalent to the solution of the original steady-state problem (variational principle). The features of this functional give rise to proofs of solution existence and solution uniqueness. It is remarkable that, with the functional to be proposed here, instead of solving a system of N coupled partial differential equations, we need to look only for the minimum of a single functional.


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## Body

## 1. Introduction

Based on an original idea pointed out by Fick (in 1855) [1] and Stefan (in 1872) [2] and developed later by Atkin and Craine [3, 4] and Bowen [5], the Continuum Theory of Mixtures represents a generalization of the Classical Continuum Mechanics [6–8] in which a composite body (or system) is regarded as a superposition of N continua occupying, simultaneously, the whole volume of the body. In other words, the body is regarded as a mixture consisting of N continuous constituents.Each constituent possesses its own temperature field and exchanges energy with the other constituents of the mixture when there is no thermal equilibrium.The main subject of the present work is the energy transfer phenomenon in a rigid continuous mixture. This phenomenon (for a mixture withN constituents) will be described by a set of N partial differential equations coupled by internal source terms, which takes into account the temperatures of all the constituents and provides the thermal interaction among them.The study of this kind of phenomenon is motivated by the existence of multimaterial heterogeneous bodies (reinforced concrete bodies, carbon fiber materials bodies, porous bodies, and others), for which the usual heat transfer approach is not convenient.The main objectives of this work are to present the following:(1)
A minimum principle, suitable for the steady-state heat transfer process in a rigid continuous mixture.(2)
A proof of solution existence.(3)
A proof of solution uniqueness.A continuous mixture is defined as a superposition ofN continuous constituents, each one representing a given continuum (a given material), as suggested in Figure 1.Figure 1
Illustrating the difference between the classical continuum mechanics and the continuum mixture theory viewpoints: (a)N continua; (b) mixture with N constituents.In fact, the continuous mixture viewpoint consists of assuming that a body composed by a set ofN continua, occupying, respectively, the spatial regions Ω1,Ω2,Ω3,Ω4,…,ΩN with Ωi∩Ωj≡∅ if i≠j and with ∪i=1NΩi≡Ω, be represented by a set of N continuous constituents, all of them occupying simultaneously the entire region Ω (the whole mixture), in such a way that the original interfaces vanish, as suggested in Figure 1.This superposition allows the existence ofN distinct temperatures at each spatial point in the mixture.It is to be noticed that theseN distinct temperature fields are not independent since the energy transfer process in each constituent is affected by the energy transfer process in all the others constituents of the mixture.In order to take into account the thermal interaction among the constituents, some special fields must be considered in the balance equations. In other words, the energy transfer process in each constituent will be described by an equation which possesses a field (the internal source term) representing the amount of energy (per unit time and unit volume) supplied to it by the other ones of the mixture. This field will be denoted here byΨα (for the constituent Cα).The fieldΨα will depend on the difference between the temperature of the αth constituent and each one of the other constituents (in other words, if two constituents have, at the same point, different temperatures, then there will exist a heat exchange between them).


## 2. Energy Balance

The energy balance for each constituentCα of a given rigid continuous mixture (composed by N constituents) is mathematically described as follows [4]:(1)ραcα∂Tα∂t=−divqα+q˙α+Ψα,inΩforα=1,2,…,N,in which Ω is a bounded open set representing the region occupied by the mixture, ρα represents the mass density of Cα (ratio between the mass of the constituent Cα to the corresponding volume of mixture—different from the classical mass density), cα is the specific heat of Cα, Tα is the temperature of Cα, qα is the partial heat flux associated with Cα, q˙α is the internal heat supply (per unit time and per unit volume) for Cα, and Ψα is the energy supply arising from the interaction between Cα and the others constituents of the mixture.The internal heat supplyΨα is an internal effect. It provides the coupling among the heat transfer processes in all the constituents and represents (for each α) the amount of energy, per unit time and unit volume, supplied to the constituent Cα due to its thermal interaction with the other constituents of the continuous mixture. Hence, in addition to equation (1), the following must hold (energy balance for the mixture as a whole):(2)∑α=1NΨα=0,inΩ.


## 3. Constitutive Relations

The partial heat flux (per unit time and unit area) associated with the constituentCα is given by(3)qα=−ΛkαϕαgradTα,in which Λ is a positive-definite second order tensor (usually constant), kα is the thermal conductivity of the material represented by Cα, and ϕα is the ratio between ρα and the actual mass density of the material represented by Cα. The fields ϕα are such that(4)∑α=1Nϕα=1,inΩand0<ϕα<1inΩ,for anyα.The fieldsϕα are always known, for any α.The internal heat supplyΨα is the sum of the heat supplies coming from all the constituents to the constituent Cα. In other words, we may write(5)Ψα=∑β=1NΓα⟶β,in which Γα⟶β is the heat supply from Cβ to Cα. Hence,(6)Γα⟶β=−Γβ⟶α⇒Γα⟶α=0.As a consequence of the Second Law of Thermodynamics [9], the sign of Γα⟶β, at each point, depends only on the sign of the difference Tβ−Tα at this point. If Tβ>Tα, Γα⟶β must be positive (since Cα will receive energy from Cβ). In other words, the sign of Γα⟶β is the same sign of Tβ−Tα. This fact induces the following constitutive relationship:(7)Γα⟶β=RαβTβ−Tα,Rαβ=Rβα>0,for anyαandβ.The simplest case is the one in whichRαβ is assumed to be a constant for any α and β.It is remarkable that since (7) holds, equation (2) is automatically satisfied.


## 4. Boundary Conditions

It will be assumed that each constituent of the mixture exchanges energy with the environment by convection [10] (Incropera and Dewitt, 1996). In this way, on ∂Ω, we have the following boundary condition:(8)qα·n=hαTα−T∞,on∂Ωforα=1,2,…,N,in which hα is a nonnegative constant (convection heat transfer coefficient), T∞ is a temperature of reference, and n is the unit outward normal defined on ∂Ω. It will be assumed that, for at least one constituent, hα be positive on some nonempty subset of ∂Ω.It is to be noticed that, whenhα is assumed zero for a given constituent, this constituent does not exchange energy with the environment (insulated boundary).


## 5. The Steady-State Mathematical Description

Inserting (3) in (1) and in (8) and considering (7) and (5), we have the following description for the steady-state heat transfer phenomenon in a rigid continuous mixture:(9)divΛkαϕαgradTα+∑β=1NRαβTβ−Tα+q˙α=0,inΩ,−ΛkαϕαgradTα·n=hαTα−T∞on∂Ωforα=1,2,…,N,in which the unknowns are the temperature fields T1,T2,…,TN.It will be shown now that these unknowns may be obtained from the minimization of a convex and coercive functional, ensuring the existence and the uniqueness of the solution.


## 6. The Minimum Principle

If we assume thatΛ,kα,ϕα,Rαβ and hα do not depend on the unknowns T1,T2,…,TN, and the problem represented by the set of Npartial differential equations and N boundary conditions is equivalent to the minimization of the following convex and coercive functional [11]:(10)Iw1,w2,…,wN=∑α=1N∫Ω12kαϕαΛgradwα·gradwαdV−∑α=1N∫Ωq˙αwαdV++∑α=1N∑β=αN∫Ω12Rαβwβ−wα2dV+∑α=1N∫∂Ω12hαwα−T∞2dA,in which the admissible fields wα must belong to the Sobolev space H1Ω [12].Minimizing a functional, like the one presented in (10), instead of solving a system of partial differential equations, represents an enormous advantage since it provides less effort when compared with traditional methods (like finite difference). In addition, a minimum principle, like the one represented by (10), provides easy and powerful tools for demonstrating the uniqueness and the existence of the solution. Variational tools are always welcome [13, 14].But, first of all, it is mandatory to show that the functional defined in (10) is the correct choice for the considered (original) problem. In other words, we must show the equivalence between (9) and the minimization of Iw1,w2,…,wN.In order to show that the solution of problem (9) is equivalent to the minimization of the functional Iw1,w2,…,wN defined in (10), we begin evaluating the first variation of Iw1,w2,…,wN, denoted here by δI.The first variation ofI is given as follows [15]:(11)δI=∑α=1N∫ΩΛkαϕαgradwα·gradδwαdV−∑α=1N∫Ωq˙αδwαdV++∑α=1N∑β=αN∫ΩRαβwβ−wαδwβ−δwαdV+∑α=1N∫∂Ωhαwα−T∞δwαdA,or, employing Green’s identity [16],(12)δI=∑α=1N∫Ω−divkαϕαΛgradwαδwαdV+∑α=1N∫∂ΩkαϕαΛgradwα·nδwαdA++∑α=1N∫∂Ωhαwα−T∞δwαdA−∑α=1N∫Ωq˙αδwαdV+∑α=1N∑β=αN∫ΩRαβwβ−wαδwβ−δwαdV.Since(13)α≤β≤N,1≤α≤N⇔1≤α≤β,1≤β≤N.The last term of (12) may be rewritten as follows:(14)∑α=1N∑β=αN∫ΩRαβwβ−wαδwβ−δwαdV=∑α=1N∑β=αN∫ΩRαβwβ−wαδwβdV+∑β=1N∑α=1β∫Ω−Rαβwβ−wαδwαdV,or replacing α by β in the last term and taking into account that Rαβ=Rβα (and that wα=wβ if α=β),(15)∑α=1N∑β=αN∫ΩRαβwβ−wαδwβ−δwαdV=∑α=1N∑β=1N∫Ω−Rαβwβ−wαδwαdV.So, (12) may be rewritten as(16)δI=−∫Ω∑α=1NdivkαϕαΛgradwα+∑β=1NRαβwβ−wα+q˙αδwαdV++∫∂Ω∑α=1Nhαwα−T∞+kαϕαΛgradwα·nδwαdA.The extrema of the functionalI are obtained for the fields uαα=1,2,…,N such that wα≡uα gives rise to δI=0.ImposingδI=0 and taking into account that the fields δw1,δw2,…,δwN are arbitrary and independent, we have that the fields uα must satisfy (Euler–Lagrange equations and natural boundary conditions):(17)divΛkαϕαgraduα+∑β=1NRαβuβ−uα+q˙α=0,inΩ,−Λkαϕαgraduα·n=hαuα−T∞on∂Ωforα=1,2,…,N.It is easy to see that problems (17) and (9) are equivalent. In other words, the extrema of I are reached for the solutions of (9) (or (17)).


## 7. Uniqueness

It will be shown now that the solution of (17) (or (9)) is unique and corresponds to a minimum of the functional I. In other words, it will be shown that if T1,T2,…,TN satisfies (9), then(18)IT1,T2,…,TN<Iw1,w2,…,wN,providedTα≠wαfor someα.Aiming to this, it is sufficient to prove that the functionalI is strictly convex, which means it satisfies the following inequality [17]:(19)Iθw¯+1−θv¯<θIw¯+1−θIv¯,for anyθ∈0,1,for anyv¯≠w¯,in which w¯=w1,w2,…,wN and v¯=v1,v2,…,vN.The first step for proving (19) is to show that, for θ∈0,1, the expression θIw¯+1−θIv¯−Iθw¯+1−θv¯ is always nonnegative.Aiming to this, let us write the above expression as follows:(20)θIw¯+1−θIv¯−Iθw¯+1−θv¯==θ∑α=1N∫Ω12kαϕαΛgradwα·gradwαdV−∑α=1N∫Ωq˙αwαdV++∑α=1N∑β=αN∫Ω12Rαβwβ−wα2dV+∑α=1N∫∂Ω12hαwα−T∞2dA++1−θ∑α=1N∫Ω12kαϕαΛgradwα·gradwαdV−∑α=1N∫Ωq˙αvαdV++∑α=1N∑β=αN∫Ω12Rαβvβ−vα2dV+∑α=1N∫∂Ω12hαvα−T∞2dA−−∑α=1N∫Ω12kαϕαΛgradθwα+1−θvα·gradθwα+1−θvαdV−−∑α=1N∫Ωq˙αθwα+1−θvαdV++∑α=1N∑β=αN∫Ω12Rαβθwβ+1−θvβ−θwα+1−θvα2dV++∑α=1N∫∂Ω12hαθwα+1−θvα−T∞2dA.Reordering some terms and taking into account that(21)θ∑α=1N∫Ωq˙αwαdV+1−θ∑α=1N∫Ωq˙αvαdV=∑α=1N∫Ωq˙αθwα+1−θvαdV,we may rewrite (20) as follows:(22)θIw¯+1−θIv¯−Iθw¯+1−θv¯=θ1−θ∑α=1N∫∂Ω12hαwα−vα2dA++θ1−θ∑α=1N∫Ω12kαϕαΛgradwα−vα·gradwα−vαdV++θ1−θ∑α=1N∑β=αN∫Ω12Rαβwβ−vβ−wα−vα2dV.It is easy to see that (22) is nonnegative for any θ∈0,1. Now, we must prove that, for any v¯≠w¯, expression (22) is positive valued.Aiming to this, let us suppose that, for someα, the difference wα−vα is not a constant. In this case, since kαϕα is positive valued for any α and Λ is a positive-definite tensor, we have(23)∑α=1N∫Ω12kαϕαΛgradwα−vα·gradwα−vαdV>0.On the contrary, if the differencewα−vα is a nonzero constant, we have that(24)∑α=1N∫∂Ω12hαwα−vα2dA>0.Therefore, the positiveness of (22) is proven. This ensures (19). Since (19) holds, the functional I is strictly convex and, consequently, its minimum (if exists) is unique. Thus, the uniqueness of the solution of (9) is proven.The only question now is to ensure the existence of the solution.


## 8. Solution Existence

A sufficient condition for the existence of the minimum ofI(and consequently for the existence of a solution to (9)) is the coerciveness of the functional. Since the functional I is strictly convex, the coerciveness is ensured if the following holds [18]:(25)limγ⟶∞1γIγw¯=+∞,for anyw¯such thatw¯≠0,in which the above norm is defined as(26)w¯=∑α=1N∫Ωgradwα2+wα2dV.The limit in (25) may be expressed as follows:(27)limγ⟶∞1γIγw¯=limγ⟶∞1γ∑α=1N∫Ω12kαϕαΛgradγwα·gradγwαdV−∑α=1N∫Ωq˙αγwαdV++∑α=1N∑β=αN∫Ω12Rαβγwβ−γwα2dV+∑α=1N∫∂Ω12hαγwα−T∞2dA==limγ⟶∞γ∑α=1N∫Ω12ΛkαϕαΛgradwα·gradwαdV−∑α=1N∫Ωq˙αwαdV++γ∑α=1N∑β=αN∫Ω12Rαβwβ−wα2dV+γ∑α=1N∫∂Ω12hαwα−T∞γ2dA.In order to demonstrate the coerciveness, let us assume that somewα is not a constant. In such case, we have that(28)∫Ω12kαϕαΛgradwα·gradwαdV>0for someα.Therefore, sinceΛ is positive-definite, (25) holds.On the contrary, if all thewα are equal to a given constant (that must be nonzero), we have that(29)∑α=1N∫∂Ω12hαwα2dA>0.As previously assumed, at least onehα is nonzero. So, again, (25) holds.The last possibility arises when we assume that all thewα are constants, but at least one of them is different from the others. In such cases, we have, for a given α,(30)∑β=αN∫Ω12Rαβwβ−wα2dV>0,and, once more time, (25) holds.In fact, in order to reach a limit different from+∞, we must have(31)∑α=1N∫Ω12kαϕαΛgradwα·gradwαdV+∑α=1N∑β=αN∫Ω12Rαβwβ−wα2dV+∑α=1N∫∂Ω12hαwβ2dA=0.Nevertheless, the above equation holds, if and only ifwα=C≡0. But, in this case, we would have w¯=0, and this contradicts (25).Hence, the convex functionalI is coercive [18]. In other words, it admits at least one minimum. Since the functional is strictly convex, this minimum is unique [19].


## 9. Final Remarks

The existence and the uniqueness of the solution of (9) has proven. In addition, the equivalence between the variational principle (the minimum principle) and the original problem was proven too. This fact provides a useful tool for numerical simulations by means, for instance, of a finite element approximation.It is to be noticed that, assuming that the internal suppliesq˙α belong to L2Ω and that Ω has the cone property [12], the fields Tα which minimize the functional I are continuous and bounded. Such feature was expected for these quantities.



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*Source: 6623995-2021-02-22.xml*