Abstract

The subject of this paper is the steady-state heat transfer process in a rigid mixture with 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 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 coupled partial differential equations, we need to look only for the minimum of a single functional.

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 continua occupying, simultaneously, the whole volume of the body. In other words, the body is regarded as a mixture consisting of 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 with constituents) will be described by a set of 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 of 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 of continua, occupying, respectively, the spatial regions … with if and with , be represented by a set of continuous constituents, all of them occupying simultaneously the entire region (the whole mixture), in such a way that the original interfaces vanish, as suggested in Figure 1.

This superposition allows the existence of distinct temperatures at each spatial point in the mixture.

It is to be noticed that these 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 by (for the constituent ).

The field will depend on the difference between the temperature of the 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 constituent of a given rigid continuous mixture (composed by constituents) is mathematically described as follows [4]:in which is a bounded open set representing the region occupied by the mixture, represents the mass density of (ratio between the mass of the constituent to the corresponding volume of mixture—different from the classical mass density), is the specific heat of , is the temperature of , is the partial heat flux associated with , is the internal heat supply (per unit time and per unit volume) for , and is the energy supply arising from the interaction between and the others constituents of the mixture.

The internal heat supply is an internal effect. It provides the coupling among the heat transfer processes in all the constituents and represents (for each ) the amount of energy, per unit time and unit volume, supplied to the constituent 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):

3. Constitutive Relations

The partial heat flux (per unit time and unit area) associated with the constituent is given byin which is a positive-definite second order tensor (usually constant), is the thermal conductivity of the material represented by , and is the ratio between and the actual mass density of the material represented by . The fields are such that

The fields are always known, for any .

The internal heat supply is the sum of the heat supplies coming from all the constituents to the constituent . In other words, we may writein which is the heat supply from to . Hence,

As a consequence of the Second Law of Thermodynamics [9], the sign of , at each point, depends only on the sign of the difference at this point. If , must be positive (since will receive energy from ). In other words, the sign of is the same sign of . This fact induces the following constitutive relationship:

The simplest case is the one in which is assumed to be a constant for any and .

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 , we have the following boundary condition:in which is a nonnegative constant (convection heat transfer coefficient), is a temperature of reference, and is the unit outward normal defined on . It will be assumed that, for at least one constituent, be positive on some nonempty subset of .

It is to be noticed that, when 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:in which the unknowns are the temperature fields .

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 that and do not depend on the unknowns , and the problem represented by the set of partial differential equations and boundary conditions is equivalent to the minimization of the following convex and coercive functional [11]:in which the admissible fields must belong to the Sobolev space [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 .

In order to show that the solution of problem (9) is equivalent to the minimization of the functional defined in (10), we begin evaluating the first variation of , denoted here by .

The first variation of is given as follows [15]:or, employing Green’s identity [16],

Since

The last term of (12) may be rewritten as follows:or replacing by in the last term and taking into account that (and that if ),

So, (12) may be rewritten as

The extrema of the functional are obtained for the fields such that gives rise to .

Imposing and taking into account that the fields are arbitrary and independent, we have that the fields must satisfy (Euler–Lagrange equations and natural boundary conditions):

It is easy to see that problems (17) and (9) are equivalent. In other words, the extrema of 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 . In other words, it will be shown that if satisfies (9), then

Aiming to this, it is sufficient to prove that the functional is strictly convex, which means it satisfies the following inequality [17]:in which and .

The first step for proving (19) is to show that, for , the expression is always nonnegative.

Aiming to this, let us write the above expression as follows:

Reordering some terms and taking into account thatwe may rewrite (20) as follows:

It is easy to see that (22) is nonnegative for any . Now, we must prove that, for any , expression (22) is positive valued.

Aiming to this, let us suppose that, for some , the difference is not a constant. In this case, since is positive valued for any and is a positive-definite tensor, we have

On the contrary, if the difference is a nonzero constant, we have that

Therefore, the positiveness of (22) is proven. This ensures (19). Since (19) holds, the functional 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 of (and consequently for the existence of a solution to (9)) is the coerciveness of the functional. Since the functional is strictly convex, the coerciveness is ensured if the following holds [18]:in which the above norm is defined as

The limit in (25) may be expressed as follows:

In order to demonstrate the coerciveness, let us assume that some is not a constant. In such case, we have that

Therefore, since is positive-definite, (25) holds.

On the contrary, if all the are equal to a given constant (that must be nonzero), we have that

As previously assumed, at least one is nonzero. So, again, (25) holds.

The last possibility arises when we assume that all the are constants, but at least one of them is different from the others. In such cases, we have, for a given ,and, once more time, (25) holds.

In fact, in order to reach a limit different from , we must have

Nevertheless, the above equation holds, if and only if . But, in this case, we would have , and this contradicts (25).

Hence, the convex functional 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 supplies belong to and that has the cone property [12], the fields which minimize the functional are continuous and bounded. Such feature was expected for these quantities.

Appendix

A. Numerical Example

In order to illustrate the use of the functional defined in (10) for obtaining numerical results, let us consider a particular isotropic rigid binary mixture; for instance, a reinforced concrete body or simply a clay body structured by steel strings.

In such case, problem (9) reduces to

Assuming that is the identity tensor (isotropic mixture) that the body is represented by the interval , , , , , , , , and are constants we have (dependence only on the rectangular Cartesian coordinate ), and we have (A.1) reduced to

In this case, the functional defined in (10) reduces to