10.1 Introduction

In this chapter we develop the variational formulation to model the steady‐state heat transfer in rigid continuum media, that is, in bodies where the spatial position of particles remains invariant.

According to the roadmap established in Chapter 3, the first step in the construction of a variational model consists of defining the kinematics for the model, that is, the motion actions that particles can execute. In the heat transfer problem, the temperature is the primary scalar field that characterizes the average kinetic energy of molecules, and so it characterizes the kinematics in this problem. The temperature in a body is constrained to satisfy certain conditions, and therefore it is possible to define the concept of admissible variations of the temperature fields. With this, it is possible to introduce the generalized strain action operator, which in this context is denoted by (in Chapter 3 it was denoted by ), which leads us to the conception of the virtual internal power. From there, the characterization of generalized internal and external forces will follow, and then the application of the Principle of Virtual Power will be exploited to define the concept of equilibrium for the system.

Let us consider a body occupying a bounded and regular region ¹, with boundary , in the three‐dimensional Euclidean space . As a consequence, spatial coordinates coincide with the material (reference) coordinates admitted for the particles. That is, the spatial configuration and the material configuration of the body agree. For the heat transfer problem in bodies whose deformations are substantial, the developments of this chapter remain valid, but the reader has to keep in mind that the configuration of the body referred to in what follows is the spatial (actual, deformed) configuration the body occupies in space once the mechanical equilibrium has been achieved.

10.2 Kinematics

Making the analogy with the velocity field in the mechanics of continuum media, the scalar field called temperature, and denoted by , becomes the primal (also primary) variable in the heat transfer problem.²

We call the set of all possible temperature scalar fields that are sufficiently regular and can be defined over the body . Here, the set , endowed with the usual operations of addition and multiplication by a real number, becomes a vector space. By regular it is understood that is smooth enough such that all operations to be executed over this field are well‐defined.

Consider that there exists a portion of the boundary, called , where the temperature is prescribed. Then, we define the set

(10.1)

where is the value of the prescribed temperature on , as illustrated in Figure 10.1.

The set contains all admissible temperature fields for the problem under study. Moreover, we define the vector subspace associated with with fields satisfying homogeneous boundary conditions on , that is

(10.2)

It is easy to verify that is the translation of the subspace . The elements are called virtual thermal variations, or admissible temperature variations. With the previous elements we can write

(10.3)

where is arbitrary. Now, consider that the regularity of the temperature fields is such that we can express the temperature at a point close to using a Taylor expansion as follows

(10.4)

Calling to the vector field and for all points in a sufficiently small neighborhood of we can admit that the following holds

(10.5)

In addition, the field is called constant, or rigid, if it verifies

(10.6)

or equivalently

(10.7)

Thus, another fundamental ingredient in the present model is the vector field which represents the temperature spatial gradient (or thermal gradient). In particular, the set of all sufficiently regular fields is denoted by .

We will say that is a thermal gradient, or compatible gradient, if it is possible to determine such that

(10.8)

Thus, we can define the operator which for each assigns a thermal gradient .

The kernel of operator , denoted by , is then characterized by

(10.9)

and consists of all rigid thermal actions. All the elements and kinematical concepts introduced so far have a full parallel with the fundamental ingredients introduced in Chapter 3.

10.3 Principle of Thermal Virtual Power

As in the mechanics of continuum media, let us admit that external thermal loads that exert some action over the body are characterized by linear (and continuous) functionals . Hence, the external thermal loads are elements of the space dual to , here represented by . We will call the value this functional takes in the external thermal power, which is then given by

(10.10)

where is the duality product .

In this section we will see that these external thermal loads can be explicitly characterized by making use of the Principle of Thermal Virtual Power, which will be enunciated next.

First, let us define the internal thermal stresses through proper linear (and continuous) functionals defined over the space of temperatures and of thermal gradients . The value taken by such functionals at point is termed the internal thermal power, or simply . Once again, as in the mechanics domain, we will introduce a series of hypotheses that will allow us to find a representation for such a functional.

First, let us assume that the functional has the following general representation

(10.11)

The second hypothesis corresponds to admitting that is null for all rigid thermal actions (uniform ), that is

(10.12)

which yields

(10.13)

Since the expression above must hold for every part of the body , it implies

(10.14)

Thus, we have

(10.15)

With this procedure, we have arrived at a duality between the thermal gradient and the heat flux.³

We then conclude that internal thermal stresses are linear (and continuous) functionals defined over , that means they are elements in the space dual to , here designated by . By virtue of the Riesz representation theorem we can put these functionals in correspondence with a vector field, say , which is named as the heat flux vector field. Therefore, the value of this functional at is given by

(10.16)

We have then defined the spaces , , and and the dual products (linear forms) and which provide a correspondence between these spaces. Now, we define the adjoint operator , called the thermal equilibrium operator, as

(10.17)

At this stage we have all the elements required to formalize the variational principle which rules the problem, and which we call the Principle of Thermal Virtual Power (PTVP). Then, we say that a body , under the action of the external thermal loads , is at static⁴ thermal equilibrium if

(10.18)

which implies that the external thermal virtual power is nullified for all rigid thermal actions and, in addition, satisfies

(10.19)

that is, the sum of the internal thermal virtual power and of the external thermal virtual power equals zero for all admissible virtual thermal actions.

The first part of the PTVP enables us to establish which are the external thermal loads compatible with the thermal equilibrium. Note that this first part makes sense provided the existence of for a given at equilibrium is demonstrated. This is, in fact, the reason to show the following result.

According to the previous results we conclude that

(10.28)

which means

(10.29)

Hence, the theorem 10.1 and the concept of equilibrium for rigid thermal actions allow us to obtain the second part of the PTVP.

In this manner, we arrived at the point in which we have been given all the necessary elements to formulate the variational equations and retrieve from them the associated Euler–Lagrange equations which govern the steady‐state heat transfer problem in strong form. Hereafter, we will provide a more concrete notation to these ingredients, specifying the context of three‐dimensional heat transfer with . Therefore, from the PTVP we have

(10.30)

If and are sufficiently regular fields, we can integrate by parts as follows

(10.31)

where we have used the fact that over , where with (see Figure 10.1).

Placing (10.31) into expression (10.30) provides the characterization for the element

(10.32)

This result tells us that the external thermal loads compatible with the kinematical model consist of a thermal load per unit volume, denoted by , and a thermal load per unit area, denoted by .

In diverse applications of practical interest, where the material surrounding the solid body is a flowing fluid, the load depends upon the temperature of the solid boundary, , of the fluid temperature at remote locations, say , and of the properties of the fluid and of the type of surface, lumped into a parameter , through a relation . We call this physical phenomenon the convective heat exchange, and the surface where this exchange takes place is denoted . If we consider we obtain the following representation for

(10.33)

Then, the PTVP consists of the variational equation

(10.34)

To extend the PTVP to the case of unsteady heat transfer it is enough to replace by where is the density for the material in and is the specific heat, being in the case of solids.

Thus, in order to find the local form of the variational problem (10.34) we have to find the associated Euler–Lagrange equations, whose abstract form is as we have seen in (10.20). We proceed with the integration by parts for the left‐hand side of equation (10.34), yielding

(10.35)

Using now standard variational arguments (fundamental theorem of the calculus of variations), we find

(10.36)

(10.37)

(10.38)

which are the Euler–Lagrange equations sought for the variational equation (10.34).

Notice that, even in situations in which the fields are not regular enough so that integration by parts can be safely performed as we did, the PTVP still holds. In this regard, consider, for example, that there exists an internal surface, say , which divides the body into two parts, each of which features smooth fields so that integration by parts can be pursued. In this case, from the integration by parts procedure emerges an additional term of the form . This indicates that the system of external thermal loads could account for a load per unit area defined over the internal surface , say . Hence, the equilibrium in such case implies

(10.39)

Let us now characterize the reactive flux by releasing the kinematical constraints from space . To do this, note that (10.20) is equivalent to the following

(10.40)

Using the PTVP yields

(10.41)

and, after integration by parts, it becomes

(10.42)

As a direct consequence, we have that

(10.43)

which, in local form, implies

(10.44)

The reactive flux is the normal component of the heat flux vector field over the part of the boundary in which the temperature has been constrained.

Finally, the reader will find in Figure 10.2 the main concepts and results which characterize the variational formulation.

10.4 Principle of Complementary Thermal Virtual Power

The previous section dealt with the primal variational form of the steady‐state heat transfer problem, characterized through the PTVP, which establishes a connection between the internal thermal stress and the external thermal load so that (steady‐state) equilibrium is achieved.

As in the mechanics of deformable bodies, for the heat transfer problem it is also possible to introduce a problem which can be regarded as dual to the primal problem, in the sense that it provides a characterization of the compatibility of the thermal gradient vector field , that is, to determine whether there exists such that . To formulate this problem we first define the following sets

(10.45)

which is the set of all internal thermal stresses that are equilibrated with the external thermal load , and

(10.46)

which is the vector subspace of whose elements are internal thermal stresses (heat flux fields) at thermal equilibrium with the null thermal load.

As before, it is important to note that if and then

(10.47)

because

(10.48)

This result shows us that is a translation of , that is

(10.49)

With the previous definitions, we can now enunciate the Principle of Complementary Thermal Virtual Power (PCTVP). We say that the vector field is thermally compatible for an external thermal load at equilibrium if and only if

(10.50)

for some .

To see this result, consider first that is compatible, then such that . Since is a translation of , that is, , with arbitrary, we have . Then

(10.51)

which results in the very definition of the space .

Suppose now that

(10.52)

where . Hence

(10.53)

and we conclude that

(10.54)

Moreover, from (10.46) we also know that

(10.55)

and then

(10.56)

Replacing (10.56) into (10.54) yields

(10.57)

which means or, equivalently, such that .

For the case under analysis, expression (10.50) corresponds to

(10.58)

so by making use of integration by parts we find

(10.59)

where we make use of the fact that in , on and on , and on .

Therefore, we conclude that the PCTVP acquires the following variational form

(10.60)

10.5 Constitutive Equations

In our road towards the formulation of a field problem for the heat transfer problem there is still a missing aspect concerning the material behavior in the present context. This constitutive material response determines the way in which the heat flux is related to the thermal gradient . That is, we have to look for an operator that maps the space into its dual . A typical constitutive model is that given by a linear relation, which states that

(10.61)

where is a second‐order tensor and the sign indicates that the heat flux opposes the thermal gradient. Next, we further assume that the tensor satisfies the following properties

(10.62)

(10.63)

(10.64)

Let us now define the thermal gradient energy density function as

(10.65)

Notice that from the properties of it follows that

(10.66)

where the equality is verified for .

Thus, the heat flux related to the thermal gradient through the constitutive equation is such that

(10.67)

In turn, a Taylor's expansion around the point provides

(10.68)

which gives

(10.69)

while the equality is verified for . In this case we conclude that is a strictly concave function.

In the case of materials featuring nonlinear constitutive behavior we assume the following property for holds

(10.70)

with the equality being valid for , and , . Let be a thermal gradient whose heat flux is . Then

(10.71)

and integrating the process that takes us from to we arrive at

(10.72)

and so we get

(10.73)

where we have recovered the concavity of function from the monotonicity property stated by (10.70).

By virtue of the positive‐definiteness of tensor , we can calculate the inverse , and we can find the inverse constitutive equation as follows

(10.74)

therefore it is possible to introduce the complementary thermal gradient energy density function as

(10.75)

where the equality is verified for . Furthermore,

(10.76)

and also

(10.77)

which allows us to conclude that function is also concave in the space .

From the properties of and we verify that

(10.78)

where the equality is satisfied if and are related by means of the constitutive equation.

10.6 Principle of Minimum Total Thermal Energy

The goal of this section is to transform the steady‐state heat transfer problem into a minimization problem with a corresponding cost functional. Therefore, we proceed analogously to the steps presented in Chapter 4.

Since function is concave, from (10.73) we can write

(10.79)

where the equality holds for all . Integrating (10.79) in the domain occupied by the body , we obtain the following

(10.80)

Replacing this result into the PTVP expressed by equation (10.30) gives

(10.81)

Let us define the Total Thermal Energy as follows

(10.82)

then after arranging terms we get

(10.83)

In this way, we have arrived at the Principle of Minimum Total Thermal Energy, which consists of finding that minimizes the Total Thermal Energy functional, that is

(10.84)

10.7 Poisson and Laplace Equations

Consider the constitutive model most widely used in practice for this kind of problem. Such a model is known as the Fourier law and it establishes that the heat flux and the thermal gradient are related as follows

(10.85)

where is characterized by a single (positive) scalar field , called the material thermal conductivity, and is the second‐order identity tensor.

Then, we rewrite the PTVP given by (10.34) for the particular choice of a material ruled by constitutive equation (10.85)

(10.86)

The Euler–Lagrange equations are obtained by integrating by parts the left‐hand side in the above expression and using standard arguments from the calculus of variations

(10.87)

(10.88)

(10.89)

Let us further simplify the presentation for the case of thermally homogeneous materials, that is, for materials featuring a constant conductivity , and also for homogeneous essential boundary conditions over the whole boundary, that is, over . Then, the problem is reduced to the widely known Poisson problem

(10.90)

(10.91)

If now we consider a given over the boundary , and that , we arrive at the Dirichlet problem for the Laplace equation

(10.92)

(10.93)

which characterizes the so‐called harmonic functions.

Next, we present the variational problem (10.86) in the format of a minimization problem. Thus, and noting that for the Fourier law the thermal energy density function is given by

(10.94)

the total thermal energy functional (10.82) gives

(10.95)

Finally, after rearranging terms, the minimization problem standing from (10.84) amounts to finding such that

(10.96)

Note that the term becomes quadratic when paired with , this gives rise to the term of the form in the expression above. Here we have simplified the presentation by directly modifying the cost functional to account for this fact, and to ensure that the first Gâteaux derivative of functional in (10.95) actually corresponds to the variational equation (10.86). An alternative more systematic way to do this involves postulating that the internal energy is not only composed by integrated over the body domain , but also by a surface energy of the form , defined over the boundary . In turn, the external system of body forces must include a heat flux of the form .