G. THE CONSTITUTIVE EQUATION FOR THE HEAT FLUX VECTOR – FOURIER’S LAW

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G. The Constitutive Equation for the Heat Flux Vector – Fourier’s Law



the success of our guess by comparing measured and predicted temperature fields in real

fluids.

A reasonable initial guess is the simplest assumption that is consistent with the relationship (2–64); namely, that the heat flux vector, q(x, t), depends linearly on∇θ(x, t):

q = −K · ∇θ.



(2–65)



Here, K is a second-order tensor that is known as the thermal conductivity tensor, and

the constitutive equation is known as the generalized Fourier heat conduction model for

the surface heat flux vector q. The minus sign in (2–65) is a matter of convention; the

components of K are assumed to be positive whereas a positive heat flux is defined as going

from regions of high temperature toward regions of low temperature (that is, in the direction

of −∇θ).

The reader may well be curious why the particular linear form of (2–65) was chosen

because at least one other vector function is linear in ∇θ , namely β ∧ (∇θ ), where β is a

constant vector. To provide a complete answer, it is necessary to introduce two important

principles that all constitutive relations are expected to obey. The first, which is frequently

taken for granted, may be called coordinate invariance. This principle states simply that

the form of a constitutive equation must be invariant under orthogonal coordinate transformations. Underlying this principle is the obvious fact that a change in orientation or sense

of the coordinate system cannot influence the relevant physical processes and thus should

not influence the form of the constitutive equation. The second invariance requirement of

a constitutive equation is that it must also remain unchanged under a transformation in the

frame of reference of the observer, even if the frame of the observer (or the fluid) is accelerating with respect to an inertial frame. This is usually thought of as being a consequence of

the intuitive notion that the mechanical or thermal properties of a material element cannot

depend on any motion of the person observing the material and is called the principle of material objectivity. Material objectivity is a stronger requirement than coordinate invariance,

but is relevant only for constitutive equations that involve dynamical variables, such as u.

Returning to the form of the constitutive equation for q, we have seen that there are two

distinct possibilities that are linear in∇θ , namely (2–65) and β ∧ (∇θ ). In this case, the

principle of coordinate invariance is sufficient to distinguish between these two possibilities.

The reader who is experienced with vector and tensor analysis may immediately recognize

that β ∧ (∇θ ) is not an acceptable form because it consists of the vector product (or cross

product) of two vectors and is thus a pseudo-vector. A key property of a pseudo-vector is

that it changes sign if we invert the coordinate axes from a right- to a left-handed coordinate

system whereas a true vector is invariant to this transformation. In particular, if we define L as

the coordinate transformation matrix (L · LT = I when the transformation is orthogonal),

then a pseudo-vector transforms according to the rule B = (det L)L · B whereas a true

vector transforms according to A = L · A. The vector formed as the cross product of the

two vectors β ∧ (∇θ ) changes sign on inversion of coordinates and it is thus a pseudo-vector.

The heat flux vector, on the other hand, is a true vector that is invariant to such changes of

coordinate systems. One condition for satisfying coordinate invariance is that all terms in

any equation involving vectors or tensors must have the same “parity” – that is, they must

all be either true vectors or they must all be pseudo-vectors. Because q is a true vector, the

only choice for the form of a constitutive equation that is linear in ∇θ and involves ∇θ only

at the present moment in time and the same point in space as q, is (2–65). Although the

same result can be obtained formally by application of a coordinate transformation to the

terms, q, K · ∇θ , and β ∧ (∇θ ), we will be content here to accept the conclusion based on

the qualitative arguments previously outlined.

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Basic Principles



It is important to emphasize that the mathematical constraint imposed by coordinate

invariance addresses only the selection of an allowable form of a constitutive equation,

given the physical assumption, based on an educated guess, that there is a linear relationship

between q and ∇θ . Whether the resulting constitutive equation captures the behavior of

any real material is really a question of whether the physical assumption of linearity is

an adequate approximation. In fact, in the generalized Fourier heat conduction model,

Eq. (2–65), there are several additional physical assumptions that must be satisfied, besides

linearity between q and ∇θ :

1. The surface heat transfer process is assumed to be local, in the sense that the flux

associated with the fluid at some point depends on only the temperature gradient at the

same point.

2. The surface heat transfer process is assumed to be instantaneous; the heat flux at a

point depends on only the temperature gradient at that point at the same instant of time.

In particular, there is no dependence on the thermal history of the fluid element that

currently occupies the point in question.

3. The fluid is assumed to be homogeneous. The form of the relationship between the

heat flux q and the temperature gradient ∇θ is the same at all points. Furthermore, the

only dependence of q on position x is due to the possible dependence of the so-called

thermal conductivity tensor K on the thermodynamic state variables (say, p and θ ) or

the dependence of ∇θ on spatial position.

4. When there is no temperature gradient, the surface heat flux is identically zero.

It should be emphasized that all the preceding points are simply assumptions underlying

the assumed constitutive form (2–65). We can make no claim on the basis of continuum

mechanics alone that these assumptions or the basic linearity of (2–65) will necessarily be

satisfied by any real fluid.

Fortunately, in view of the simplicity of (2–65), comparison between predicted and

measured data for the heat flux and temperature gradient shows that the general linear form

does work extremely well for many common gases and liquids. However, the majority of

these materials exhibit one additional characteristic that leads to further simplification of

the constitutive form (2–65) – they are isotropic. This means that the magnitude of the heat

flux at any point is dependent on only the magnitude of the temperature gradient, not on its

orientation relative to axes fixed in the material. A common material that is not isotropic in

this sense is wood, because a temperature gradient of given magnitude in wood generally

produces a larger heat flux if it is oriented along the grain than it does if it is oriented across

the grain. In the absence of motion, almost all common fluids will be isotropic (an exception

is a liquid crystalline material).18 If the fluid is made up of molecules and/or particles that

are not spherical (or spherically symmetric), the orientations of these molecules or particles

will generally be random at equilibrium as a consequence of random (Brownian) motions.

Hence, when seen from the spatially averaged continuum viewpoint, such a fluid will be

isotropic.

A mathematical statement of the property of isotropy is that the constitutive equation must be completely invariant to orthogonal rotations of the coordinate system. For

the constitutive form (2–65), it can be shown that this condition will be satisfied if and

only if

K = kI,

where k is a scalar property of the fluid that is known as the thermal conductivity.

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(2–66)



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H. Constitutive Equations for a Flowing Fluid – The Newtonian Fluid



Thus, for an isotropic fluid that exhibits a linear, instantaneous relationship between the

heat flux and temperature gradient, the most general constitutive form for q is

q = −k∇θ.



(2–67)



This is known as Fourier’s law of heat conduction. We may note that the inequality (2–56)

imposes a restriction on the sign of the thermal conductivity k. In particular, in the absence

of fluid motion, (2–56) reduces to the simple form

−



q · ∇θ

≥ 0.

θ2



It follows from this inequality and the constitutive form (2–67) that

k



∇θ

θ



2



≥ 0.



Hence, assuming k = 0, we see that the thermal conductivity must be positive, k > 0.

Although this simplified version of Fourier’s heat conduction law is well known to be

an accurate constitutive model for many real gases, liquids, and solids, it is important to

keep in mind that, in the absence of empirical data, it is no more than an educated guess,

based on a series of assumptions about material behavior that one cannot guarantee ahead

of time to be satisfied by any real material. This status is typical of all constitutive equations

in continuum mechanics, except for the relatively few that have been derived by means of

a molecular theory.

H. CONSTITUTIVE EQUATIONS FOR A FLOWING

FLUID – THE NEWTONIAN FLUID



In the previous sections, we discussed constitutive approximations for the stress and surface

heat flux in a stationary fluid, where u ≡ 0. In view of the molecular origins of q, there

is no reason to expect that the basic linear form for its constitutive behavior should be

modified by the presence of mean motion, at least for materials that are not too complicated

in structure. Of course, this situation may be changed for materials such as polymeric liquids

or suspensions, because in these cases the presence of motion may cause the structure to

become anisotropic or changed in other ways that will affect the heat transfer process. We

will return to this question in Section J.

The constitutive equation, (2–60), for the stress, on the other hand, will be modified for

all fluids in the presence of a mean motion in which the velocity gradient ∇u is nonzero.

To see that this must be true, we can again consider the simplest possible model system of

a hard-sphere or billiard-ball gas, which we may assume to be undergoing a simple shear

flow,

u = γ yix

˙

with velocity in the x direction and a gradient of magnitude γ in the y direction. The

˙

parameter γ is called the shear rate. A flow of this type, which has the velocity in a single

˙

direction, is called unidirectional. In fact, in the next chapter, we will consider the general

class of unidirectional flows. Now let us consider a surface in this fluid whose normal n is

parallel to ∇u (that is, n ≡ i y ). In this case, any interchange of molecules across the surface

will result in a transfer of mean momentum; that is, the faster-moving fluid on one side of

the surface will appear to be decelerated as one of its molecules is exchanged for a slower

moving molecule from the other side of the surface, whereas the slower-moving fluid will

appear to be accelerated by the same process. Hence, from the continuum point of view in

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Basic Principles



which this transfer of momentum is modeled in terms of equivalent surface forces (stresses),

we see that the surface-force vector in a moving fluid must generally have a component that

acts tangent to the surface, and this is fundamentally different from the case of a stationary

fluid in which the only surface forces are pressure forces that act normal to surfaces. We

may also note, in the case of a hard-sphere gas, that the rate of momentum transfer by means

of random molecular motions is proportional to ∇u. On the other hand, we know that T

must reduce to the form (2–60) when ∇u = 0, both for a hard-sphere gas and for other,

more complicated (real) fluids.

We conclude, based on the insight that we have drawn from the hard-sphere gas model

and our general understanding of the molecular origins of T, that

T + pI = ␶ (∇u, terms involving higher-order spatial derivatives)



(2–68)



for real fluids, where ␶ (∇u . . .) could be either a function of current and local values of

∇u or a functional that includes a dependence on both previous and current values. The

second-order tensor ␶ is usually known as the deviatoric stress.

To obtain a specific form for ␶(∇u . . .), we again require a guess. However, some general

properties of ␶ can be deduced that do not depend upon a specific constitutive form, and we

begin by discussing these general properties. First, it is obvious from (2–58) and (2–68) that

␶(∇u . . .) = 0 for ∇u = 0, provided that a large-enough time increment has passed after

setting ∇u = 0. The requirement that ␶ → 0 asymptotically for t

1 is necessary because

all fluids are not “instantaneous” in the sense that ␶ depends on only the current values of

∇u. It is known, for example, that fluids exist where the deviatoric stress, ␶(∇u . . .), vanishes

only if ∇u has been zero for a finite period. Such fluids are said to possess a “memory” for

past configurations and are typified by polymer solutions in which the molecular structure

can return to an equilibrium state only by means of diffusion processes that require a finite

period of time. A second general property of ␶ is that it is symmetric in the absence of

external body couples. This follows directly from (2–68) and the fact that T is symmetric

in the same circumstances, as shown in Section C. A third general property is that ␶ must

depend explicitly on only the symmetric part of ∇u, rather than on ∇u itself. We have

already noted in (2–49) that the symmetric part of ∇u is called the rate-of-strain tensor and

is usually denoted as E. It might seem, at first, that this third property would follow from

the fact that ␶ is symmetric, but this is not the case.

There are two proper explanations, one based on physical intuition and the other based

on the principle of material objectivity. The latter is discussed in many books on continuum

mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis

of this is that contributions to the deviatoric stress cannot arise from rigid-body motions –

whether solid-body translation or rotation. Only if adjacent fluid elements are in relative

(nonrigid-body) motion can random molecular motions lead to a net transport of momentum.

We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change

of the length of a line element connecting two material points of the fluid (that is, to relative

displacements of the material points), whereas the antisymmetric part of ∇u, known as the

vorticity tensor Ω, is related to its rate of (rigid-body) rotation. Thus it follows that ␶ must

depend explicitly on E, but not on Ω:

␶ = ␶(E, . . . , ).



(2–69)



To prove our assertions about the physical significance of the rate-of-strain and vorticity

tensors, we consider the relative motion of two nearby material points in the fluid P, initially

at position x and Q, which is at x + δx. We denote the velocity of the material point P as

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