H. CONSTITUTIVE EQUATIONS FOR A FLOWING FLUID – THE NEWTONIAN FLUID

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



u and that of Q as u + δu. Now a Taylor series approximation can be used to relate u and

u + δu, namely,

u + δu = u + (E + Ω) · δx + O(|δx|2 ).



(2–70)



The symbol O( ) is known as the order symbol. It indicates the magnitude of the error in

truncating the Taylor series approximation after only two terms, which is negligible in this

case since |δx|2

|δx|. It follows from (2–70) that the material point Q moves relative to

the material point P with a velocity

δu = E · δx + Ω · δx + O(|δx|2 ).



(2–71)



Now, the length of the line element connecting P and Q is

|δx| = (δx · δx)1/2 ,

and the rate of change in this length is thus proportional to

δx · δu = δx · [E · δx + Ω · δx + O(|δx|2 )],



(2–72)



where δu = [D(δx)/Dt]. However, because Ω is antisymmetric,

δx · Ω · δx ≡ O,

so that

1 D

(|δx|2 ) = δx · E · δx + O(|δx|3 ).

(2–73)

2 Dt

Thus the rate of change of the distance between two neighboring material points depends

on only the rate-of-strain tensor E, i.e., on the symmetric part of ∇u. It can be shown in

a similar manner that the contribution to the relative velocity vector δu that is due to the

vorticity tensor Ω is the same as the displacement that is due to a (local) rigid-body rotation

with angular velocity ␻/2, where

␻ = ␧ : Ω.



(2–74)



For future reference, we also note that

␻=∇



∧



u.



(2–75)



The vector ␻ is known as the vorticity vector. It can be calculated from u by means of either

(2–74) or (2–75).

To proceed beyond the general relationship (2–69), it is necessary to make a guess of

the constitutive behavior of the fluid. The simplest assumption consistent with (2–69) is

that the deviatoric stress (at some point x) depends linearly on the rate of strain at the same

point in space and time, that is,

␶ = A : E.



(2–76)



Here, A is a fourth-order tensor that must be symmetric in its first two indices,

Aijkl ≡ Ajikl ,



(2–77)



because ␶ is symmetric according to the constraint (2–41). The constitutive relation (2–76)

is analogous to the generalized Fourier model (2–65) for the heat flux vector q. Like the

generalized Fourier model, it assumes that the fluid is local, instantaneous, homogeneous,

and invariant to rotations or inversions of the coordinate axes.

An additional physical assumption that is satisfied by many fluids is that the structure is

isotropic even in the presence of motion. For an isotropic fluid, the constitutive equation is

completely unchanged by rotations of the coordinate system. It can be shown, by use of the

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Basic Principles



methods of tensor analysis, that the most general fourth-order tensor with this property10

is

Aijpq = λδi j δ pq + μ(δi p δ jq + δiq δ j p ) + ν(δi p δ jq − δiq δ j p )



(2–78)



where λ, μ, and ν are arbitrary scalar constants and δi j is the ij component of the identity

tensor I, that is,

δi j =



1

0



i= j

,

i= j



where i, j = 1, 2, 3.



(2–79)



Because the tensor A must also satisfy the symmetry condition, (2–77), it follows that

ν ≡ 0.

Substituting (2–78) into (2–76), we see that the most general constitutive equation for

the total stress T that is consistent with the linear and instantaneous dependence of the

deviatoric stress ␶ on E, plus the assumption of isotropy, is

T = (− p + λ tr E)I + 2μE.



(2–80)



Expressed in component form using Cartesian tensor notation, this equation is

Ti j = (− p + λE pp )δi j + 2μE i j .

Fluids for which this constitutive equation is an adequate model are known as Newtonian

fluids. We have shown that the Newtonian fluid model is the most general form that is linear

and instantaneous in E and isotropic. If the fluid is also incompressible,

tr E = ∇ · u = 0,

and the constitutive equation further simplifies to the form

T = − pI + 2μE.



(2–81)



The coefficient μ that appears in this equation is known as the shear viscosity and is a

property of the fluid.

We have, of course, said nothing about the physical reality of the assumptions of isotropy

or of a linear, instantaneous dependence of T on E. It is possible, insofar as continuum

mechanics is concerned, that no fluid would be found for which these are adequate assumptions. Fortunately, in view of the simplicity of the resulting constitutive model, (2–80) or

(2–81), experimental observation shows that the Newtonian constitutive assumptions are

satisfied for gases in almost all circumstances and for the majority of low- to moderatemolecular-weight liquids, providing that ||E|| is not extremely large and does not change too

rapidly with respect to time. Polymeric liquids, suspensions, and emulsions do not generally

satisfy the Newtonian assumptions and require much more complicated constitutive equations for T. We shall briefly discuss these latter fluids in the next two sections. An extremely

important fact is that Newtonian fluids are also generally found to follow Fourier’s law of

heat conduction in the isotropic form, (2–67).

It has been emphasized repeatedly that continuum mechanics provides no guidance in

the choice of a general constitutive hypothesis for either the heat flux vector q or the stress

tensor T. On the other hand, it was noted earlier that (2–41) and (2–56), derived respectively

from the law of conservation of angular momentum and the second law of thermodynamics,

must be satisfied by the resulting constitutive equations. It thus behooves us to see whether

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I. The Equations of Motion for a Newtonian Fluid – The Navier–Stokes Equation



these two constraints are satisfied for the Newtonian and the Fourier constitutive models

that have been proposed in this and the preceding section. In the absence of external body

couples, the constraint from angular momentum conservation requires only that the stress

be symmetric, and this is obviously satisfied by the Newtonian model for any choice of λ

and μ. The second constraint, from the second law of thermodynamics, requires

q · ∇θ

T : E + p(∇ · u) −

≥ 0.

θ2

Substituting for T from (2–80) and for q from (2–67), we find

k

(∇θ )2 ≥ 0.

θ

(2–82)

If we consider the special case of an isothermal, incompressible fluid, inequality (2–82)

becomes

2

λ + μ (tr E)2 + 2μ

3



E−



1

tr E I : E −

3



1

tr E I

3



+



2μ(E : E) ≥ 0.

Obviously, if the Newtonian constitutive model for an incompressible fluid is to be consistent

with the second law of thermodynamics, we require that the viscosity be nonnegative, that

is,

μ ≥ 0.



(2–83)



This is, in fact, the most significant result that can be obtained for incompressible Newtonian

fluids from the second-law inequality. If we do not restrict ourselves to incompressible or

isothermal conditions, inequality (2–82) can be satisfied for arbitrary motions and arbitrary

temperature fields only if

2

λ + μ ≥ 0, μ ≥ 0, and k ≥ 0.

(2–84)

3

The quantity [λ + (2/3)μ] is commonly called the bulk viscosity coefficient. Besides the

inequalities (2–84), no further information appears to be attainable for a Newtonian fluid

from the constraint (2–56).

I.



THE EQUATIONS OF MOTION FOR A NEWTONIAN

FLUID – THE NAVIER–STOKES EQUATION



Let us now return to the equations of motion for a Newtonian fluid. With the constitutive

equation, (2–80) [or (2–81) if the fluid is incompressible], the continuity equation, (2–5)

[or (2–20) if the fluid is incompressible], and the Cauchy equation of motion, (2–32), we

have achieved a balance between the number of independent variables and the number of

equations for an isothermal fluid. If the fluid is not isothermal, we can add the thermal

energy equation, (2–52), and the thermal constitutive equation, (2–67), and the system is

still fully specified insofar as the balance between independent variables and governing

equations is concerned.

In this section, we combine the Cauchy equation and the Newtonian constitutive equation

to obtain the Navier–Stokes equation of motion. First, however, we briefly reconsider the

notion of pressure in a general, Newtonian fluid.

The physical significance of pressure, as it first appeared in the constitutive equation for

stress in a stationary fluid, (2–60), is clear. This is the familiar pressure of thermodynamics.

When a fluid is undergoing a motion, however, the simple notion of a normally directed

surface force acting equally in all directions is lost. Indeed, it is evident on examining the

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Basic Principles



Newtonian constitutive equation, (2–80), that the normal component of the surface force

or stress acting on a fluid element at a point will generally have different values depending

on the orientation of the surface. Nevertheless, it is often useful to have available a scalar

quantity for a moving fluid that is analogous to static pressure in the sense that it is a measure

of the local intensity of “squeezing” of a fluid element at the point of interest. Thus it is

common practice to introduce a mechanical definition of pressure in a moving fluid as

1

p ≡ − tr T.

3



(2–85)



This quantity has the following desirable properties. First, it is invariant under rotation

of the coordinate axes (unlike the individual components of T). Second, for a static fluid

−1/3 · tr T = p, the thermodynamic pressure. And third, p has a physical significance

analogous to pressure in a static fluid in the sense that it is precisely equal to the average value

of the normal component of the stress on a surface element at position x over all possible

orientations of the surface (alternatively, we may say that 1/3 · tr T is the average magnitude

of the normal stress on the surface of an arbitrarily small sphere centered at point x).

The definition (2–85) is a purely mechanical definition of pressure for a moving fluid, and

nothing is implied directly of the connection for a moving fluid between p and the ordinary

static or thermodynamic pressure p. Although the connection between p and p can always

be stated once the constitutive equation for T is given, one would not necessarily expect

the relationship to be simple for all fluids because thermodynamics refers to equilibrium

conditions, whereas the elements of a fluid in motion are clearly not in thermodynamic

equilibrium. Applying the definition (2–85) to the general Newtonian constitutive model,

(2–80), we find

2

p = p − λ + μ ∇ · u,

3

1

2

␶=T+

¯

tr T I = 2μE − μ(∇ · u)I.

3

3



(2–86)



Only if the fluid can be modeled as incompressible does the connection between p and p

simplify greatly for a Newtonian fluid. In that case,

p ≡ p;



␶ = ␶ = 2μE.

¯



(2–87)



So far, we have simply stated the Cauchy equation of motion and the Newtonian constitutive equations as a set of nine independent equations involving u, T, and p. It is evident

in this case, however, that the constitutive equation, (2–80), for the stress [or equivalently

(2–86)] can be substituted directly into the Cauchy equation to provide a set of equations

that involve only u and p (or p). These combined equations take the form

ρ



∂u

2

+ u · ∇u = ρg − ∇ p + ∇ · (2μE) − ∇ · [μ(div u)I] .

∂t

3



(2–88)



If the fluid can be approximated as incompressible and if the fluid is isothermal so that the

viscosity μ can be approximated as a constant, independent of spatial position (note that

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