I. THE EQUATIONS OF MOTION FOR A NEWTONIAN FLUID – THE NAVIER–STOKES EQUATION

P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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

50



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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



any dependence of μ on pressure is generally weak enough to be neglected altogether), this

equation takes the form

∂u

+ u · ∇u = ρg − ∇ p + μ∇ 2 u.

∂t



ρ



(2–89)



This is the Navier–Stokes equation of motion for an incompressible, isothermal Newtonian

fluid. Two comments are in order with regard to this equation. First, we shall always assume

that ρ and μ are known–presumably by independent means – and attempt to solve (2–89)

and the continuity equation, (2–20) for u and p. Second, the ratio μ/ρ, which is called the

kinematic viscosity and denoted as ν, plays a fundamental role in determining the fluid’s

motion. In particular, it can be seen from (2–89) that the contribution to acceleration of

a fluid element (Du/Dt) that is due to viscous stresses is determined by ν rather than

by μ.

Finally, we note that it is frequently convenient to introduce the concept of a dynamic

pressure into (2–89). This is a consequence of the utility of having the pressure gradient that

appears explicitly in the Navier–Stokes equation act as a driving force for motion. In the

form (2–89), however, a significant contribution to ∇ p is the static pressure variation∇ p =

ρg, which has nothing to do with the fluid’s motion. In other words, nonzero pressure

gradients in (2–89) do not necessarily imply fluid motion. Because of this it is convenient

to introduce the so-called dynamic pressure P, such that ∇ P = 0 in a static fluid. This

implies

−∇ P ≡ ρg − ∇ p.



(2–90)



Note that this equation requires that the sum ∇ p − ρg be expressible in terms of the

gradient of a scalar P. This is possible only if ρ is constant – that, is the fluid must be both

incompressible and isothermal. In this case,

ρ



∂u

+ u · ∇u = −∇ P + μ∇ 2 u.

∂t



(2–91)



Equation (2–91) would seem to imply that the gravity force ρg has no direct effect on the

velocity distribution in a moving fluid, provided the fluid is incompressible and isothermal

so that the density is constant. This is generally true. An exception occurs when one of

the boundaries of the fluid is a gas–liquid or liquid–liquid interface. In this case, the actual

pressure p appears in the boundary conditions (as we shall see), and the transformation of

the body force out of the equations of motion by means of (2–90) simply transfers it into the

boundary conditions. We shall frequently use the equations of motion in the form (2–91),

but we should always keep in mind that it is the dynamic pressure that appears there.

Finally, it was stated previously that fluids that satisfy the Newtonian constitutive equation for the stress are often also well approximated by the Fourier constitutive equation,

(2–67), for the heat flux vector. Combining (2–67) with the thermal energy, (2–52), we

obtain.



ρC p



Dθ

θ

=−

Dt

ρ



∂ρ

∂θ



p



Dp

+ p(∇ · u) + (T : E) + ∇ · (k∇θ ).

Dt



(2–92)

51



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



Basic Principles



For an incompressible Newtonian fluid this take the simpler form

ρC p



θ

Dθ

=−

Dt

ρ



∂ρ

∂θ



p



Dp

+ 2μ(E : E) + ∇ · (k∇θ ),

Dt



(2–93)



which we shall use frequently in subsequent developments.



J.



COMPLEX FLUIDS – ORIGINS OF NON-NEWTONIAN BEHAVIOR



Experience has shown that the Newtonian fluid model subject to the previously discussed

assumptions, provides a very good approximation under most flow conditions for gases

and small-molecule liquids such as water. The reason for this is that the random, thermally

driven motions within such materials at moderate temperatures are sufficiently vigorous

that they completely overcome any tendency of the forces associated with flow to produce a molecular configuration state that differs significantly from the isotropic, homogeneous state of statistical equilibrium. There is net displacement of mass associated with

the mean (continuum-level) flow, but virtually no change in the “shapes,” orientation distributions, or other statistical features of the configuration of the material at the molecular

level.

There is, however, a vast body of materials whose behavior as liquids undergoing flow

do not satisfy the assumptions for a Newtonian fluid. This includes many polymeric liquids,

suspensions, multifluid blends, liquids containing surfactants that tend to form particle-like

micelles when they are present at high concentrations, and many others. As we shall subsequently discuss from a qualitative point of view, these fluids exhibit more complicated

macroscopic properties and have historically been lumped together under the general designation of non-Newtonian fluids. In the more recent literature, they have also been called

complex liquids.20

This book is focused almost entirely on the dynamics of, and transport processes in,

Newtonian fluids. Nevertheless, the class of complex or non-Newtonian fluids is extremely

common, especially in chemical engineering applications, and it is probably useful to spend

some time discussing what is responsible for the departures from Newtonian behavior. We

shall see that the applicability of the Newtonian fluid model is based on a combination of the

intrinsic characteristics of the fluid and the characteristics of the flow. Specifically, a fluid

may exhibit behavior that can be described accurately by the Newtonian fluid model under

one set of flow conditions but appear as non-Newtonian under a different set of conditions.

One motivation for discussing the origins of non-Newtonian behavior is that this will help us

to understand when we might expect the Newtonian fluid model to apply. A second is that it

will give us a very crude and qualitative understanding of the behavior that is characteristic

of non-Newtonian materials.

Let us begin by discussing a specific example, namely, a dilute solution of rigid rodlike

macromolecules in a solvent. By dilute, we mean that the macromolecules are far enough

apart that they do not interact directly in the flow. An analogous system is a suspension of

rod-shaped colloidal particles. In both cases, we suppose that the macromolecules/particles

are small enough to experience significant random rotational motion that is due to Brownian

motion, so that their orientation and position in the absence of flow is a random function of

time. To describe the microstructural state of the solution/suspension, we therefore require a

probability density function (p, x p ), which prescribes the probability of finding a particle

with its principal axis oriented in a direction p at a position x p . In the following discussion,

we assume that the macromolecules/particles are always uniformly dispersed with respect to

spatial position. Hence the probability of a particle being at some position can be specified

52



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



J. Complex Fluids – Origins of Non-Newtonian Behavior



in terms of a concentration, say c, that is assumed to be independent of position. Then

specification of the microstructural state requires only the orientation distribution function,

N (p; x, t).

The probability of finding a particle/macromolecule with an orientation within a solid angle

dp of some orientation p is

N (p; x, t)dp.

In general, this probability will be a function of both position x and time t. However,

at equilibrium, all orientations are equally probable. Hence the equilibrium orientation

distribution, normalized so that the total probability of orientation in some direction is

unity, is just

1

N (p; x, t) =

.

4π

In general, we seek a description of the solution/suspension from a macroscopiccontinuum perspective. Hence we generalize our notion of a material point to be a volumetric

region that is small enough compared with the (macroscopic) length scales of a flow domain

to be considered as a point, but large enough to contain a statistically meaningful number

of particles/macromolecules. Any macroscopic property of the suspension/solution at a

material point will then depend on the orientation distribution at that point and time. In

the equilibrium state, the orientation probability is uniform, as already noted, and hence

the material at equilibrium is statistically isotropic, in spite of the fact that individual particles/macromolecules are rodlike in shape.

Now, let us think qualitatively about what happens to the material when it is subjected

to a flow. For simplicity, let us imagine that the flow is a simple shear flow

u = γ y ix ,

˙



(2–94)



with a shear rate γ . Simple shear flow is a good approximation to the flows generated in

˙

many rheometers – namely, the instruments designed to measure the flow properties of

liquids – and thus it is an appropriate choice for the present discussion.

When an individual particle/macromolecule is subjected to a flow of the type (2–94),

it will tend to rotate. In general, the rotational motion of a rod in a shear flow is complex,

though a complete theory for rodlike ellipsoids had been developed already in the early

1900s by Jefferey (1922). 21 For present purposes, let us simply consider a rod that lies in

the plane of the flow (i.e., the x–y plane). This rod will rotate. However, the rate of rotation is

not uniform. If the rod is oriented in the y direction (across the flow), it is intuitively obvious

that the hydrodynamic torque on it will be much larger than if the same rod were oriented

with its axis in the x direction (parallel to the flow). Hence it will rotate more rapidly in the

cross-flow orientation than when it is aligned with the flow. Even though the rod will continue

to tumble over and over, it will spend relatively longer in orientations that are near alignment

with the flow and less time in orientations that are across the flow. Thus the action of the flow

is to produce a nonuniform orientation distribution, with larger N values in the flow-aligned

direction and smaller values away from this direction. Of course, Brownian motion is still

present – i.e., “rotational diffusion” – and this continues to try to maintain the orientation

distribution in a uniform or completely random state. The orientation distribution that results

can be viewed as a consequence of the competition between the flow-induced tendency

toward a nonuniform orientation distribution and the tendency of Brownian diffusion to

maintain a uniform state. The strength of the flow effect will clearly be proportional to γ ,

˙

whereas the diffusion process is characterized by a so-called rotational diffusivity D. Both

of these quantities have units of inverse time. D, in particular, provides a measure of the

53



10:7