Peube J-L. Fundamentals of fluid mechanics and transport phenomena

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Physics of Energetic Systems in Flow 133

comprised a solid structure beforehand. The polymerization of a liquid progressively

increases its viscosity until it becomes solid.

The properties of fluids may thus depend on their history, or on the way we

excite them (this is the case for quicksand, yoghurt stirred with a spoon, soils

liquefaction, etc.), and the chemical transformations which they undergo (cooking of

food, polymerization during the generation of plastics, for example). Such behaviors

have significant practical importance, as they are often encountered in the chemical

industry, in the food industry and in many other natural phenomena. The complete

representation of these properties is a difficult problem which is beyond the scope of

this book; we will here limit ourselves to the presentation of just some of the laws

governing the complex viscous behavior of simple 1D flows.

The general form of the relation between the viscous stress tensor and the strain-

rate tensor must be invariant in changing reference frames (in particular, geometric

space is homogenous and isotropic). Moreover, a fluid is often a body with isotropic

physical properties (with the exception of liquid crystals, nematic and smectic

liquids, etc.), so the relationship between the viscous stresses characterized by the

tensor

and the strain rates characterized by the tensor

should not have any

favored direction. This means that the two tensors should have the same principal

axes. The general expression of the laws governing non-linear fluid behavior, which

obeys the necessary invariance in changing reference frames, is beyond the scope of

this book and we will limit ourselves in a first instance to the study of 1D flows, and

then to a more general study of Newtonian fluids which correspond to the linear

approximation of irreversible thermodynamics. For other cases the discussion

becomes quite complex and the reader should refer to texts concerning the rheology

of non-Newtonian fluids ([FRE 64], [GER 94], [TAN 00], [VER 97]).

Finally, recall that the action of the stress tensor on matter is equivalent to the

existence of the volume source

Vdiv (see section 2.1.3.2) which can be written:

divpgraddiv  [3.40]

3.4.1.3.

Physical origin of viscosity

A fluid is comprised of matter which does not have any particular structure: the

relative positions of the particles which make up the matter are not fixed with

respect to any reference structure. These particles can move around freely, rather

like a person moving around in a crowd which is confined to a closed space: this

person can have a relatively autonomous motion in the crowd, but he or she must

follow the general motion.

134 Fundamentals of Fluid Mechanics and Transport Phenomena

Macroscopic forces in a fluid are either forces induced by a force field at the

microscopic level (intermolecular forces), or the result (or manifestation) of

momentum transfers at the molecular level (thermal agitation). For an ordinary fluid

at macroscopic rest, the effects of molecular forces and thermal agitation reduce to

pressure forces.

A simple physical rationale can be used to explain the origin of viscous

phenomena. Consider a gas in which the molecules are sufficiently sparsely

distributed for their interactions to be entirely localized in the vicinity of the

intermolecular collisions. Consider schematically two layers P and Pƍ in the vicinity

of molecules, of average velocities

u and u + du (Figure 3.9). The forces exerted

directly (from a distance) by each of the layers on the other are zero, on account of

the preceding assumption. In addition to their mean velocities, all of the molecules

are subjected to a thermal agitation whose action does not have any privileged

direction. In particular, the molecules of the lower layer move through the upper

layer and vice versa. While a molecule moves from one layer to the other due to a

transverse motion it conserves its longitudinal momentum. Thus, the molecules of

the lower layer Pƍ, of mean velocity

u, are slower than those of the upper layer P into

which they arrive. This layer, P, will exert a force on these slower molecules in

order to increase their velocity to

u+du. The molecules thus accelerated will

obviously have exerted an opposing force on the layer P which is opposite to the

velocity. An analogous effect is produced in the reverse direction for molecules

which descend into the plane Pƍ. This constant exchange means that the lower layer

slows the upper layer, while the upper layer transports the lower layer.

u+du

Figure 3.9.

Viscous friction and momentum transfer

Let us propose a simple model, supposing that the molecules are exchanged

between two layers separated by a distance equal to the mean free path

. Let

c be

the mean transfer velocity of molecules towards the plane P (mean quadratic

velocity). A molecule of mass

m moving from the lower layer to the upper layer

arrives with a momentum deficit equal to

mdu



. Now, over the distance

, the

variation of the velocity is equal to

dydu

. The mass flux of the molecules, per

unit surface, arriving from the lower layer into P is equal to

22 ccnM

(where

Physics of Energetic Systems in Flow 135

n is the volume number of moles), where the factor 1/2 comes from the simplifying

assumption that half of the molecules move from the lower to the upper layer. This

results in a force per unit surface, exerted by these molecules on the plane P, equal

 . We note that the friction force increases with the temperature in

the same way as the mean velocity does.

In other words, the viscous stress

is proportional to the velocity gradient, and

the order of magnitude of the proportionality coefficient (the dynamic viscosity

defined in section 3.4.3) is:

OUP

[3.41]

The preceding rationale can be rigorously applied in the context of kinetic gas

theory. It cannot be directly applied to liquids in which intermolecular forces act

directly between the fluid layers. The effect of viscous friction is thus directly

related to these forces and to the geometry of the molecules, particularly when these

are complex (long macromolecules which may be oriented, stretched or broken

under the shearing action of a velocity gradient). Furthermore, these mechanisms

show that an increase in temperature, on account of the increased thermal agitation,

will decrease the effect of these intermolecular forces and thence the level of viscous

friction.

The origin of viscosity is thus seen to be a microscopic momentum-mixing

phenomenon. At the macroscopic scale, the same phenomenon is encountered in

turbulent flows where considerable fluctuations occur. However, the scale of these

fluctuations is no longer that of the mean free path or the intermolecular distance,

but that of the flow. This leads to considerable difficulties which completely change

the nature of the problem ([MAT 00], [TEN 72]).

3.4.2. Viscometric flows

3.4.2.1. Introduction

We consider here the flow of fluids of constant specific mass in which the

acceleration is everywhere equal to zero. The trajectories and streamlines are thus

straight lines traveled over at constant speed by fluid particles which possess a

uniform motion. The conservation of the mass requires that these straight lines be

parallel, all convergence or divergence of these lines obviously implying an

acceleration or deceleration of particles in the medium which is supposed

continuous. The corresponding dynamic equations are obtained by writing, as in

136 Fundamentals of Fluid Mechanics and Transport Phenomena

fluid statics, that the sum of the external forces acting on the fluid is zero. In the

presence of a mass force of density

and using expression [3.39], we obtain:

0or0 





divpgradg

[3.42]

If the force

g derives from a potential U, we can, in such flows, replace the

pressure with the driving pressure

gzpp

g 0



(z, vertical upwards coordinate;

see section 2.2.1.4.1):

0or0 





divpgrad

[3.43]

As the axis Ox is taken parallel to and in the same direction as the velocity

u(y,z)

of the flow, the strain-rate tensor in Cartesian coordinates is reduced to two

components:

zxxzyxxy

HHHH

Let us assume that for a fixed flow geometry, the shear velocities do not lead to

normal and transverse stresses (no effect analogous to a Poisson coefficient in

elasticity). All that remains therefore are the viscous shear stresses,

and W

Using these assumptions with the equations for viscous fluids, we see that there is

no variation of the driving pressure in the directions

Oy and Oz. There remains only

a single equation in the direction

Ox for the force balance:

zyx

xyg





0 [3.44]

We can immediately see that as the viscous stresses are only functions of the

coordinates (y,z), the

driving pressure is necessarily a linear function of the

direction x.

The gradient dxdp is equal to the ratio

/'

, where 'p

is the

pressure drop observed over the distance L (for example, in a tube (Figure 3.10)).

Supposing that we know the driving pressure, then equation [3.44] contains two

unknown viscous stresses which can only be calculated if the physical law which

relates them to the velocity gradient is given.

Flows of this type will often involve plane or cylindrical boundaries aligned with

Ox, and whose viscous properties are independent of physical factors which are a

Physics of Energetic Systems in Flow 137

function of the x-direction. These flows are fully immersed, i.e. without free

surfaces.

3.4.2.2.

2D viscometric flows

Axisymmetric or 2D plane problems can be easily solved, as there is only a single

unknown viscous stress

, which we will denote simply as

in order to simplify

notations. Equation [3.44] can thus be integrated and the

distribution of viscous

stresses is independent of the phenomenological laws governing

the viscous stresses.

For a 1D flow with plane symmetry we obtain:

ydx



By integrating we can show that the distribution of viscous stresses is a linear

function of the

y-direction:

const

[3.45]

We obtain an analogous result for an axisymmetric problem of axis O

x, either

writing equation [3.44] with cylindrical coordinates or making a balance of forces

exerted from the exterior on the cylinder whose radius is

r (Figure 3.10b):



rrr

WSS

By integrating with respect to the radial coordinate r, we obtain the distribution

of viscous stress:



const



[3.46]

Under the assumption that there is no variation of the driving pressure in the medium,

the viscous stress

 is constant. The flow can thus only be produced by the viscous

entrainment of the fluid due to friction on some cylindrical walls which are moving at a

different velocity. In the 2D plane, a Couette flow is produced between the two plane

surfaces (Figure 3.10a); such a flow is also generated by two circular cylinders

undergoing axial displacement, but these configurations are of little practical interest.

The flow between two fixed planes or in a cylindrical tube requires a driving

pressure gradient.

138 Fundamentals of Fluid Mechanics and Transport Phenomena

In the preceding cases, the viscous stress distribution is independent of the laws

of viscosity

. In particular, these results will be valid for turbulent flows for which

there is no local physical law for the stresses ([COU 89], [MAT 00], [TEN 72]).

When a fluid is

purely viscous, the phenomenological law takes the form:

 

ruyu 'or'

[3.47]

Substituting this relation into the expression for the viscous stress distribution

[3.45] or [3.46], we obtain a differential relation which permits the calculation of the

longitudinal velocity distribution

u and of the mass flux across a cross-section. The

volume flow rate in a circular cylinder can therefore be obtained by integration over

the cross-section; for example, for the problem of revolution in the circular cylinder

of radius R:

rudrq

3.4.2.3.

The Couette flow

Assuming that there is no variation of the driving pressure in the medium, it is

straightforward to verify that with the preceding assumptions the viscous stress is

constant. For the plane geometry, we have a viscous flow generated by entrainment

of the fluid by friction, which occurs between a fixed plane

y=0 and a mobile plane

y=e which is moving at velocity V (Figure 3.10a). The velocity field u(y) is parallel

to the axis

Ox and the viscous stress tensor reduces to the components

WWW

yxxy

which cause the fluid situated above the plane to exert a shear stress

on the fluid situated beneath the plane. This tension is constant in the thickness of

fluid contained between the two planes.

W

(a)

u(y)

R

(b) (c)



Figure 3.10.

Couette flow: (a) principle, (b) Couette rheometer,

Physics of Energetic Systems in Flow 139

The practical realization of such a flow involves two co-axial cylinders, one of

radius

R, which is fixed, and a second, of radius R+e, which is animated by a

rotational motion of angular velocity

about its axis (Figure 3.10b). The moment of

the torque measured on the fixed cylinder gives the value of the viscous stress.

Another realization is the plane-cone rheometer which is comprised of a cone

turning over a plane (Figure 3.10c); a fluid placed between these is then studied.

Locally this gives a Couette flow. As the thickness

e between the walls and the

velocity of the cone wall are proportional to the distance from the axis, the shear

velocity is constant at all points.

These two devices are particularly well-adapted to studying the

phenomenological law of a fluid whose

viscous stress is only a function of the shear

velocity

gradient uƍ(y). The result of this is that the shear velocity is also constant

and the velocity distribution is linear regardless of the phenomenological laws

governing the fluid studied (see Figure 3.10),

y being the distance of a point situated

between the two cylinders from the inner cylinder:



Vyu

The Couette flow allows us to obtain in

a direct and simple manner the viscous

law of a fluid for viscometric flows. As the general law for the viscous behavior of

fluids is of a tensorial nature, the preceding relationship is a simplified law, which is

uniquely valid for this kind of flow.

3.4.2.4.

Principal physical laws for viscous behavior of a fluid

3.4.2.4.1. Introduction

We have just discussed a means of simultaneously measuring the velocity

gradient



yu'

and the viscous stress

, in other words, means of establishing the

phenomenological law for the viscous behavior of a fluid. Fluids comprised of

molecules of a sufficiently small size generally have a linear behavior, viscous stress

being proportional to shear velocity (Newton’s law). Fluids containing

macromolecules, particles which are more or less solid in suspension, that is to say

pasty substances, may have relatively varied viscous laws. In general, the non-

linearity of the viscous law is closely related to the complexity of the molecular

structure. The viscous behavior may be a function of time, in other words of the

mechanical excitations and motions which the fluid has most recently undergone.

140 Fundamentals of Fluid Mechanics and Transport Phenomena

Due to the tensorial nature of the viscous stresses and strain-rates, there may

exist transverse effects associated with a 1D effect (normal stresses for a shear

velocity). This is comparable to the properties of solids (Poisson coefficient in

elasticity). This type of effect is mainly manifest in liquids with complex structures.

Furthermore, elastic effects may occur for relatively weak stresses in viscoelastic

bodies; these may or may not be time dependent: the body may present the

properties of a solid for a stress whose modulus is less than some level



, beyond

which it becomes liquid (Figure 3.11b and Figure 3.11c). In what follows we will

only discuss some simple cases from the vast domain of rheology.

3.4.2.4.2.

Time-dependent fluids

We will distinguish:

 the thixotropic (Figure 3.11a) situation in which the excitations and the

imposed stresses reduce the viscosity (ketchup, yoghurt, gels, drilling mud,

quicksand, etc.);

 rheopexy, which is less frequently encountered (Figure 3.11b), which

corresponds to an opposite effect, where the viscosity increases with agitation

(suspensions of Indian corn starch in water).

The time dependence effect of the viscous behavior leads to hysteresis

phenomena (Figure 3.11c).

(b) (c)(a)



u'(y) u'(y) u'(y)

Figure 3.11.

Hysteresis cycles: thixotropic fluid (a) without or

(b) with elastic properties; (c) rheopexic fluid with elastic properties

These effects are obviously associated with local modification for the particles or

macromolecules which are more or less in contact at the microscopic level. The

explanation for thixotropy is schematically as follows. In suspensions of solid grains

in liquid, the grains, after a sufficiently long duration of immobility, finish by

coming into contact with one another; this leads to solid friction. After a certain

level of agitation (vibration, flow) and the resultant stresses imposed on the

ensemble, all solid-solid contact has been undone, and a lubrication occurs such that

the grains now slide over one another when a velocity gradient is imposed. This

Physics of Energetic Systems in Flow 141

corresponds to the phenomenon of “quicksand”. Further to these purely mechanical

effects, physico-chemical interactions may occur between the grains, leading to a

structural macroscopic organization. This is progressively destroyed under the

effects of a velocity gradient. The possible complexity of these phenomena is such

that explanations of the same nature may, depending on the circumstances, often

lead to inverse effects (rheopexy).

3.4.2.4.3.

Purely viscous fluids

Their viscous behavior is characterized by a single physical law which is

independent of time for a fluid of a given structure and composition. The most

commonly encountered types of fluid are:

7he Newtonian fluid which obeys a linear law (curve 1 of Figure 3.12):

' ( ) ( : the coefficient of dynamic viscosity)

W P Puy [3.48]



The Ostwald-de Waele fluid. Behavior is no longer linear for fluids containing

molecules with complex structures or particles in suspension and of strong

concentration:

)(')('

yuyum



[3.49]

When the viscous stress increases faster than the velocity gradient (

n>1) the fluid

is said to be dilatant or rheothickening (curve 3 of Figure 3.12); in the opposite case

(

n<1), it is known as pseudoplastic or rheofluidifying (curve 4 of Figure 3.12). The

first case occurs in concentrated suspensions of solid particles which are in contact

with one another, when the velocity gradients are steep. The second situation is

observed in flows of polymers of high molecular mass, whose linear molecules are

more or less intertwined and trap a certain quantity of water. Under agitation, the

velocity shear aligns the molecules along their axes, thus freeing the trapped water.

u'(r)

(1)

(2)

(3)

(4)







Figure 3.12.

(1) Newtonian fluid, (2) Bingham,

(3) rheothickening, (4) rheofluidifying

142 Fundamentals of Fluid Mechanics and Transport Phenomena

 The Bingham fluid. This kind of a fluid is characterized by a cohesion such

that if the modulus of the stress

W is less than a given threshold W



 no strain rate

exists in the fluid, which therefore remains rigid; the sliding of liquid layers over

one another only occurs when the stress exceeds the threshold value, and the

relationship between the stress and the velocity gradients then becomes linear (curve

2 of Figure 3.12):



WP WW!W

WdW

Byu

[3.50]

This law is used in particular for the flow of pastes and certain types of mud.

As discussed above, fluids containing solid particles in suspension have a

complex behavior. However, when the solid particles are in weak volumic

concentration

c, the fluid thus constituted is Newtonian: its viscosity

P

is modified

with respect to that of the pure fluid according to the (Einstein) relation, valid for

any 3D flow:

P P c

3.4.2.5.

Poiseuille flow

3.4.2.5.1. Flow in a circular cylinder

The flow is here due entirely to a pressure gradient and the conduit is assumed

sufficiently long for the flow to become fully established, in other words such that

phenomena associated with the tube entrance are not present. Furthermore, the fluid

is assumed to be incompressible such that pressure variations do not lead to changes

in the fluid density.

We will consider the flow of a fluid in a circular tube of radius R (using the

notation of Figure 3.13). The flow, directed in the positive direction along the axis

Ox, leads to a constant negative axial pressure gradient, corresponding to a drop in

the driving pressure equal to

for a length L of the pipe.