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Appendix AFluid Mechanics: Physical Principles

A.1 Introduction

The aim of fluid mechanics is to describe the characteristics of flows using systems ofequations that account for the forces acting on the material and its intrinsic behaviour.In the case of simple fluids, i.e., Newtonian fluids, in the laminar regime, i.e., whenthe flow velocities are not too high, the theoretical basis is by now perfectly wellunderstood and many kinds of flow can be successfully described. However, themechanics of complex fluids, i.e., non-Newtonian fluids, provides a vast field forfuture research that has still only been touched upon. The difficulty here is twofold: theconstitutive laws are not well established and their complexity makes them difficultto handle from the point of view of calculating flows.

In this appendix, we shall review the general principles of fluid mechanics, whichactually sit within the broader framework of continuum mechanics. To simplify thedescription, we shall focus on tools developed to deal with fluid materials, sincethese are the main subject of the present book. However, it should be noted thatthese considerations fall somewhat short when dealing with yield stress fluids, sincesuch materials may be solid or liquid depending on the circumstances. We shall aimin the following presentation to emphasise the physical origin of the principles ofcontinuum mechanics in a way that is consistent with the rest of the book.

Once we can provide an accurate description of the interactions between con-stituent elements, we can in principle predict the flow characteristics of a fluid madeup of an ensemble of such components as a function of the forces applied to it. If thematerial only contains a few components, the task is relatively straightforward. Whenthe system under investigation comprises a very large number of components, whichis clearly the case when we consider an arbitrary fluid, it is extremely difficult, evenwith the help of modern computing, to predict the motion of each component anddeduce from that the macroscopic features of the flow. So in order to obtain a goodphysical grasp of the relevant phenomena and prepare for real practical applications,it is crucial to develop analytical tools able to describe the collective properties ofan ensemble of components, i.e., a fluid, in motion. This is the whole point of theformalism and the tools of continuum mechanics.

P. Coussot, Rheophysics, Soft and Biological Matter, 289DOI: 10.1007/978-3-319-06148-1,© Springer International Publishing Switzerland 2014

290 Appendix A: Fluid Mechanics

A.2 Flow Variables

Since fluid mechanics was developed mainly in order to describe the flows of simplefluids such as air and water, whose properties do not depend on their deformationhistory, the usual starting point is to consider the velocity of the fluid at a giventime and place. This is the so-called Eulerian description. This velocity u is thevelocity of the component that happens to be passing through this point at thisprecise time. The aim then is to describe the time dependence of the local velocitydistribution within the material. Another possibility is to follow the velocity of eachfluid component. This is the Lagrangian description of the motion. The aim in thiscase is to describe the trajectory of each of these components, starting from theirinitial position. This kind of description is clearly well suited to studying limiteddeformations, or materials with some preferred configuration which never stray farfrom that configuration. In fact certain classes of material such as viscoelastic fluidsor yield stress fluids have instantaneous properties that depend on deformations theyhave undergone relative to a specific configuration. For these materials, it may beuseful to use both kinds of description, depending on the state in which the materialhappens to be. Here we apply a rather general definition of what constitutes a fluid,according to which they are materials capable of deforming at will without losingtheir mechanical properties, even though their instantaneous behaviour may dependon the history of deformations they have undergone previously.

Two other variables characterise the state of a fluid: the density, which indicatesthe number of components per unit volume, and the temperature, which gives an ideaof the degree of agitation of the components. We shall also use a Eulerian approach todescribe these variables. In short, we seek to predict the evolution in space and timeof three variables characterising the state and motions of the material, viz., its densityρ, its velocity u, and its temperature T . These variables are functions of the time tand the local position x with components (x, y, z) relative to a fixed orthonormalframe (i, j, k). The components of the velocity in the same frame are (u, v, w).

A.3 Continuity of the Medium

It is not as easy as one would think to specify the velocity of a given material com-ponent located at some precise point. This implicitly assumes that the material is ajuxtaposition of infinitely small components, which is obviously never really the case.In practice, when we use ordinary methods to measure the value of one of the abovevariables at a given point, we actually measure the average value of this variableover a small volume containing a certain number of fluid components. In an ordinaryliquid, this means that we record the average velocity of some 1015 molecules. More-over, a measurement of the density must necessarily concern a volume containing atleast enough components to ensure that the result does represent the average valuein the neighbourhood of the given point, while remaining small enough relative tomacroscopic variations of this variable, on the scale of the observer. In continuummechanics, the values of the velocity, density, and temperature variables are therefore

Appendix A: Fluid Mechanics 291

q

l

Local discontinuitesMacroscopic variations

Fig. A.1 Dependence of an average physical property q over a certain volume of material on thesize l of this volume

ensemble averages over representative volume elements Ωe. It would be difficult tospecify this representative volume element a priori because its dimensions must jus-tify the preconditions for the continuous medium hypothesis, and these preconditionsdepend in part on the mechanical behaviour of the material.

To facilitate the mathematical description, we require these variables, which arealso functions of the space and time coordinates, to be continuously differentiable(and hence also continuous) with respect to those coordinates. In this way we can usea single set of equations involving space and time derivatives to describe the motionof the fluid at each point (appending specific relations between the variables on eitherside of any discontinuity surfaces). For this to work, the difference �ι between thevalues of variables associated with neighbouring points, i.e., associated with twoneighbouring volume elements, must be very small compared with the difference�I in the values of this variable at two points separated by a typical macroscopiclength scale of the flow. Likewise, time variations in the value of a variable at aspecific point must be very small compared with variations over the same time ofthe macroscopic difference mentioned above.

From this definition, we thus find that the continuous medium hypothesis is nec-essarily associated with a range of observation scales. Indeed, if we observe on ourscale the flow of a simple liquid made up of a very large number of molecules, it iseasy to divide space up into a multitude of very small volumes, each containing manymolecules. It seems natural then to consider that, from one small volume to the next,the change in the variable is likely to be small compared with macroscopic variationsin this variable. But if we use such a small observation scale that the groups of mole-cules used to measure a variable no longer contain more than a few components, themedium is likely to be discontinuous, because from one group to another, the density,velocity, or temperature may vary significantly (see Fig. A.1). For ordinary liquids,just as for colloidal suspensions, this problem will clearly only arise in very specificflow situations, since the scale of observation is generally very big compared withthe size of the components making up the fluid. In the most favourable cases, the


spatial variations of the given variable ι are linear. We then have �ι = �I/N , whereN is the number of volume elements of the fluid making up a distance equal to thecharacteristic scale L of the flow. In this case, the continuity hypothesis �ι � �Ifor the variable ι is valid whenever N > 10.

Up to here we have only considered a geometric criterion for the continuity of themedium. Significant differences in the velocity from one group of components to aneighbouring one can be observed even when these groups are very small comparedwith the volume of suspension, simply due to the collective behaviour of these groups,i.e., due to the mechanical behaviour of the suspension. For example, if the givenmaterial is organised in such a way that a flow separates the material into two moreor less rigid parts which slide over one another, the relative speed of neighbouringcomponents situated on either side of the slip surface will be of the same order asthe macroscopic flow speed. The continuum assumption is no longer valid.

For simple liquids under ordinary conditions, this kind of problem occurs veryrarely thanks to the disorder and thermal agitation that prevail in such media.For certain complex fluids, the problem is more delicate because the spatial vari-ations of a variable ι, which depend on the distribution of forces over the fluidcomponents and their reactions against these forces, are not linear. A case in point isthe yield stress fluid in certain flow configurations where the velocity may vary veryquickly over a relatively small thickness. Under such conditions the macroscopicvariations in the velocity are of the same order as those at the boundaries of thisregion of rapid variations and if the continuous medium hypothesis is to be justified,there must be a sufficient number of volume elements in the fast variation zone of thevelocity. The problem is that the criterion depends on the flow conditions because thethickness sheared tends to zero when the stress at the wall tends to the yield stress ofthe suspension. With a confined granular suspension in frictional flow (see Sect. 7.5),the situation is much more critical because, whatever the dimensions of the sample(for a given particle size), rupture surfaces, or more precisely, shear bands will tendto form beyond a certain deformation. In this case, there is no simple criterion forthe continuity of the material.

These examples show that it is not possible to state universal criteria for thecontinuity of a fluid, especially if they must encompass concentrated suspensions.Only necessary conditions, in particular geometric conditions, can be put forward.Continuity depends on characteristics of the material and the flow, which meansthat the continuity hypothesis must always be confirmed with hindsight. In the restof this book, we assume in principle that the material is continuous, treating anydiscontinuity surfaces as singularities in the flow.

A.4 Forces

The motion of a fluid is determined in part by the external forces on it and by themutual forces between its constitutive components. These forces fall into two cate-gories. The first includes the so-called body or volume forces, such as electromagneticforces, gravitational forces, or inertial forces, which usually vary slowly with distance

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Fig. A.2 How forces betweenfluid components lead tosurface forces ds

i

i'Fii'

(a)

(b)

(relative to the size of the fluid components) and have very long range. Consider arepresentative volume element Ωe containing a certain number of fluid components,i.e., liquid molecules and solid particles for a suspension, and such that the mediumcan be treated as continuous when we describe the changes in the values of the vari-ables by dividing space up into like volumes. The external force on this volume isobtained by summing all the external forces Fi on each component of Ωe:

FΩe =∑

Fi . (A.1)

We then define the external force density at a point of space by b = FΩe/Ωe. Anarbitrary volume Ω can be divided into a certain number of these volume elements.When b is constant throughout Ω , the external force applied to this volume is given by

Fvol = bΩ. (A.2)

When the only body force is gravity g, we have b = ρg.The second category of forces are surface forces Fsurf , which act between two

components very close together but decrease rapidly with distance. For an ordi-nary liquid, this mainly concerns van der Waals forces, which are essentially forcesbetween neighbouring molecules. For a suspension, these may be forces betweenmolecules of the interstitial liquid (see Chap. 2), or indeed interaction forces betweenthe particles, some of which may be electrostatic (see Chap. 5) and could then haveeffects over a certain distance through the liquid. However, the magnitude of theseforces still decreases rapidly with distance and it seems reasonable to neglect theiraction beyond the nearest neighbouring particle. Otherwise, it would still be possibleto specify volumes (which would then be called elements) which encompass severalfluid components such that the force due to this kind of interaction and exerted by sucha volume on the surrounding volumes would not extend beyond nearest neighbours.

Consider a surface element ds within the fluid, centred on a space point x. Wewould like to evaluate the force exerted by the fluid situated on one side (B) ofthis surface on the fluid situated on the other side (A). Given the short range of thesurface forces discussed above, only those fluid components located very close to thissurface will suffer any force due to fluid components located along the surface on theother side (B). To a first approximation, each component i suffers a force Fi i ′ due toeach component i ′ situated opposite it on the other side of the surface (see Fig. A.2).

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Fig. A.3 Decompositionof the stress vector into atangential component τ anda normal component σn withrespect to the surface element ds

t

σn

τ

The total force exerted by the fluid situated on side B on the fluid situated on side Acan then be written in the form of a vector dF such that

dF =∑

i

Fi i ′ . (A.3)

If Fi i ′ is roughly constant, then dF is proportional to the area ds. It thus makessense to consider the variable defined by the vector tσ(x) = dF/ds. When the areais big enough, tσ is the average value of the force per unit area exerted by eachcomponent on the component opposite it along the given surface. However, as forthe other variables, if the variable tσ(x) is to be continuously differentiable, the areaconsidered around x must be neither too big, in which case the changes in tσ from onepoint to another would be of the same order as the changes between the boundariesof the sample, nor too small (of the order of the area of a few fluid components), inwhich case the changes in tσ from one point to another would be associated with theintrinsic discontinuities in the fluid. The minimum area dse that can be used for thispurpose will be referred to as a surface element.

Analogously with the volume element, some geometric arguments can be putforward regarding the evaluation of this surface element, but its exact value will alsodepend on the mechanical properties of the fluid. However, assuming this surfaceelement known, we can now define the stress vector t at a point x in the following way:

t = limds→dse

dFds

. (A.4)

Note that we have assumed here that the components do not exert torques on oneanother.

The stress vector defined in this way can be decomposed into a term along thenormal n to the surface element and a term perpendicular to n, viz., t = σn + τ (seeFig. A.3). The first term is a normal stress expressing forces that tend to separatethe elements from one another or push them together. The magnitude of this termis proportional to the way the fluid components are packed together and also to theattractive or repulsive forces between these components. The second term is the shearstress, which arises because fluid components situated on one side of the surface tend


Fig. A.4 Small cube ofmaterial for calculating thetorque exerted on the z axis(along the vector k) whichpasses through the centre O ofthe cube

k

j

i

σyx

xy

O

σxy

σyx

σ

to drag along components located on the other side in a slipping motion in the planeperpendicular to the direction of n.

Decomposing relative to the three axes of the frame, the stress vector given by(A.4) on the facet with normal i can be written

tx = σxx i + σxyj + σxzk, (A.5)

where σxx = σ. Likewise for the facets with normals j and k, the stress vectors onthese are ty = σyx i + σyyj + σyzk and tz = σzx i + σzyj + σzzk, respectively.

Consider now a volume of fluid with characteristic length r . Applying Newton’ssecond law ma = Fext to this volume and taking into account the two kinds of forceidentified above, we obtain

ρr3a = Fvol + Fsurf = r3b + r2�t, (A.6)

where a is the average acceleration of the volume elements and �t is the sum ofthe stress vectors on the outer surface of the given volume (the surface forces insidethe volume cancelling out in pairs). When r tends to zero in the limit allowed by thecontinuous medium hypothesis, (A.5) gives �t = 0 to first order. Since this is truefor any parallelepiped, we deduce that the stress t−x on the facet with normal −i isminus the stress on the opposite facet (with normal i). An analogous result can beobtained in the other directions. We thus deduce that t−x = −σxx i − σxyj − σxzk,with analogous expressions for t−y and t−z .

Consider now a cubic volume for which the normals to the different faces areparallel to the three axes of the frame (i, j, k) (see Fig. A.4). The total torque onthis volume along the k axis is the sum of the torques due to the stresses on eachface. The torques due to all the normal stresses have zero resultant because thesestresses induce forces on either side of the z axis. The shear stresses σxz , σzx , σzy ,and σyz induce forces along the z axis which do not give rise to any torque about


Fig. A.5 Tetrahedron used todeduce the stress vector on asurface element with arbitrarynormal n

j

ik

n

1/c

1/b

1/ads

this axis. The total torque is due solely to the shear stress σxy and σyx , and it isproportional to 2(σxy − σyx ). There is no reason to expect a nonzero couple on thematerial components resulting solely from the stresses. This would cause them tospin in some particular direction. We deduce therefore that

σxy = σyx . (A.7)

This argument can be repeated for the torque around the other axes, so we deduce ofcourse that σxz = σzx and σzy = σyz .

To find the stress vector on a facet with arbitrary normal ai + bj + ck, weconsider the stress vector t = σa i + σbj + σck on the tetrahedron constructed byjoining the origin and the plane with equation ax + by + cz = 1 by means of threestraight lines along each of the three axes (see Fig. A.5). If we know the differentcomponents of the stress identified above, we can calculate the stress vector for eachof the three facets lying in the planes x = 0, y = 0, and z = 0. When the volume ofthe tetrahedron tends to zero, (A.6) tells us once again that �t = 0, and with (A.7)and similar, this implies that

σa = aσxx + bσxy + cσxz,

σb = aσxy + bσyy + cσyz,

σc = aσxz + bσyz + cσzz .

(A.8)

This shows that we can express the force exerted on a small element ds of the materialwith arbitrary outward unit normal n in the form

dF = t ds = (Σ · n) ds, (A.9)

where Σ is a symmetric tensor according to (A.7) and the analogous relations. Thisis the stress tensor. Its components in the upper right triangle including the diagonalare σxx , σxy , σxz , σyy , σyz , and σzz . Note that (A.9) specifies a sign convention forthe stress because we chose the outward normal, oriented from the inner face to theouter face, to calculate the stress exerted by the outside on the given surface. It shouldalso be noted that the expression for this tensor depends on the choice of frame. In


another frame obtained by a rotation of the initial frame represented by a matrix R,it takes the form ΣR = RΣRT. However, the tensor Σ has three invariants, whichdo not depend on the choice of frame, namely, its determinant, its trace, and anotherinvariant defined by

Σ� = 1

2

[(trΣ)2 − tr

(Σ2

)]. (A.10)

Simplifying, we can say that this variable expresses the strength of the shear. Thereis always a special frame in which only the diagonal components of the stress tensorare nonzero. These components are in fact the eigenvalues of this tensor, referred toas the principal stresses. Naturally, these do not depend on the frame, which explainsthe frame independence of the three invariants named above.

Note finally that in general a volume of liquid will only deform significantly ifdifferent stresses are applied to its various facets. For this reason, the stress tensor isdecomposed into a pressure term which deals with stresses that cannot (in general)induce a deformation of the fluid and a term called the deviatoric stress tensor whichdeals with stresses that tend to deform the fluid. Bearing in mind the above remarks,we define the pressure term p in the general case as the average of the principalstresses:

p = −1

3trΣ . (A.11)

In this case we have

Σ = −pI + T, (A.12)

where T is the deviatoric stress tensor.

A.5 Mass Conservation

The principle of mass conservation expresses the fact that, when there are no masssources, matter is conserved during any thermodynamic transformation. Taking intoaccount the fluid motions, this means for example that, within a fixed volume Ω ofspace with surface area Σ , the rate of change of the total mass is equal to the flux offluid components through the outer surface of the given volume. The volume of suchcomponents crossing a surface element ds during a time dt is (u dt ·n)ds. It followsthat

d

dt

⎛

⎝∫

Ω

ρ dv

⎞

⎠ = −∫

Σ

(u · n)ρ ds. (A.13)

Since (A.13) holds for each volume element of the fluid sample, we deduce that


Fig. A.6 Main forces actingon fluid components: volumeand surface forces

dv

g

ds

n

Ω

Σ

(Σ.n)ds

∂ρ

∂t+ ∇·(ρu) = 0. (A.14)

This is the local expression of mass conservation. For a gas, there may be very largechanges in the density, but for a liquid, and a fortiori a solid, the density varies rela-tively little and a simplified form of (A.14) is often used in practice, namely ∇ ·u = 0.This simplification is generally valid also for liquid–solid mixtures that are not intwo-phase flow (i.e., the average velocity of the solid does not differ from the aver-age velocity of the liquid), since the density of each phase is roughly constant. Thesituation for gas–solid mixtures is somewhat different. The behaviour of the mixtureand the density are often dictated by the solid phase, whereupon the presence of thegas can often be ignored. As a consequence, the density of these mixtures may varysignificantly when the solid particles move closer together or move further apart,whether this be due to migration through the gas or because the gas is compressedor expanding.

A.6 Momentum Conservation

The principle of momentum conservation results directly from Newton’s secondlaw according to which any change in the motion of a body induces a force, andconversely. More precisely, the acceleration of a rigid body is equal to the sum of theforces acting on it divided by its mass. For a fluid, this principle applies to each ofits constitutive components. Consider a volume Ω of fluid subject to a body force ofdensity b and all the stresses exerted on the various fluid facets that can be identifiedwithin this volume and on its outer surface (see Fig. A.6). In this case, the sum ofexternal forces acting on the various components of this volume is


Fext =∑

i

bΩi +∑

facets

Σ · n ds. (A.15)

In the second term on the right-hand side, the stresses in the fluid which act on eitherside of a facet lying entirely within the given volume will of course cancel out inpairs, leaving only those stresses acting on the boundary surface Σ of the volume.

To calculate the acceleration of the given volume element, one must take intoaccount the fact that the velocity u is not necessarily the velocity of the same fluidelement from one moment to the next since it is the velocity associated with a givenposition at a given time. The acceleration of a fluid element is in fact

a(x, t) = limdt→0

u(x + dx, t + dt) − u (x, t)

dt, (A.16)

where dx = u dt is the path travelled by the fluid element located at x at time t duringthe time interval dt . In this way we can follow a given fluid element, with the result

a(x, t) = ∂u∂t

+ (u · ∇)u. (A.17)

Summing the expressions of Newton’s second law for each component within thegiven volume and using (A.15) and (A.17), we find eventually

∫

Ω

∂(ρu)

∂tdv +

∫

Σ

ρu(u · n) ds =∫

Ω

b dv +∫

Σ

(Σ · n) ds. (A.18)

This expression is valid for any fluid volume. We thus deduce the local form of theprinciple of momentum conservation:

ρ∂u∂t

+ ρ(u · ∇)u = b + ∇ · Σ . (A.19)

A.7 Temporal Fluctuation

Since fluids comprise a disordered assemblage of components which may in principlehave many different forms, the changes in the variables associated with volumeelements will not be regular, but will fluctuate around average values. In practice,however, these values are measured over finite periods of time so the fluctuationswill not generally be observed. For example, the instantaneous velocity at a givenpoint can be expressed in the form

u = u + u′, (A.20)


where u is the time-averaged value of the velocity and u′ the fluctuation. We assumethat u is the value measured by the experimenter if the measurement takes a finitetime to carry out, longer than the minimum time θ∗ required for the average of thevariable over this period to be very close to the average over a much longer time, butshorter than a time beyond which significant changes would occur due to the flow.For a flow in which the characteristic time for temporal evolution of the averagevelocity is Θ , we may adopt the following definition:

u = u(θ) = 1

θ

θ∫

0

u dt, (A.21)

which is constant for θ∗ < θ < Θ . Averages and fluctuations of other variables canbe defined in a similar way.

The equations for mass and momentum conservation are valid in principle onlywhen we introduce instantaneous values for the variables. The corresponding expres-sions for the variables are found by time-averaging these equations over a suitablelength of time. During this operation, all the terms proportional to one variable or totwo variables whose fluctuations are assumed to be uncorrelated give similar termsinvolving the average value of the variable. On the other hand, in the momentumconservation equation there is a term which is not linear in the velocity or one of itsderivatives. This is the term ρ(u · ∇)u in (A.19). Its average is not the product of theaverage of each factor. Indeed, we have

(u · ∇)u = (u · ∇)u + (u′ · ∇)u′. (A.22)

Under suitable assumptions (the system is ergodic and the time-averaging opera-tion commutes with the differentiation operators), it can be shown that ρ(u′ · ∇)u′can also be written in the form ∇ ·

(ρu′ ⊗ u′

). We then observe that, in terms of

the average values of the variables (denoted hereafter without the bar to simplify),the momentum conservation equation can be written in the same form as the localexpression, provided that we use the modified stress tensor defined by

Ξ = Σ − ρu′ ⊗ u′. (A.23)

It should be remembered that the variables are understood here as average values overrepresentative volume elements. The tensor Ξ is a case in point. Quite generally, thedefinition (A.23) allows one to include in the stress tensor a term due to momentumtransport through the fluid. A single expression can thus express the resistance toflow due to friction or contacts between fluid components which give rise to the stresstensor Σ , and at the same time the resistance due to turbulence in the flow, thermalagitation (see Sect. 2.2.2), or Brownian motion (see Sect. 5.2) of these components.Note finally that a pressure term due to velocity fluctuations, viz.,

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1

3tr

(ρu′ ⊗ u′

),

can be included in (A.23).

A.8 Turbulence

When the speed of flow of a fluid is gradually increased, we find that the temporalfluctuations in the local values of the various variables around the average becomeever greater until they eventually predominate. This is the phenomenon known asturbulence. Since in particular the local velocities fluctuate in all directions aroundthe average velocity, this tends to accelerate diffusion within the fluid. Turbulence isthen easy to observe with the help of this effect. A dye injected into a turbulent flowwill diffuse much more quickly in directions perpendicular to the direction of theaverage velocity of the fluid. Turbulence occurs when the inertia of the fluid com-ponents becomes too great in comparison with the viscous energy dissipation dueto the average motion. The phenomenon can then be quantified using a dimension-less number, the Reynolds number, calculated from the ratio of the correspondingenergies. A general expression for the Reynolds numbers is ρV 2/τ , where V is theaverage flow speed and τ a typical stress value within the system. Note that the limit-ing value of this number, beyond which turbulence becomes significant, depends onthe initial conditions and boundary conditions of the flow, because this phenomenonhas a distinctly nonlocal nature.

The problem with turbulence is that, in the present state of knowledge, there is nodirect way to determine the evolution of the fluctuation terms in the velocity in termsof the other (average) variables. In particular, due to the nonlocal nature of turbulence,it is no longer possible to determine any counterpart of the constitutive law of thefluid in the turbulent regime by means of independent experiments and to predictthe flow characteristics under other conditions. The ‘state’ of turbulence cannot bedissociated from the flow characteristics. In the general case, in order to model aflow in a new geometry, the second term of Ξ must be estimated using relativelycomplex procedures. The constitutive laws of materials like those discussed in thisbook correspond to laminar flow, i.e., scenarios in which turbulence can be neglected.

A.9 Solution of the Flow Problem

In addition to the above equations, one must also know sufficient initial conditionsand boundary conditions for the problem to be well posed and possess a uniquesolution from the mathematical standpoint. Simplifying somewhat, one must knowat each point a total of three components among the components of the velocity orthe stress tensor, taken relative to independent axes. In practice, this is often the case.


For example, we know, or assume given, the flow rate upstream of a free surface flowand also the geometry in which the flow takes place (the velocity is therefore zeroalong the walls). Likewise for a flow in a duct, the flow rate or pressure differencebetween the ends of the duct is given and we assume in general that the velocityvanishes on the walls.

We now consider a flow problem at constant temperature, for which the temporalfluctuations in the velocity are negligible and the initial conditions and boundaryconditions are given. In this case, we may assume that the motion obeys (A.14) and(A.19). These constitute four equations in the ten unknowns ρ, u, v, w, σxx , σxy ,σxz , σyy , σyz , and σzz . It is not therefore possible in principle to solve any flowproblem without further input, i.e., without further relations between these variables.The relations in question are known as the constitutive law of the material.

When there are no temperature variations, the constitutive law is often given inthe form of six expressions specifying Σ as a function of ρ and u. Here we shall bemainly concerned with the characteristics of the constitutive law when there is noheat exchange. The problem is of the same kind, although more complex, when weconsider flows in which temperature variations cannot be neglected. New variablesT , e, and q then come in and one requires the principle of energy conservation toexpress their dynamics (see Appendix B). These must also be related together andwith the other variables by the constitutive law. When the temporal fluctuations inthe velocity due to internal agitation of the fluid components cannot be neglected,the extra tensor in (A.23) can be determined directly as a function of the local stateof the material and the flow characteristics. In this case, (A.23) is the constitutivelaw of the material. This is no longer true when the temporal fluctuations are due toturbulence (see Sect. A.8).

When the constitutive law of the material is known, there is then no theoreticalreason why the flow problem should not be solved. In practice, however, the relevantsystem of equations is very complex and even a numerical solution can be verydifficult to find. It is often necessary to simplify the problem by assuming that certainphenomena or characteristics of the flow play only a minor role.

A.10 Constitutive Laws

A.10.1 General Considerations

When heat exchange can be neglected, the constitutive law of a material as wehave identified it is a relation between the stresses and the fluid motion. From thephysical standpoint, we thus need to determine the forces between two groups offluid components located on either side of a surface as a function of the motion ofthese groups. These forces will depend on the motion of the two groups, but also onthe history of their motion which may have influenced their state. This notion of stategenerally plays no role when we are dealing with an ordinary liquid under typical


flow conditions, and in particular, when considering sufficiently long periods of timecompared with the characteristic time of thermal agitation, for then the moleculeswill forget their prior movements almost instantaneously, they are in the same stateof thermal agitation, and they interact among themselves in a similar way.

On the other hand, the state of the system may play a crucial role for materialsin which the components are much bigger than the molecules of an ordinary liq-uid, exerting interactions on one another that vary significantly depending on priordeformations or relative positions. This state can in principle be characterised by alocal variable. Note that for some kinds of material it may be preferable to use atensor rather than a scalar. But whatever the situation, given the short range of themutual actions of the different fluid components, the stresses on a small fluid volumewill depend in particular on the history of the motion in the close vicinity of thisvolume. This is the principle of local action.

Many different kinds of constitutive laws have been proposed in theory, some verysimple and some very complex. Among the latter, very few have been confirmed byexperiment. In practice, it seems more useful to refer to a simple and physicallyreasonable scheme which can include the main laws actually observed. To this end,we assume that the local stresses at a point depend only on the state of the system andthe deformations (or strains) at the given time in a representative volume elementcontaining this point. We shall thus require mathematical tools for quantifying thedeformations.

A.10.2 Strain Rate Tensor

The key feature of the components of a fluid is their ability to move relative to oneanother. When the fluid flows and there is no slipping on the walls, mass movement ofthe whole fluid is impossible and it is compelled to deform. In this case, the velocitydiffers from one point to another, i.e., the velocity gradient tensor ∇u is nonzero.This tensor contains terms associated with elongation of the material. These areterms on the diagonal, viz., ∂u/∂x , ∂v/∂y, and ∂w/∂z. For example, when a fluidis subjected to a flow such that only these terms are nonzero, and in addition theyare actually constant, this implies that each velocity component is proportional tothe distance of the given point from the origin. As a consequence, a fluid sampleundergoes steady extension or contraction in the various directions as time goes by.In contrast, the off-diagonal terms of the velocity gradient tensor are associated withshear. If for example only the term ∂u/∂y is nonzero, the velocity only varies inthe y direction and fluid layers parallel to the plane Oxz slip over one another. Thematerial is thus sheared in the y direction. However, if ∂v/∂x is also nonzero andequal to −∂u/∂y, the fluid motion is in fact a rotation of all the fluid componentsabout a fixed point. This type of mass movement does not in principle give rise to anyparticular interactions between neighbouring components and should not thereforeaffect the stresses within the fluid.


When we study the constitutive law, it is thus natural to focus on the strain ratetensor D, defined as the symmetric part of ∇u, which only retains terms correspond-ing to the relative motions due to shear:

D = 1

2

(∇u + ∇uT

). (A.24)

The components of this tensor are not affected by rotations since, any time twoterms symmetric relative to the diagonal of ∇u have opposite values, as in the aboveexample, the term corresponding to them in D is equal to their sum and hencevanishes. Note that, when the fluid density is constant, (A.14) implies that

tr D = ∇ · u = 0. (A.25)

A.10.3 Simplified Form of the Constitutive Law

Finally, taking into account the various simplifications made up to now, the consti-tutive law of a material should express the stress tensor at each point as a functionof the strain rate tensor and the instantaneous state of the fluid. The latter could alsobe expressed using a tensor that would take into account any anisotropy in this state,but it usually suffices to describe it with a single variable λ. Then we have

Σ = f (D,λ). (A.26)

To complete the constitutive law, a further equation must be adjoined to (A.26) todescribe how λ depends on the flow history. The latter thus comes in through thestate the fluid has reached at the given time.

In fact, in order to determine or at least characterise the constitutive law of amaterial experimentally, one must in principle measure the velocity field and thestress field within the material during a flow. In practice, it is often only possible tomeasure on the one hand an average flow velocity in a cross-section or the angularvelocity of a given surface, and on the other, the torque on a device, a pressuredifference, or a force applied to the fluid as a whole. Under such conditions, wecannot deduce the tensor form of (A.26) directly. It is therefore particularly useful toarrange for conditions such that the form of the tensor D is as simple as possible. Inthis way, the number of relations between the components of D and the componentsof Σ that actually need to be measured can be drastically reduced. It is precisely theaim of rheometry (see Chap. 8) to realise simple flows of this kind, in which the fluiddeformations are controlled in the best possible way by varying them over as widea range as possible. In the following we shall consider two kinds of simple flow:simple shear, which is a viscosimetric flow, and elongation, which is not.

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A.10.4 Simple Shear

We are often interested in the situation known as simple shear, for which there is aframe of reference in which the tensor D has the form

D = γ̇

2

⎛

⎝0 1 01 0 00 0 0

⎞

⎠ , (A.27)

with γ̇ the shear rate. This kind of expression is obtained for example for the perma-nent shear of a fluid between two parallel planes in relative translational motion (seeFig. 1.5) at speed V . Indeed, when there are no other external forces on the fluid (andin particular, gravity can be neglected), it is easy to show, by expressing the balanceof forces on parallelepiped test volumes with two faces parallel to the slip plane, thatthe shear stress is constant in all planes parallel to the solid surfaces. In addition, bysymmetry, the velocity cannot have a nonzero component in any other direction thanthe relative velocity of the planes. There is no particular reason why this velocityshould vary within a given plane, so the fluid layers associated with these planes slipover one another in the direction parallel to the planes. Finally, as the stress tensordoes not vary from one plane to another, the strain rate tensor will also be constant,since these two tensors are related by (A.26). The only nonzero term in D, namely∂u/∂y, is constant between the two solid surfaces, and we thus deduce the followingvelocity distribution

u = γ̇y, v = 0 = w. (A.28)

Moreover, according to (A.28), the shear rate is γ̇ = V/H , where H is the thicknessof the sheared fluid. It can also be shown that the shear stress along each plane isgiven by τ = F/A, where F is the force applied to the planes in the direction ofmotion and A the sheared area.

When D can be written in the form (A.27), it can be shown using the principlesof rational mechanics that, in the same frame, the stress tensor has the form

Σ =⎛

⎝σ11 σ12 0σ12 σ22 00 0 σ33

⎞

⎠ . (A.29)

The pressure is related to the density by an equation of state (especially for gases) orelse we assume that the density is constant, whence the pressure can be treated as anunknown variable instead of ρ. Under these conditions, assuming that the pressurep = (σ11 + σ22 + σ33)/3 has been determined independently, the stress tensor onlyinvolves three unknowns. We then define the shear stress

τ = σ12, (A.30)

and the first and second differences between the normal stresses

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N1 = σ11 − σ22, N1 = σ22 − σ33. (A.31)

The apparent viscosity η is defined by

η = τ

γ̇. (A.32)

For this simple shear, we thus have conditions such that the different variables char-acterising the stress tensor, viz., τ , N1, and N2, are functions of γ̇ alone, whichfor its part completely determines the tensor D. It is now easy to understand theunderlying aims of rheometry, which reduces the search for a tensorial expres-sion like (A.26) to finding three relations between scalars. In this particular case,we can in principle fully determine the form of (A.26) by carrying out suitablemeasurements of tangential and normal forces. The main flow geometries provid-ing a way to carry out such measurements are described in detail in Chap. 8. Itis important to note that this simplifying approach can conceal a certain numberof problems. For example, the simplest general tensor relation (A.26) is usuallyinferred from the expression determined under simple shear conditions, when infact several tensor relations compatible with this expression may in principle bepossible.

A.10.5 Elongation

Elongational flows are sometimes also used to characterise the behaviour of a fluid.In this case, all the tangential components of the strain rate tensor (i.e., those relatedto a shear) are zero, and in an appropriate frame D can be expressed in the form

D =⎛

⎝b 0 00 c 00 0 d

⎞

⎠ , (A.33)

where b, c, and d are three constants such that b + c + d = 0 according to (A.14).For a simple uniaxial elongation, the velocity profile is

u = ax, v = −a

2y, w = −a

2z, (A.34)

where a is a constant. This type of flow is obtained if we succeed in stretching a cylin-der of fluid at exponential speed. We than have b = a and c = −a/2 = d. Moreover,the stress tensor has no nonzero tangential components and the elongational viscosityis by definition

ηE = N1(a)

a. (A.35)

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Note that a simple shear can also be described as the composition of a rota-tional motion at speed γ̇/2 [according to the decomposition of the velocity gra-dient tensor in (A.24)] and a plane elongation at rate γ̇/2 in the direction x + y,since

D·(x + y) = γ̇

2(x + y), D·(x − y) = − γ̇

2(x − y).

A.10.6 Energy Dissipation

If we know the forces and deformations within a fluid, we can calculate the powerdissipated by the internal forces in a given volume Ω . This is given by

P =∑

i

dFi ·ui , (A.36)

where i labels a contact between two contiguous surface elements of area dσ and nor-mal ni in the volume Ω and ui is the relative velocity of these surfaces. By definition,the local force can be written in the form dFi = Σ · ni dσ and the relative velocityis ui = ∇u · dxi , where dxi is the vector joining the centres of the fluid elementson either side of the given surface. The power dP dissipated within a representa-tive volume element around a given point can be calculated by summing the powersdissipated over surfaces with normals oriented in the three possible directions i, j,and k. This gives dP = tr (Σ · D) dΩe. From this we deduce the power dissipatedwithin an arbitrary volume in the form

P =∫

Ω

tr (Σ · D) dv. (A.37)

A similar result is obtained by a more rigorous method in which the principle ofenergy conservation is written out in detail (see Sect. B.1). This expression takeson a particular meaning when we study constitutive laws since it represents energydissipation due to the resistance to relative motion of the fluid components, so it doesindeed correspond to viscous dissipation.

A.10.7 Main Types of Behaviour

The constitutive law of a Newtonian fluid has the general form

Σ = −pI + 2μD, (A.38)

where μ is the (constant) viscosity of the fluid.


The constitutive law for a simple (non-thixotropic) yield stress fluid in the steadystate can now be written quite generally as

√−T� < τc ⇐⇒ D = 0,

√−T� > τc =⇒ Σ = −pI + τcD√−D�

+ F(D�)D,(A.39)

where F is a positive function of the second invariant D�. The Bingham modelcorresponds to F = 2μB, where μB is called the plastic viscosity. The Herschel–Bulkley model corresponds to

F(D�) = 2nk(√−D�

)1−n ,

where k and n are two parameters associated with the material. Under simple shear,the constitutive law (A.39) has the form

τ < τc =⇒ γ̇ = 0,

τ > τc =⇒ τ = τc + f (γ̇),(A.40)

where f is a positive function of γ̇. With the Herschel–Bulkley model, we havef (γ̇) = K γ̇n . We also sometimes use the Casson model, for which

f (γ̇) = K γ̇ + 2√

K τcγ̇.

Note that the above expression (A.39) neglects possible elastic and viscous effectsin the solid regime, and elastic effects in the liquid regime. These aspects may nev-ertheless play a significant role in the fluid behavior, in particular for transient flows.

Further Reading

1. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press,Cambridge (1967)

2. Truesdell, C.: A First Course in Rational Mechanics. Academic Press, San Diego(1991)

3. Coleman, B.D., Markowitz, H., Noll, W.: Viscometric Flows of Non-NewtonianFluids, Springer Tracts in Natural Philosophy, vol. 5. Springer, Berlin (1966)

4. Piau, J.M.: Fluides non-newtoniens, Techniques de l’Ingénieur A710, 1–16,A711, 1–24 (1979)


5. Ferguson, J., Kemblowski, Z.: Applied Fluid Rheology. Elsevier, Amsterdam(1991)

6. Tanner, R.I.: Engineering Rheology. Clarendon Press, Oxford (1988)

Appendix BElements of Thermodynamics

B.1 First Law

In thermodynamics, we assume that the total energy of a system can be written asthe sum of the macroscopic kinetic energy K associated with average motions ofthe components of the system with respect to some given frame, and its microscopicinternal energy. This internal energy U is the sum of the potential energies of inter-action between the constituents and their kinetic energy on the local scale. U is afunction of the state of the system, but since it is difficult to describe this state indetail, we generally describe only the changes in this function during transformationsof the system.

The first law of thermodynamics postulates that the change in the total energyduring a transformation is the sum of the heat supplied to the system and the workdone by external forces on the system. The heat Q is the energy associated with arise in temperature or a change of phase (and hence state) of the system components.The work done W is the energy transmitted by application of a force when it givesrise to a displacement of the macroscopic elements of the system. The power is therate of change of the work per unit time.

The first law can be written in the form

dU

dt+ dK

dt= Pe + δ Q

δ t, (B.1)

with

U =∫

Ω

ρe dv, K = 1

2

∫

Ω

ρu2 dv, Pe =∫

Ω

ρb · u dv +∫

Σ

(Σ · n)·u ds,


312 Appendix B: Elements of Thermodynamics

where e is the internal energy density and Pe the power due to work by externalforces. The rate of flow of heat δ Q/ δ t into the volume Ω can also be written inthe form

δ Q

δ t=

∫

Ω

r dv −∫

Σ

q · n ds, (B.2)

where r is the rate of heat production per unit volume, e.g., due to chemical reactions,and q the local heat flux (heat per unit area and per unit time). We may now rewrite(B.1) using these variables, with the result

d

dt

∫

Ω

ρ

(e + u2

2

)dv =

∫

Ω

(b · u + r) dv +∫

Σ

[(Σ · n) ·u − q · n

]ds. (B.3)

Now the kinetic energy theorem, which is a reformulation of the conservation ofmomentum, tells us that the change in kinetic energy K is equal to the sum of thework done by external forces and the work done by internal forces, so

dK

dt= Pi + Pe, (B.4)

where

Pi(u) = −∫

Ω

tr (Σ · D) dv (B.5)

is the power generated by the work done by the internal forces.Equations (B.2), (B.3), (B.4), and (B.5) imply the following local form of the

principle of energy conservation:

ρde

dt= tr (Σ · D) + r − ∇ · q. (B.6)

The left-hand side of (B.6) is the rate of change of the internal energy of the system.The last two terms are related to heat transfer in the usual sense, i.e., resulting fromtemperature gradients. The first term on the right-hand side plays an important rolewhen we study the flow of viscous fluids, since it corresponds to the power dissipatedby viscous friction per unit volume.

For a material at rest, the change in the kinetic energy of the system is zero and,for a small transformation, the first law takes the form

dU = δ Q + δ W, (B.7)

where W is the work done by external forces.

Appendix B: Elements of Thermodynamics 313

For a simple fluid, and in particular for Newtonian fluids, undergoing a very slowtransformation, the quantity δ W can be expressed more precisely. Indeed, the termassociated with body forces in the power generated by external forces is zero becausethe speeds are negligible. In addition, since the deformation rates are also very small,only the pressure term in the constitutive law is significant, i.e., Σ = −pI, and thepower generated by external forces is therefore

Pe =∫

Σ

(−pI · n) ·u ds = −pdΩ

dt,

whence we have δ W = −p dΩ . This no longer holds for complex fluids, and inparticular, yield stress fluids, for which anisotropic stress may remain when thesystem is at rest or undergoing slow deformations.

B.2 Entropy

For a system in a given macroscopic state, the microscopic state, i.e., the distributionof energy states of the microscopic components of the system, is not precisely defined.In other words, many different microscopic states will be compatible with the givenmacroscopic state. If the number of these microscopic states is Z , the entropy of thesystem is defined by

S = kB ln Z . (B.8)

Given the equivalence of states for which similar components could exchange energylevels, statistical information suffices to describe the changes in the entropy of thesystem. Note in passing that the entropy is additive. This follows directly from theabove expression as a logarithm of the number of states. Consider two systems A andB, with entropies SA and SB , respectively, and involving Z A and Z B microscopicstates, respectively. The total system A + B has Z A Z B possible microscopic states,so it has entropy

S = kB ln(Z A Z B) = kB ln Z A + kB ln Z B = SA + SB .

When the states are defined by a probability distribution ψ(r) such that the probabilityof having a state between r and r + dr is equal to ψ(r) dr , the number of statesbetween r and r + dr is nψ(r) dr , where n is the total number of states. The entropyof the system is therefore kB ln [nψ(r) dr ], whence, up to addition of kB ln(n dr)

which may be considered constant, the entropy is given by

S = kB ln ψ(r). (B.9)

314 Appendix B: Elements of Thermodynamics

B.3 Second Law

The temperature can be defined from the entropy and energy of the system:

1

T= ∂S

∂U

∣∣∣∣Ω

. (B.10)

It follows that, during a reversible infinitesimal transformation, we have dS = δ Q/T ,where δ Q is the heat transfer to the system. Furthermore, in an isothermal system,i.e., at constant temperature, we have the following expression for the entropy, up toa constant:

Sisoth = Uisoth

T. (B.11)

The second law of thermodynamics tells us that the entropy of a closed systemincreases to a maximum when equilibrium is established. This implies in particularthat, for a system in contact with a single heat source, we have Q/T < 0.

B.4 Free Energy

The free energy F , also called the Helmholtz free energy, is a function of state,changes of which correspond to the work that must be done to get the system fromthe initial state to the final state by a reversible transformation at constant temperature.So if we carry out an operation during which the infinitesimal work δ W is done tothe system at constant temperature, the resulting change in the free energy is

dF = δ W. (B.12)

Since dU = δ Q + δ W and dS = δ Q/T , it follows that dF = dU − T dS.Consider now a system in contact with a thermostat which ensures that the tem-

perature is constant and the combined system plus thermostat is isolated. In thissituation, the total entropy Stot = S + Sisoth is maximal by the second law. The totalenergy of this ensemble is the sum of the energies of the system and the thermostat,so Utot = U + Uisoth. We deduce that the expression Stot = S + (Utot − U )/T ismaximal at equilibrium. Since the total energy of the combined system is constant,this means that the free energy function F = U − T S is minimal, or put anotherway, at constant temperature,

dF = dU − T dS = 0. (B.13)

Since the total differential of a function f (x, y) has the form

d f = ∂ f

∂x

∣∣∣∣y

dx + ∂ f

∂y

∣∣∣∣x

dy,

Appendix B: Elements of Thermodynamics 315

we may deduce several useful relations from the above results. We note first thatdU = T dS − pdΩ , and also dS = dU/T + pdΩ/T , whence

p = T∂S

∂Ω

∣∣∣∣U

, (B.14)

and since dF = dU − T dS − SdT = −SdT − pdΩ in the general case, we deducethat

p = − ∂F

∂Ω

∣∣∣∣T

. (B.15)

B.5 Energy Distribution

Consider a small part A of a large system A0 which has energy E0 and which isthermally isolated. We assume that the interaction energy between A and the rest ofthe large system A′ is small enough to be able to consider that E0 is the sum of theenergies E and E ′ of each of the subsystems, so that E ′ = E0 − E . We seek theprobability p(E) that the small system is in a microscopic state of energy E � E0when equilibrium is reached. The number of possible states in this situation resultsfrom the degrees of freedom left to the rest of the system, i.e., the number Z(E0 − E).This implies that

ln p(E) = ln Z(E0 − E) = 1

kBS(E0 − E) ≈ 1

kBS(E0) − E

kB

dS

dE0. (B.16)

Since we know by definition that dS/dE0 = 1/T , we find

p(E) = α exp

(− E

kBT

). (B.17)

This is known as the Boltzmann distribution. The constant α is determined by thenormalisation condition, i.e., the sum of the probabilities of occupying the differentavailable energy levels must be equal to 1.

In the above discussion, we assumed the existence of a discrete distribution ofenergy levels. For a continuous distribution of energy states E , we must introducea probability density. The probability that the system is in one of its energy statesbetween E and E + dE is then given by

p(E) dE = α exp

(− E

kBT

)dE . (B.18)

Index

AAdhesion energy, 41, 85, 203Amphiphilic molecule, 214Anisotropy effects, 104, 109Apparent viscosity, 53Archimedes’ principle, 90, 93, 116

BBagnold number, 242Bancroft’s rule, 216Body forces, 292Born repulsion, 40Brittleness, 3, 72, 74Brownian diffusion, 162Brownian motion, 18

in colloids, 158, 159, 167, 181, 183Buoyancy, 90, 93, 116

CCapillary number, 211Chemical bonds, 40Coalescence, 26, 202, 212, 213, 216Cohesive energy, 65, 203Colloid, 15, 21, 157, 199

aggregation, 158, 190–192attractive system, 158, 179, 190, 198

concentrated pasty regime, 194, 195concentrated regime, 192dilute regime, 194liquid regime, 195, 198semi-dilute regime, 194

depletion interactions, 176, 177diffusion limited aggregation, 191elastic modulus, 194, 195electrostatic forces, 170, 172

flocculation, 178pasty–hydrodynamic transition, 198reaction limited aggregation, 191repulsive system, 158, 179, 180, 190

concentrated pasty regime, 185, 188concentrated regime, 183, 184dilute regime, 184glassy regime, 184, 185liquid regime, 188, 190semi-dilute regime, 184

sedimentation, 165, 167shear thinning, 197solid–liquid transition, 187stability, 16, 173, 175, 178, 179thermal agitation, 157–160thixotropy, 158, 198van der Waals forces, 167, 169yield stress, 189, 194, 196–198

Colloidal interaction, 13, 16, 180, 198Compressibility, 71Compression modulus, 71Concentration, 12

critical, 88, 100, 103, 109effects, 97, 103non-uniform, 109, 111–113of polymers, 136

Concentration regimesin colloids, 180, 199in emulsions and foams, 220, 230in polymers, 137, 140, 147, 155in suspensions, 99, 103

Concentric cylinder rheometer, 265, 267Cone–plate rheometer, 265Configuration effects, 15, 238, 241Conformation, 122Constitutive law, 302–308

Newtonian fluid, 307


318 Index

simplified, 304yield stress fluid, 308

Continuous medium hypothesis, 82, 83, 95,96, 291, 292

Coulomb criterion, 247Coulomb model, 31Cox–Merz law, 152Creaming, 94, 165

in emulsions, 207Cross-linking, 142, 144

DDebye length, 172, 178Density, 84, 85, 290Depletion interaction, 176, 177Deviatoric stress tensor, 297Diffusion, 162, 165

rotational, 164Diffusion coefficient, 162Dilatancy, 100, 239, 240Dissipation, 55, 307Double layer, 170, 172Drag, 90

in yield stress fluid, 117on droplet or bubble, 206on particle, 89, 91torque, 91

Duct, flow in, 268, 269Ductility, 3, 72, 73

EElastic regime, 61Elasticity, 3, 5, 67, 194, 195

of polymers, 21, 22, 143, 145, 146, 152–155

Electrostatic forces, 170, 172Elongation, 68, 69, 306–307Emulsification, 208, 215Emulsifier, 216Emulsion, 26, 28, 201, 230

adhesion energy, 203coalescence, 202, 212, 213, 216cohesive energy, 203concentrated regime, 221, 225, 230continuous phase, 201creaming, 207dilute regime, 221, 222, 224dispersed phase, 201liquid regime, 226, 229ripening, 202, 204, 212, 216, 218sedimentation, 207semi-dilute regime, 221, 224, 225

solid regime, 226–228stability, 212, 218van der Waals forces, 214yield stress, 26, 28, 228, 229

Entropy, 42, 46, 48, 313of polymer chain, 21

Evaporation, 66Eyring model, 63, 64

FFirst law of thermodynamics, 311–313Floc, 192, 193, 195Flocculation, 178Flory’s theorem, 140Flow curve, 264, 265, 274, 275, 279, 281,

286, 288decreasing, 276, 277

Foam, 29, 30, 201, 230compact regime, 222continuous phase, 201dispersed phase, 201dry, 222of emulsion, 212, 218stability, 212, 218

Fractal, 192, 193, 195Fracture, 72

brittle, 5, 74ductile, 5

Free energy, 314–315of polymer configurations, 127, 132, 133of polymer interactions, 133, 135

Friction coefficient, 242, 244, 252, 254, 255,258

effective, 256

GGaseous state, 36, 42, 43, 55Gel, 195Glass, 4

transition, 76, 79temperature, 76

viscosity, 79Glassy state, 37, 43, 76, 79Granular materials, 30, 33, 231, 258

Bagnold number, 242collisional regime, 242, 251, 253collisions, 236, 238configuration, 238, 241Coulomb criterion, 247critical state, 246–248dilatancy, 239, 240frictional contacts, 235, 236

Index 319

frictional regime, 242, 243, 251internal friction angle, 243jamming, 238, 240Leighton number, 242lubricated contacts, 233, 235lubrication regime, 242maximum rest angle, 250, 251restitution coefficient, 238roughness, 232, 235, 244settling, 240solid fraction, 240Stokes number, 242surface roughness, 234

HHeat, 311Herschel–Bulkley model, 116, 117, 189,

229, 308Hydrodynamic interaction, 82, 94, 99–101,

103, 108Hydrogen bond, 41Hydrophilic–lipophilic balance (HLB), 215Hydrostatic pressure distribution, 93, 116

IIdeal gas, 38, 47–49Inclined plane, 286, 288Instability of flow, 274, 276–277Interaction potential, 38Interface, 202, 203

pressure difference across, 203Interfacial tension, 66, 203Internal energy, 311Interstitial liquid, 82

JJamming, 100, 103

in colloids, 185in granular materials, 238, 240

KKinetic theory, 50, 55Krieger–Dougherty model, 103, 109, 111,

118, 181, 225

LLaminar flow, 9Latent heat

of evaporation, 66

of sublimation, 75Leighton number, 242Lennard-Jones potential, 40, 41Liquid, 5, 10

interstitial, 82polar, 170, 172

Liquid state, 36, 42, 55, 66

MMass conservation, 297–298Maximum packing fraction, 87–89, 98, 111Mean free path, 46Migration, 109, 110, 274

and slipping, 113induced concentration effect, 112under flow, 111, 112under simple shear, 110

Momentum conservation, 298–299Monomer, 121Monomer volume fraction, 136

NNewtonian fluid, 6, 55, 63, 65

constitutive law, 307Normal stress, 294

OOrientation effects, 13, 14, 96, 104, 107Osmotic pressure, 165

PParallel disk rheometer, 263, 265Paste, 115, 117, 118

stability, 116, 117Péclet number, 181Percolation threshold, 11Phase separation, 10Plastic, 21Plasticity, 5, 72Poisson coefficient, 71Polymer, 21, 26, 121, 156

adsorption, 173, 175apparent chain length, 123, 126chain extension, 127, 128concentrated regime, 136, 139, 140, 150,

154conformation, 122correlation length, 138cross-linked, 142, 144dilute regime, 136, 137, 147, 150

320 Index

effective volume fraction, 136elastic modulus, 143, 145, 146, 152–155entanglement, 140, 141free energy, 127, 132, 136gel, 142good solvent, 136in solution, 23, 132, 136interactions with solvent, 132mass fraction, 136melt, 23persistence length, 129, 132persistence time, 132poor solvent, 136radius of gyration, 126, 127, 175relaxation modulus, 149, 156relaxation time, 146, 148, 154repeat unit, 121reptation, 25, 153, 154segment, 129semi-dilute regime, 136, 138, 139, 154temperature effects, 155, 156theta solvent, 135viscoelasticity, 23, 25, 145, 146, 148,

150, 151, 154viscous modulus, 146, 150

Power, 311Pressure, 48, 49

hydrostatic distribution, 93, 116in kinetic theory, 50, 52

RRelaxation time, 25, 78

liquid, 61polymer, 146, 148, 154

Representative volume element, 83, 84Reptation, 25, 153, 154Restitution coefficient, 238Rheometer

concentric cylinder, 265, 267cone–plate, 265Couette, 265gap, 262parallel disk, 263–265

Ripening, 202, 204, 212, 216, 218Roughness, 232, 234, 235, 244, 271, 278Rubber plateau, 151, 153

SSecond law of thermodynamics, 314Sedimentation, 94, 95

in colloids, 165, 167in emulsions, 207

Settling, 240Shear bands, 274, 276Shear box, 244Shear modulus, 69–71, 195

in emulsion, 227Shear rate, 52Shear stress, 6, 53, 294Shear thickening, 15, 113, 115Shear thinning, 22

in colloids, 197in polymer solutions, 146

Silicone oil, 23Simple shear, 7, 8, 52, 305–306

for rheometry, 261Slip on walls, 271, 273Solid, 2, 5Solid fraction, 12, 89, 97, 100, 101, 104, 111

in granular materials, 240non-uniform, 109, 111, 113

Solid state, 36, 42, 66, 75Squeeze test, 282, 286Stability

of colloid, 16, 173, 175, 178, 179of emulsion, 212, 218of paste, 116, 117of suspension, 92, 95

Stokes number, 242Strain rate tensor, 303–304Stress tensor, 296

invariants, 297Sublimation, 75Surface forces, 293Surface tension, 66Surfactant, 18, 26, 207, 212, 214–216, 219

Bancroft’s rule, 216Suspension, 10, 15, 81, 118

concentrated regime, 100dilute regime, 99–101, 108, 117in yield stress fluid, 115, 118semi-dilute regime, 99, 108, 118stability of, 92, 95viscosity, 95–98, 101–105, 108, 109,

112, 113, 117

TTemperature, 38, 49, 155

of glass transition, 76Thermal agitation, 36–38

in colloids, 157–160Thermodynamics, 311–315

first law, 311–313second law, 314

Index 321

Thixotropy, 18, 21in colloids, 158, 198

Torque, 91Turbulence, 301Turbulent flow, 9

VVan der Waals equation of state, 58Van der Waals forces, 6, 16, 39

in colloids, 167, 169in emulsions, 214

Velocity gradient tensor, 303Viscoelasticity, 23, 25, 78

of polymer, 145, 146, 148, 150, 151, 154Viscosity, 6

apparent, 53of gas, 52, 55of glass, 79of liquid, 63, 65of suspension, 95–98, 101–105, 108,

109, 112, 113, 117

Viscous dissipation, 55, 307Volume forces, 292Vulcanisation, 25

WWeissenberg effect, 26Wetting, 86Work, 311

YYield stress, 16, 18, 20, 275, 276, 281

colloid, 189, 194, 196–198emulsion, 26, 28, 228, 229

Yield stress fluid, 16, 185, 226constitutive law, 308drag on particle, 117rheometry, 262, 279, 281, 288solid–liquid transition, 281suspension in, 115, 118

Young’s modulus, 143

appendix a fluid mechanics: physical principles978-3-319-06148-1/1.pdf · appendix a fluid...

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