Mass and entropy transport in a suspension of rigid particles

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Mass and entropy transport in a suspension of rigid particles

D. Lhuillier

To cite this version:

D. Lhuillier. Mass and entropy transport in a suspension of rigid particles. Journal de Physique, 1986,

47 (10), pp.1687-1696. �10.1051/jphys:0198600470100168700�. �jpa-00210365�

1687

Mass and entropy transport ^{in a} suspension ^of rigid particles

D. Lhuillier

Laboratoire de Mécanique Théorique (*), Université P. et M. Curie, ⁴ place Jussieu, 75230 Paris Cedex 05,

France

(Requ ^{le 3 mars} 1986, accepté ^{le 26} juin ¹⁹⁸⁶⁾

Résumé.

En utilisant la thermodynamique ^des phénomènes irréversibles, ^{on fait une} analyse ^du transport ^de

masse et d’entropie ^{dans une} suspension. La méthode utilisée est une extension de celle appliquée par Einstein et Batchelor à l’étude de la diffusion brownienne. On montre que ^le couplage (dû ^aux relations de symétrie d’Onsager) qui existe pour ^un mélange à ^deux composants, ^{subsiste encore} pour ^une suspension, ^mais qu’il ^est

cette fois dû ^au rôle simultané joué dans les deux transports par ^la vitesse relative entre phases. ^{Les résultats}

les plus remarquables ^{sont :} i) l’absence d’effet Soret lié à la tension interfaciale si les particules suspendues

sont rigides ; ii) ^la dépendance du coefficient de diffusion avec les masses volumiques de chacune des phases ; iii) ^{une forme correcte} de la force de Coriolis agissant ^{dans le mouvement relatif ;} iv) ^la possibilité pour la diffusion de Faxen (diffusion ^{dans les} gradients de vitesse inhomogènes) ^de jouer ^{un rôle} important pour peu que la sédimentation soit négligeable.

Abstract.

Mass and entropy transport ^{in a} suspension ^are analysed in the framework of irreversible

thermodynamics. ^Our approach extends the one used by Einstein and Batchelor for the study of Brownian diffusion. We show that the well-known coupling (due ^{to the} Onsager symmetry relations) ^{which occurs in a} two-component mixture, ^{also occurs in a} two-phase suspension, ^{as a} consequence of the joint ^role played by

the relative velocity ^{in mass and} entropy transport. ^Most noticeable results are i) ^{if the} suspended particles ^are rigid, surface tension cannot lead to any thermo-diffusion (Soret effect) ; ii) ^the general expression ^{for the}

diffusion constant depends ^on the densities of the two phases ; iii) ^{in a} rotating suspension, the Coriolis force

acting ^on the relative motion is not the one commonly ^{found in} the literature and iv) ^diffusion by ^non- homogeneous velocity gradients (Faxen’s diffusion) may become important whenever sedimentation is

negligible.

J. Physique ⁴⁷ (1986) ^{1687-1696 OCTOBRE} 1986,

Classification

Physics ^Abstracts

05.60

47.55K

1. Introduction.

The transport properties ^{of a} suspension ^of particles

have recently ^{received a lot of} attention. These studies have dealt not only ^{with momentum} transport in shear flow but also with ^mass transport ^{in relation} with sedimentation [1, 2] and concentration diffusion

[3, 4]. ^At variance, ^the thermo-diffusion problem

has not yet ^{received a definite answer. One knows}

that in a two-component system (defined ^{as one in}

which the « particles ^{» have atomic} size), ^{the mass}

flux is coupled ^{to the} entropy ^flux. What remains of that coupling ^{for a} dispersed two-phase system ⁱⁿ which each particle ^{contains a lot of atoms ? It is the}

purpose of this paper ^to give ^{an answer for} rigid particles, ^{and to} reconsider ^mass and entropy ^trans- port ⁱⁿ suspensions ^{with the} help ^of irreversible

thermodynamics. Our chief aim is the study ^of

thermo-diffusion ^or Soret effect but, ^{as we shall} see,

our analysis ^also brings ^{out new results} concerning

(*) ^{associ6 au CNRS, U.A. 229.}

concentration diffusion as well as diffusion in rota-

ting systems.

2. Mass and entropy diffusion in a two-component mixture

Irreversible thermodynamics provides ^{us with a} description ^{of mass and} entropy ^{diffusion in a multi-} component ^mixture [5]. The main results in the case

of two components (labelled ^{1 and} 2) ^{are reminded}

below.

When no chemical reaction _occurs, the mass of each component ^is separately conserved, ^hence

where Vk ^{is the} mass-weighted average velocity ^of

component ^k. Defining ^the barycentric velocity ^V

and the mass p per unit volume of the mixture by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470100168700

the above equations ^{can be rewritten as}

where J is the mass diffusion flux defined as

The entropy S per unit volume of the mixture evolves according ^to

where is the entropy ^{diffusion flux and 0’ is the} entropy production which reads

In this expression, U k is the chemical potential ^of component k, and T is the mixture temperature. ^In the framework of (linear) irreversible thermodyna- mics, the condition of positive entropy production ^is

satisfied with

and

The Onsager symmetry principle implies

and imposes ^a coupling ^{between J and X. A} simple

dimensional analysis ^{of the} phenomenological ^coeffi-

cients L allows to rewrite the diffusive fluxes as

and

Here kT ^{is the} (positive) coefficient of thermal

conductivity, while 0 is another positive quantity

with the dimension of a time. The third coefficient

s has the dimension of a specific entropy. ^Its presence ^{in both} J and 1; is a consequence ^{of the}

Onsager symmetry but neither its sign ^nor magnitude

are restricted by thermodynamics.

3. Two-phase ^{mixtures versus} two-component ^mixtu-

res.

Consider ^a mixture which appears ^as particles ^of

component ¹ surrounded by ^{component 2. If the}

smallest particles ^{contain so many atoms that their}

continuum-mechanical description ^is justified (parti-

cle size larger ^{than 0.1 um} approximately), ^one speaks of the mixture as a dispersed two-phase mixture ; ^on the other hand, ^{if all} particles ^are

reduced to atoms, ^one speaks ^{of a} two-component mixture, ^the description ^{of which was} given ^above.

The continuum description ^{of a} two-phase ^mixture

uses averages ^over small volumes containing many particles, ^{while the} description ^{of a} two-component mixture involves averages ^over tiny volumes contai-

ning many ^atoms. The main difference between the

dynamics ^of two-phase ^and two-component ^mixtures lies in the average relative velocity ^between species ¹

and 2. In a two-phase mixture, ^the large size of the

particles allows for ^a long-lived ^relative velocity. ^In

other words, the relative velocity is raised to the status of a state variable : it appears in the Gibbs relation and obeys ^an equation ^{of motion.}

4. Mass and entropy diffusion in a simple two-phase

mixture.

In a two-phase mixture the mass conservation is

expressed ^with equations ^{similar to} (1), i.e. it has

exactly ^{the same form as for a} two-component mixture. In particular, ^{the mass} diffusion flux is still

given by (2). ^The only difference comes from the definition of V, and V2 ^{which are now the mass-} weighted averages ^{over a} volume containing many

particles.

The entropy equation ^{must have the same form as}

written in (3) since this equation ^{stems from} galilean

invariance only. Nevertheless, ^{a more detailed} expression ^{for the} entropy ^{flux can be obtained if}

one notices that the overall entropy ^{flux of a two-} phase ^{mixture can} be written as

where s gathers all the non-convective contributions

and sk stands for the specific entropy ^of phase ^k.

Writting this flux ^as the sum

where

one deduces the important ^result

Hence, ^a part ^{of X is linked to} the relative velocity

while the whole of J ^is proportional ^{to it} (cf. (2)).

As far as mass and entropy equations are concer-

ned, ^the differences between a two-component ^{and a}

two-phase ^{mixture are not} very striking. ^{We now}

1689

analyse ^more specific features of a two-phase ^mix-

ture, namely the evolution equation for the relative

velocity ^{and the} expression ^{of the Gibbs relation.}

The evolution in time of the relative velocity

is described by ^a Galilean-invariant equation

where F and R are the yet unknown force and stress

acting in the relative motion while p* is the related inertia defined as

where flk;n is the average kinetic energy of the mixture. The above equation (9) ^was recently ^studied

in the case of extremely high Reynolds numbers, ^Le.

when the flow around each particle may be approxi-

mated by the Euler equation [6]. ^{Here we are}

interested in the opposite situation of very low relative Reynolds numbers, ^when the Stokes equa- tions are relevant and all inertia terms are negligible.

In this low - W regime, the average kinetic energy is

merely

and consequently

Hence, ^{for low} enough ^relative velocity, equation (9) ^is nothing ^{but an evolution} equation ^{for the mass}

diffusion flux J (such ^an equation ^{was considered}

earlier in the frame of extended thermodynamics [7]). ^We shall later determine the low-W expressions

for F and R.

The second important difference between a two-

phase ^{and a} two-component mixture is the Gibbs relation. From the ^« microscopic ^{» Gibbs relation} (i.e. the classical _one, valid on a scale much smaller than the particle size) and the above expression ^for

the kinetic energy, ^{one can} get the so-called ^mesos-

copic ^{relation or} Gibbs relation averaged ^{over a}

volume containing many particles. ^With U;nt standing

for the average internal energy (and including ^the

kinetic energy of relative motion) ^one gets

where P k ^and Tk ^are the average pressure and

temperature ^of phase k, while ak is its volume fraction. Since particles and fluid occupy the whole available space

We shall henceforth suppose that the particles ^are rigid. _disappear ^{The ratio from the} p 1 a ^{set is then of} independant ^{constant and variables. a 1 must}

This requires the mixture to be permanently ⁱⁿ equilibrium ^{regarding a 1 and this amounts to}

or equivalently

This condition of rigidity is easy ^to understand since any difference between the average pressures would

ultimately ^{lead to an} expansion ^or contraction of the

particles [8] and break the rigidity assumption.

Besides rigidity ^{we make a second} assumption ^and

suppose that the average temperature ^{of the} particles

is always equal ^{to the} average temperature ^{of the} surrounding fluid. This implies ^{that the rate of} change ^{of the} liquid temperature ^{in the} particles

frame is always much smaller than the characteristic time for heat exchanges, ^{i. e.}

where vTl stands for the thermal diffusivity ^{of the} particles while a is the particle ^size.

The two above assumptions ^{are rewritten as}

and

where P and T are henceforth referred to as the pressure and temperature of the mixture.

For a suspension ^of rigid particles ^with high enough heat factor VT1/ a2, ^{the Gibbs} relation is thus

simplified ^from (11) ^to

where the overall entropy S ^was already ^{defined in}

(7). Combining the above Gibbs relation with the

evolution equations, ^one gets ^the following ^entropy

production :

Let us report ^{to §} 9 the discussion concerning ^{the R : VW} and viscous dissipation ^terms. Compared ^to (4) ^it

looks as if an extra term linked to W has appeared. ^{But W cannot} play simultaneously the role of a

thermodynamic force and the role of a flux (remember ^the W-dependence ^{of J and I} exhibited in (2) ^and (8)). ^In fact, ^{one must} gather ^{all the} W-dependent ^terms and the above entropy production ^{must be}

rewritten as

It will be positive provided

and

where kT is the thermal conductivity ^{of the} suspension ^{and 0 the} relaxation time of the relative velocity

motion. If we now suppose that inertia in relative motion is negligible, ^{and that} the role of R is temporarily neglected (cf. § 9), equation (9) ^{amounts to}

And from (2), (8) ^and (13) ^one gets for the diffusion fluxes

and

These expressions ^are quite ^{similar to the} expressions (5) ^and (6) ^{for a} two-component ^mixture. However,

we now interpret 0 ^as the relaxation time for the relative velocity ^{and we} get for s the simple interpre-

tation

But while the presence of s in both (5) ^and (6) ^{was a}

consequence ^{of the} Onsager symmetry, the presence

of sl - s2 ^{in both} (15) ^and (16) ^{is a} consequence ^of the joint role of the relative velocity ^{in the mass and} entropy transport. ^{We can} go further by noticing

that the Gibbs-Duhem relation for _{the pure} phase k

reads

where p 2 ^{is the mass} per unit volume of pure phase k, i.e.

Since both phases ^have locally ^{thd same} pressure and

temperature ^{one finds}

which means that the diffusive ^mass flux (15) ^is proportional ^{to the} pressure gradient only. ^{In other} words, ^mass diffusion ^{is reduced to a} baro-diffusion.

Sedimentation is nothing ^{but a} special case corres-

ponding ^{to a} hydrostatic pressure field

and accordingly

1691

It is clear that the relaxation time depends ^{on the} particle volume fraction. To agree with Batchelor’s notation ^{we now} write

The interpretation ^{of K} ( a,) ^{is readily got: if}

sedimentation occurs with no net volume flux, ^{i.e. if}

then, combining (21) ^and (22), ^{one sees that the} particles ^fall (or rise) ^{with a} _velocity

where

is the single particle falling (or rising) speed.

Let us stress that (15) ^and (19) imply the absence of any Soret effect (dependence ^{of J on} VT) ^or

concentration-diffusion (dependence ^{of J on V} « 1).

One could suggest that this is due to our neglect ^of

surface tension between the two components, ^{and to}

our neglect ^{of the} configuration entropy ^{of the}

particles. ^{Let us} examine these two points ^successi- vely.

5. Influence of surface tension

In a two-component ^{mixture the} difference of inte- ractions between the couples ^{of atoms} 1-1, 2-2 ^{and 1-}

2 manifests itself as an energy of mixing. ^{In the}

continuum description ^{of a} two-phase ^{mixture, the}

same phenomenon is taken into account in a slightly

different _way: one says that the particles ^interact

with the surrounding ^fluid through short-range ^forces

that can be accounted for with a surface energy y

depending ^{on the mean} temperature Ti ^{of the}

interface. _{If ai i} stands for the amount of interface per unit volume of the mixture, ^the thermodynamic

relations of the interface between components ^{1 and} 2 are

and

where Ui and Si ^{are the} internal energy and entropy of the interfaces per unit volume of the mixture.

According ^{to our} previous assumptions, ^{the ave-}

rage temperature Ti ^cannot be different from the

common average temperature ^{of the two} compo- nents, hence

Moreover, ^{if the} particles ^are spheres of radius a and with a constant mass, ^we have

and

In this case a is completely determined by pi and

aI’ and it is not hard to see that the above

thermodynamic ^{relations can} be rewritten ^as

and

Everything appears ^as if the particles ^{« dressed »} by

surface tension could be considered ^{as «} bare »

particles ^{with a chemical} potential

instead of and a pressure

instead of

Moreover, the interface entropy stemming ^{from the}

surface tension is obviously ^{convected with the} parti-

cles and the diffusive entropy ^{flux now} appears ^as

instead of (8). Thus, when surface tension is taken into account, ^the specific entropy ^{of the} particles

becomes

instead of

As a consequence, ^the thermodynamic force that will enter the mass diffusion flux is no more (19) ^but

According ^to the Gibbs-Duhem relations (18) ^and (24) this force is

But the assumption ^of rigid particles ^now implies

instead of

The average pressure P of the mixture is thus

and the mass diffusion flux is, ^{as in the case of} bare

particles, proportional ^{to the} gradient of pressure

only. ^{At first} sight it is rather surprising ^{that surface}

tension does not lead to any Soret effect. This is due to the mutual cancellation of two VT forces. The first comes from the gradient ^{of chemical} potential

and is the one most frequently ^{invoked in the}

literature. The second ^{one comes} from the transport of the interface entropy by ^the particles ^{and seems to}

have been overlooked _up to now. The sum of these two forces amounts to a gradient of pressure, V ( 2 y /a ) . ^{But if one} takes into account the relation (27) between pressures which is ^{a conse-} quence of the rigidity assumption, ^the resulting ^force

is exactly ^{the same as for} particles without surface tension. The absence of Soret effect is thus clearly

associated with the rigidity ^{of the} particles. ^{Since the} thermodynamic description ^{of a} suspension ^with

deformable particles ^{is much more involved, we} postpone ^{it to a} forthcoming paper.

6. The entropy ^of configuration.

In a two-component mixture, ^a _part of the total entropy ^{is due to} the many microstates having ^the

same average concentration, ^but differing ^{from each}

other by ^the spatial distribution of the two chemical

species. ^A suspension ^{also has a} configuration

entropy, but linked to the particles instead of the atoms, and defined ^as the logarithm of the number of different spatial configurations compatible ^{with a} given ^volume fraction. It vanishes not only ^{for zero} particle concentration but also when the particles ^are closed-packed ^since only ^one configuration ^is possi-

ble in that ^case.

Let us call AS the configuration entropy per unit volume of the mixture. To AS corresponds ^{an excess}

free enthalpy

Any ^correct expression ^{for AG must} satisfy ^{the well-}

known extensivity requirement

where Att k ^{is the excess chemical} potential. ^Since

the number of different configurations ^cannot depend ^on the pressure ^or the temperature ^but only

on the number of particles ^{and on} their volume

fraction, ^the only way ^to write AS in a form

compatible ^with (28) ^is

where f ^{and 9 are two} functions of the volume fraction. Hence, ^{the excess chemical} potentials ^asso-

ciated with the configuration entropy ^{take the} special

form

and

Besides (29), ^the configuration entropy ^{must also} satisfy ^{a second} property, namely

To see what it _means, we combine (29) ^and (31) ^to

get

which, ^{for a constant} temperature, implies

This is nothing ^{but a} condition of no net force acting

on the mixture, ^{as was} written in a slightly ^different

form in equation (3.13) ^of [3]. ^Thus (31) ^{means that}

the configuration entropy plays ^{no role in the} barycentric motion of the mixture. But as we shall

soon see, it plays ^a prominent role in the relative motion and hence in diffusion. Before considering

that matter in more detail, ^{let us establish a relation}

that will be useful later on ; from (30) ^and (32)

follows

where

is the excess chemical potential per particle ^{of mass}

ml

4,7r a p 1/3. It may be helpful ^{at this} point ^to

have explicit expressions ^for f and ^{9. In} principle,

the configuration entropy ^{of a} suspension ^{can be}

deduced from the entropy ^{of a} hard-sphere liquid.

The problem ^{is that no} analytical expression ^{for that} entropy ^has yet been found in the whole range 0 a, ^- a m, where a. is the close-packed ^volume

fraction. Nevertheless, ^two limiting situations have

been thoroughly investigated :

1693

1) ^{the dilute limit} (a, a.) where the virial

expension ^method (cf [3] and references therein)

leads to

where kB is the Boltzmann constant.

2) ^the high-density ^limit (al c= a m ^{where the analog of the} virial series is an expansion ⁱⁿ the relative free volume [9] with the result

The factor 3 in the above results must be replaced by ^{D in a} D-dimensional space.

Besides these exact results, ^{one can also} get ^an approximate expression for AS with the so-called lattice model in which the whole available space is divided into cells that ^can be occupied by ^one particle ^{.at most.}

The resulting expression ^for AS, supposed ^to be valid whatever is the concentration appears ^as

where nl is the number of particles per unit volume and nm its maximum value. This result ^can be transformed into a form similar to (29) ^with

The lattice-model expression ^{for AS} gives ^exact results for a one-dimensional space only. ^{For our three-}

dimensional space the lattice-model results for dilute and concentrated suspensions ^{are a bit different from}

the exact hard-sphere ^results. Anyway, the lattice model has the merit of giving easily ^{a not so bad} expression ^{of the} configuration entropy ^for arbitrary concentrations.

7. Role of configuration entropy ^{in mass and} entropy transport.

The entropy of the whole mixture is no longer (7) ^but

and the only modification brought ^{to the Gibbs relation} (12) ^{is the} replacement of ILk by ILk + 4ILk. ^A

second important modification concerns the entropy transport and is linked to the velocity with which the

configuration entropy is convected. Among ^the possible assumptions, ^{there is} only ^{one which leads to sound}

consequences : ^we shall suppose that the part of ^AS proportional ^to pl (resp. P 2) ^{moves with} V, (resp. V2).

This means that the overall entropy ^{flux is now}

where we made use of the property (29). ^{As a} consequence, the diffusive part ^{of the} entropy ^{flux is now}

instead of (8). Taking ^{into account} the above modifications brought by ^the configuration entropy ^{to the}

chemical potentials ^{and to the} entropy transport, ^{the force} acting in the relative motion is no longer (13) ^but

If we again neglect inertia in relative motion and take result (30) ^{into account we deduce}

It is thus clear that the configuration entropy ^is responsible for the concentration dependent part ^of J. ^But again ^{there is not Soret effect, due to the}

mutual cancellation of two VT forces. This cancella- tion comes from the proportionality ^{of the excess}

chemical potentials J1¡.L ^and J1¡.L 2 ^{with the} tempera- ture, ^cf. (30). ^For particles interacting ^{with each}

other through long-range ^{forces, the} configuration

entropy ^{will be} replaced by ^a configuration ^free

energy that will depend ^non linearly ^{on the} tempera-

ture. In this case one can expect ^a Soret effect even

for rigid particles.

The usual definition of the diffusion coefficient D is

Comparing ^with (35) ^and taking (33) ^{into account,}

we get ^the remarkably simple result, ^{similar to} Einstein’s for Brownian diffusion.

If we now consider the expression for the relaxation time given ⁱⁿ (22) together ^{with the} definition (33) ^of

the excess chemical potential per particle, ^{we can.}

rewrite the diffusion coefficient ^as

At first sight, ^the P 2 0 p ^{factor seems to contradict}

Batchelor’s result (6.5) ⁱⁿ [3]. In fact Batchelor’s definition of the diffusion constant refers to the

special ^case of batch sedimentation for which (23)

holds and consequently

Hence, ^{for batch} sedimentation, ^the effective ^diffu-

sion ^constant is

and we recover Batchelor’s result. Anyway, ^the general expression ^is (36) ^or (37) ^and expression (38)

is restricted to a particular (although important)

case.

By the way, let ^{us comment on a statement} made

by ^{Batchelor in} [3]. ^{In this} fundamental paper ^{it is} said that a force per unit volume acting ^{on fluid and} particles ^alike produces ^no relative motion. This statement is obviously ^{in error since} only ^{a force} per unit mass acting ^{on fluid and} particles ^{alike has that} property (think ^of gravity ^for instance). ^{If the}

external forces per unit mass were different, say g, ^and g2, ^the right-hand ^{side of} (9) would have been modified into

and a part of J would have been proportional ^to

g, - g2. ^{As a} result Batchelor’s proposal (3.14) ^for

the effective force ^{on a} particle ^{should be written as}

and not

It happens ^{that the} missing ^factor p0 compensates

for the p2/p ^factor linking Db ^{and D, and the final}

result got by Batchelor for Db ^{is correct.}

8. Intluence of rotation.

When written in a frame rotating ^with angular velocity ^{n relative to a Galilean frame, the} equation (9) ^for the relative velocity is transformed into

And when neglecting ^{both inertia} and the role of R, ^this equation merges into

when (10) ^{is taken into account. We deduce from}

1695

(13) ^and (19) ^{that the relative} velocity ^{is now} given by

with 6 given ⁱⁿ (22). ^{And if we also} neglect ^inertia

and viscous shear stress in the overall momentum

equation, ^we get

where V is the barycentric velocity. ^A settling

process in ^a rotating system involves the solution of

(39) ^and (40) compatible ^{with the two} mass conserva-

tion equations [2]. Notice that the so-called Coriolis force is 2 ft x V for the « true » velocity V, ^{but that} it is Q x W for a relative (i. e. Galilean-invariant) velocity, ^{and not 2 n x W as} often found in the literature.

9. Faxen’s diffusion.

It is worth noticing that another kind of mass

diffusion is possible ^{in a} suspension, namely ^what

can be called Faxen’s diffusion : it results from the relative motion imposed ^{on a} particle moving ^{in a} non-homogeneous velocity gradient. Faxen’s result for an isolated particle ^can be written with our

notations as

and it is possible ^{to propose a} generalisation ^{of this}

result to non-dilute suspensions, again ^{with the} help

of irreversible thermodynamics.

-To ^{do this, we notice} that when the stress

R is taken into account, ^the entropy production

involves a R : V W term and because of a thermody-

namic coupling ^{with the V V force} occurring ^{in the}

viscous dissipation ^{term, we can write}

where 17 2 is the carrier fluid viscosity while m and n

are two unknown functions of the volume fraction.

When inertia in relative motion may be neglected,

the former result (14) ^is transformed into

where F given ⁱⁿ (13). ^{If we} notice that ^a continuum

description ^{of the} suspension implies

where a is the particle size, ^{it is not difficult to}

convince oneself that the V W part ^{of R is} always negligible ^as compared ^to F, ^and consequently ⁽⁴¹⁾

amounts to

Extracting ^{W from that} equation, ^{the mass diffusion}

flux becomes

instead of (15). ^{The new} contribution to J is the ^non- dilute form of Faxen’s diffusion. From the expression (22) for 0 it is clear that this term is all the more

important ^{than the} particle ^{size is} larger, ^{but cer-} tainly requires the absence of any sedimentation

(i. e. Po

p) ^{to be observed.}

10. Conclusions.

If one may neglect inertia in relative motion, ^the general features of mass and entropy transport ⁱⁿ suspensions ^{are rather} simple. Except for Faxen’s diffusion which is specific ^of large particles, ^the general form of the mass and entropy ^{fluxes was}

shown to be the same for a suspension ^{and a two-} component mixture. The Onsager symmetry ^mani- fests itself in a suspension ^{as a} consequence of the

joint ^role played by ^{the relative} velocity ^{in both mass}

and entropy transport. ^{It was} possible ^{to conclude}

that for a Soret effect to be observed, ^the particles

should be deformable ^or interact through long-range

forces (other ^than hydrodynamic forces). ^In particu- lar, ^{there is no} Soret effect in ^a suspension ^of rigid particles. ^{We have also} reconsidered the concentra- tion-diffusion in suspensions ^{with the} help ^{of the so-}

called configuration entropy, ^{and we} showed that the true diffusion constant depends ^{on the} particle

and fluid densities, ^a result that can prove important

for diffusion in flows with non-zero volume flux.

Lastly ^{a new and correct} expression ^was given ^for

the Coriolis force in the relative motion ; ^{it remains}

to find a situation where this term is not negligible !

Acknowledgments.

^I would like to thank

D. Levesque and J. J. Weiss (LPTHE Orsay) ^for

helpful discussions on the hard-sphere liquid.

References

[1] BATCHELOR, ^G. K., ^J. Fluid. Mech. 52 (1972) ^245.

[2] SCHNEIDER, W., ^{in Lecture Notes in} Physics ^n° 235, Meier G. and Obermeier F. eds. (Springer- Verlag, Berlin) 1985, p. 326.

[3] BATCHELOR, ^G. K., ^J. Fluid Mech. 74 (1976) ^1.

[4] ^{RUSSEL, W. B., Ann.} Rev. Fluid. Mech. 13 (1981) ^425.

[5] ^{LANDAU, L. and} LIFCHITZ, E., Mécanique ^des fluides (Mir, Moscou) 1971, ^{ch. 6.}

[6] LHUILLIER, D., ^{Int. J.} Multiph. ^{Flow 11} (1985) ^427.

GEURST, ^J. A., Physica ^{A 129} (1985) ^233.

[7] MÜLLER, I., in Lecture Notes in Physics ^n° 199, ^J.

Casas-Vasquez, ^{D. Jou and} G. Lebon eds.

(Springer-Verlag, Berlin) ^1984, p. 32.

[8] LHUILLIER, D., ^J. Physique ⁴⁶ (1985) ^1325.

[9] ^{SALSBURG, Z. W. and WOOD, W. W., J. Chem.} Phys.

37 (1962) ^798.

ALDER, ^B. J., HOOVER, ^W. G. and YOUNG, ^D. A.,

J. Chem. Phys. ⁴⁹ (1968) ^3688.