Micropolar theory from the viewpoint of mesoscopic and mixture theories Текст научной статьи по специальности «Физика»

УДК 539.3

Micropolar theory from the viewpoint of mesoscopic and mixture theories

W.H. Müller1 and E.N. Vilchevskaya23

1 Institute of Mechanics, Technische Universität Berlin, Berlin, 10587, Germany

2 Institute for Problems in Mechanical Engineering RAS, St. Petersburg, 199178, Russia

3 Peter the Great Saint-Petersburg Polytechnic University, St. Petersburg, 195251, Russia

This paper takes a nontraditional look at micropolar media. It emphasizes the idea that it may become necessary to abandon the concept of material particles if one wishes to describe micropolar matter in which structural changes or chemical reactions occur. Based on recent results presented by Ivanova and Vilchevskaya (2016) we will proceed as follows. First we shall summarize the theory required for handling such situations in terms of a single macroscopic continuum. One of its main features are new balance equations for the local tensors of inertia containing production terms. The new balances and in particular the productions will then be interpreted mesoscopically by taking the inner structure of micropolar matter into account. As an alternative way of understanding the new relations we shall also attempt to use the concepts of the theory of mixtures. However, we shall see by example that this line of reasoning has its limitations: A binary mixture of electrically charged species subjected to gravity will segregate. Hence it is impossible to use a single continuum for modeling this kind of motion. However, in this context it will also become clear that the traditional Lagrangian way of describing motion of structurally transforming materials is no longer adequate and should be superseded by the Eulerian approach.

Keywords: micropolar media, spatial description, rational mixture theory, characteristics, transport equations

Микрополярная среда с точки зрения мезоскопической теории и теории смесей

W.H. Müller1, E.H. Вильчевская2'3

1 Берлинский технический университет, Берлин, 10587, Германия

2 Институт проблем машиноведения РАН, Санкт Петербург, 199178, Россия

3 Санкт-Петербургский политехнический университет Петра Великого, Санкт Петербург, 195251, Россия

В статье представлен нетрадиционный подход к описанию микрополярных сред. При описании сред, в которых происходят структурные изменения или химические реакции, в ряде случаев нужно отказаться от традиционного представления о материальной частице. На основании общей теории, предложенной ранее Е.А. Ивановой и E.H. Вильчевской (2016), сформулированы основные соотношения для описания данных ситуаций с точки зрения единого макроконтинуума. Главной особенностью данной теории являются дополнительные балансовые соотношения для тензоров инерции с источниковыми членами, отвечающими за структурные изменения среды. Необходимость формулировки новых балансовых соотношений, в частности, интерпретация входящих в них источниковых членов, демонстрируется исходя из рассмотрения репрезентативного объема на мезоуровне с учетом микроструктуры полярной среды. Также предложен альтернативный способ описания новых соотношений в рамках теории смесей. Показано, что применение предложенной теории имеет ряд ограничений. В частности, бинарная среда, состоящая из электрически заряженных частиц, находящихся в поле тяготения, разделяется на фракции. Это делает невозможным ее описание в рамках единого континуума. Также показано, что традиционный подход, основанный на лагранжевом описании, неприменим для описания структурных превращений в среде и должен быть заменен эйлеровым подходом.

Ключевые слова: микрополярные среды, пространственное описание, рациональная теория смеси, характеристики, уравнения переноса

1. Introduction

Generalized continuum theories (GCTs) are of great importance to the materials science community, the reason being the wish for describing the mechanical behavior of high performance materials with an inner structure. They

should be applicable to large and small scale applications ranging from light-weight aerospace and automotive panels down to micromechanics and microelectronic gadgets.

A particular form of GCT is the so-called micropolar theory, which concentrates on the aspect of inner rotational degrees of freedom of a material. The original mathematical formulation of this theory was "traditional" in the sense that it was based on the idea of an indestructible material particle, no matter as to whether the objective was to describe micropolar solid or micropolar fluid matter (see [1, 2, Sect. 13], [3-5], or [6], respectively). However, most

© Müller W.H., Vilchevskaya E.N., 2017

recently, this idea was abandoned in [7], which opened up micropolar theory for applications that are neither completely on the solid nor on the liquid side, namely to soils, granular and powder-like materials, and even to porous media and bubbly foams.

For the sake of conciseness we shall compile the macroscopic aspects of this new theory in the next section and point out the differences to conventional micropolar theories explicitly.

Next we shall summarize and discuss macroscopic micropolar theory from a mesoscopic point of view, which was already done in great detail in [7]. This repetition seems necessary because of several aspects: On the one hand side, it allows for a better understanding of the new features of macroscopic theory, which are, first, new balances with production terms for the rotational measures of inertia and, second, a coupling tensor for linear and angular momentum. On the other hand, it forms a basis for comparison with another way of understanding macroscopic micropolar theory, namely from the standpoint of the theory of mixtures.

We shall also present all features of a mixture theory as applied to micropolar media. In fact, the idea of using mixture theory in this context is not completely new. It was addressed much earlier for nematic crystals in a brief note by Müller [8]. However, it remained a torso at that time from the mechanical as well as from the thermodynamical point of view. In this paper we will attempt to clarify at least the mechanical issues.

We shall consider the motion of a binary, electrically charged mixture in a gravitational field, a problem that can be solved in closed form by means of the method of characteristics. We shall demonstrate that this binary type of matter will segregate into two species. Hence it is not possible to describe it as a single continuum on the macroscopic level to begin with. The example was chosen in order to point out one aspect in the mesoscopic theory as presented in [7], which is easily overlooked. It takes a continuous distribution of mixture species, in other words infinitely many, in order to describe micropolar matter with a production of rotational inertia as a single macroscopic medium. However, the example also serves for illustration of another aspect: Problems of structurally transforming micropolar matter must be treated in spatial and not in Lagrangian description.

2. Summary of macroscopic theory

Following [7] the description of the mechanics of micropolar media on a macroscopic continuum scale is based on the following balances for mass density p(x, t), specific linear momentum v(x, t) (also known as transla-tional velocity), tensor of the specific moment of inertia J(x, t) (in units of kg-m2/kg = m2), specific moment of inertia coupling tensor B(x, t) (in units of kg-m/kg = m), and angular velocity field ra(x, t):

the balance of mass Sp

ht

+ pV-v = 0,

the balance of momentum g

p—(v + B• ra) = V-o + pf + P1B • ra, (2)

ot

the balances of moment of inertia and coupling moment of inertia tensors

gj gB

— + Jxra - ra xJ = xJ, — + Bxra - raxB = xb, (3)

Ot Ot

and, finally, the balance of spin g

p—(J • ra + v • B) = V - ^ + ox + pm -5t

- vxB-ra + p(v-Xb + Xj-ra). (4)

The meaning of the various symbols is as follows:

(5)

M = M + v .V(.)

St dt

denotes the substantial (material) derivative of a field quantity, o is the (nonsymmetric) Cauchy stress tensor, f is the specific body force, xj (a second rank symmetric tensor) and xB are the productions related to the moment of inertia tensor J and to the coupling moment of inertia tensor B, respectively, ^ is the couple stress tensor, ox := £ • o is the Gibbsian cross applied to the (nonsymmetric) Cauchy stress tensor e being the Levi-Civita tensor in combination with the outer double scalar product • (cf., [9, p. 40]), and m are specific volume couples.

We shall now compare these relations with the pertinent literature on micropolar materials.

The well-known balance for the measure oftranslational inertia, i.e. the mass density, Eq. (1), remains unchanged, as to be expected. Mass is conserved, may the matter be locally spinning or not. Indeed, we find the same relation in all papers or textbooks on micropolar materials, for example [1, p. 14, 2, p. 265, 5, p. 268, 10, p. 354, 11, p. 966, etc.]. However, a caveat is in order. In all of these citations the conservation of mass is looked at from the Lagrangian perspective of a material particle. The Eulerian point of view (or spatial description), which would allow us to consider matter that is structurally changing by consolidation or defragmentation or through chemical reactions, is not even anticipated.

The situation is different in the case of the momentum balance, Eq. (2), which contains the "unusual" term B • ra. As it was put so aptly in [3, p. 46], the assumption of a vanishing coupling tensor B is "widely used" in the literature. In fact it is fair to say that there is no awareness of a coupling term in the expression for the specific kinetic energy K which serves as a potential for the specific linear momentum K— = dK/dv, and for the spin K2 = dK/dra. Hence we find in the books of one of the pioneers in micropolar media [2, p. 266]:

K = — v • v +—ra • J • ra, (6)

a result that holds, as it is known from rigid body mechanics, if the center of mass is chosen as the reference point. However, if the center of mass and the geometrical center do not coincide, as it can be the case in a nonmaterial volume element undergoing structural changes, a coupling term has to be taken into account, and we write

K = 1 v • v + v • B • ro +1 ro • J • ro.

2 2

(7)

[3

The Russian school on micropolars is aware of that in p. 45, 12, 13, 14, p. 130, 15].

Even more extensive comments are in order in context with Eqs. (3). In the presented form these equations can only be found in the recent paper by one of the authors [7]. There is a precedent to the equation for the inertia tensor J in [1, p. 15]. There it is referred to as "conservation of micro-inertia," because it does not contain the production term % j . On the macroscopic continuum level this term must be interpreted as an additional constitutive quantity. Its appearance is due to fact that there are continuum problems where the concept of a material particle or a closed system may no longer be applicable. In general, structural transformations will result in a production % J. Several examples have been provided so far in [7, 16]:

(a) A homogeneous mix of pressurized hollow spherical particles undergoing a uniform change of external pressure so that their diameter and moment of inertia changes.

(b) Particles of type (a) but initially inhomogeneously distributed in an isothermal atmosphere subjected to a barometric pressure distribution falling down and thereby transporting a flux of J into new observational points.

(c) Changes of anisotropy due to reorientation of initially randomly oriented ellipsoidal particles.

(d) Fragmentation of spherical particles in a crusher.

In this context it should once more be recalled that such

problems make the Lagrangian point of view obsolete and the Eulerian one in terms of a spatial description must be followed instead.

We finally turn our attention to Eq. (4), the balance of spin. A similar comment holds as in the case of the momentum balance (2). Due to the coupling it contains an unusual term v • B, because the specific spin follows from the expression for the kinetic energy Eq. (7) as K 2 = dK/da.

A final note must be made in context with the balances of linear and rotational inertia Eqs. (1) and (3), in particular in context with the production terms %jjb . There is yet another balance, namely the one for the particle density n(x, t):

—+ n V-v = t, (8)

St

where t(x, t) denotes the production of particles per unit volume in a point x at time t. It is due to chemical reactions or to mechanical crushing, which were already mentioned above and which makes it just another constitutive quantity, just like the other two productions. However, structural changes are not part of conventional micropolar theory and therefore this equation is never mentioned. In fact, if there is no production it can be considered as an alternative way of writing the mass balance. Therefore, as we shall see, in the extended version of micropolar theory it may serve independently.

3. Summary of mesoscopic theory

In this section we summarize some essential relations on a mesoscopic level from the paper [7]. To this end consider Fig. 1. The inset on the very left presents the continuous macroscopic point of view of the previous section. We focus on a (representative) volume element AV at a fixed position x in space containing matter carrying inertial characteristics p(x, t), J(x, t) (in units of m2), and B(x, t) (in units of m), moving with linear and angular velocities v(x, t) and a (x, t), respectively, all of which are fields. Note that AV does not refer to the volume occupied by a material point. Rather we may imagine it to be a small cell in a spatial Eulerian grid. In fact, it may even be considered as

Fig. 1. Macroscopic, mesoscopic and microscopic perspective

moving, hence allowing for an ALE-formulation, but we will not make use of this possibility explicitly in this paper.

The centerpiece illustrates the mesoscopic perspective. Within the volume element A V a sufficiently large number of "particles," i = 1, ..., N(x, t), reside. The presence of a very large number of particles is required since otherwise establishing a continuous field theory would not be possible, and fluctuations would become dominant. Furthermore note that the total particle number, N, can depend on space x and on time t. It depends on space, because it is related to A V at position x, and it depends on time, because AV is an open volume and because reactions might take place, both of which will change the total particle number N(x, t).

One of these particles, i, is depicted on the very right of the figure. It carries known time-independent inertial characteristics mt, Bt (in units of kg^m), and Jt (in units of kg • m2), the latter with respect to the geometrical center of the spatial volume element A V. Incorporating Bi is not just because we wish to complicate the theory, but rather a necessity, because a coupling term arises if the geometrical and not the center of mass is chosen as reference point. Moreover, the particle moves with individual independent linear and angular velocities vt (t) and rat (t), respectively, both of which can be time-dependent. Both are no fields but discrete quantities. This represents the mesoscopic point-of-view so-to-speak initially. We are now going to relate the macroscopic and the particle worlds by forming mesoscopic averages. The idea is to replace the particles within the elementary volume by an ensemble of identical particles each having an average mass and an average tensor of inertia. To this end the inertial characteristics of the elementary volume are assumed to coincide with those of the average particle. Moreover, the linear and angular momenta of the elementary volume consisting of the original particles are required to equal those of the elementary volume consisting of average particles.

We start by defining average inertial characteristics within AV by

N(x,t)

E m

m(x, t) :=

B(x, t) :=

N (x, t )

N(x,t )

E B

i=l

, J(x, t):=

N(x,t)

E J

i=l

' N (x, t ) '

(9)

N(x, t)

Note that all three averages are field quantities, because the total number of particles N is a field. Next we define the fields of particle and mass densities:

N ( x, t)

._ N(x, t) E m

n(x, t) :=

-, p(x, t):_

AV AV

Because of Eq. (9)j it follows that

p(x, t) = m(x, t)n(x, t).

(10)

(11)

The following relations relate the inertial tensors of the particles to the macroscopic ones by definition:

N(x,t) N(x, t)

E J E A

p(x, t)J (x, t) :_

AV

-, p(x ,t )B (x ,t ):=

i=i

AV

(12)

(13)

By virtue of Eqs. (9) and (10) we then have p(x, t)J(x, t) = n(x, t)J(x, t), p(x, t)B(x, t) = n(x, t)B(x, t). In this context a few comments are in order. Note that one of the key points in [7] is that the inertia tensors are no longer known characteristics of the medium. This is in contrast to the mainstream approach in [1, p. 15, 2, p. 265, 17, 3, p. 33], where it is assumed that the initial inertia tensor of a material particle X in the reference configuration is known, J 0( X), and the current J is fully determined by the particle's rigid body rotations. Because the inertia tensors are independent variables now additional balance equations have to be formulated. The inertia can change because of flux of inertia or because of internal structural transformations involving the inertia production. The form of the production term is an additional issue, which must be investigated. Possible forms were presented in [7] or [16]. Here we are going to clarify a possible meaning of the production term in terms of particle species and their movement as we shall see shortly. Once more it is important to say that the changes of the inertia tensor may be considered as an indicator of internal structural transformations in the material. That is why consideration of the tensor of inertia is necessary even if there is no angular velocity.

This concludes the replacement of mesoscopic inertial characteristics with macroscopic ones and vice versa. We now turn to connecting the linear and angular momenta of the macroscopic world with those of the particles. In general (see [3, Sect. 4.3, 14, Sect. 3.3]) the kinetic energy of one the particles i reads

Ki (t) = 1/2 mt v (t) • v (t) + + v.- (t) • Bi • rat (t) +1/2 rat (t) • Ji • rat (t), (14)

where Bt is a tensor coupling linear and angular velocities. This expression serves as a potential for the linear and for the angular momenta of one particle:

dK;

Ku (t) = —- = mtVi (t) + Bi • Mi (t),

' dv,-

K2,i (t )

_ K _

dro,-

(15)

v, (t) •B+ J• M,- (t).

We define the average linear and angular momenta within AV by

K

i/2(x' t) :_

N( x, t )

E KV2, (t)

(16)

N (x, t)

Note that these are both fields. Now we postulate the following connection between the macroscopic linear momentum and the spin with the particle averages from Eqs. (16):

p(v + B • a) = nKj, p(v• B + J • a) = nK2, (17)

where all quantities are fields depending on x and t. This in turn means that there are no simple relations between the particle velocities vt, and the macroscopic fields v, a and, in particular, we must conclude:

N( x, t) N( x, t)

E v (t) E at (t)

v(x, t) * Z „ , a (x, t) * '=1 . (18)

N (x, t) N (x, t)

Note that the nonequal signs turn into equal signs if B = 0. In fact, B * 0 holds only if the mass distribution within the representative volume is not homogeneous. This is hardly the case if a large number of particles are considered. We have to write for an individual particle Eq. (14), because we need to write kinetic energies of all the particles with regard to the same point, which in this case is the geometrical center of AV and this point is not a mass center for every particle. However, as soon as the average procedure is provided, B (as an average quantity) has to be close to zero, because in the case of a homogeneous distribution of mass the geometrical center becomes the center of mass of the volume. It means that the kinematic quantities of the volume are written with regard to its center of mass, and as a result B = 0.

4. A mixture theory of micropolar media

In the spirit of rational mixture theory (see [10, Ch. 6, 18, Sects. 158 and 215]) we shall now depart from the viewpoint presented in [7] and decompose the total number of particles N(x, t) within a volume element A V into a finite number of different sorts or species, characterized by Greek indices a = 1, ..., v(x, t), consisting of Na (x, t) particles each, such that v << N. Each particle within a species has exactly the same inertial properties, i.e., mass ma and tensors Ja, Ba, with regard to the geometrical center of a spatial volume element A V. In other words we now lose the aspects of particle diversity and individuality characteristic of the mesoscopic approach presented in Sect. 3. Furthermore note that no in-depth analysis and examples are provided here regarding the coupling term Ba. It is simply mentioned for consistency with the rest of the text.

Moreover, recall that mixture theory is a formal theory. As such an effort is made to establish so-called partial field quantities for which balance equations hold, for which after summation with regard to a it is required that they lead to the balances for the macroscopic fields. Also note that traditionally mixture theory does not consider polar media. Hence angular velocity or inertia tensors do not occur and the corresponding balances are neither mentioned nor discussed. However, in order to compare it with our macroscopic theory, we decided to extend the classical approach by incorporating the inertia tensor of particles and deriving the corresponding equations for mixture theory. We proceed to discuss these aspects.

4.1. Partial quantities

We start by defining average inertial characteristics within AV by

Va (x>t)

E maNa (x,t)

m(x,t):=- a=1

N(x,t)

Va (x, t)

E J a Na (x, t )

J(x, t ) :=

a=1

N(x,t)

(19)

Va (x>t)_

E B a Na (x, t)

B(x, t ):= a=1 -,

N (x,t)

all of which are field quantities, because the total number of particles N, the number of particles of species a, Na, and the number of different species va (x, t) are fields, if we allow for chemical reactions. Next we define the field of partial particle and mass densities:

na (x> 0 :

__Na (x, t)

AV

(20)

Na (x, t)ma Pa (x, t) a = mana (x, t).

Obviously we have

Va ( x,t )

N(x, t)= E Na (x, t)

a=1

and because of Eqs. (9)j and (10) we find the following relations between mixture quantities and macroscopic fields (including the particle density of Eq. (8)):

Va ( x>t ) Va ( x>t )

n(x,t )= E na (x , o, P (x ,t )= E Pa (x ,t

a=1

a=1

Va(x>t)

(21)

n(x, t)m (x, t )= E na (x, t )ma.

a=1

In contrast to these equations the following relations are not mathematical consequences. Rather they relate the macroscopic specific moment of inertia tensor and the coupling of moment of inertia tensor, J and B, by definition to J and B given by Eqs. (19)2 3 in terms of mixture related quantities:

p(x, t)J(x, t) := n(x, t)Jj(x, t),

(22)

p(x, t)B(x, t) := n(x, t)B(x, t). v 7

We write it like that in order to be able to compare with Eqs. (13). Now, in a manner similar to Eqs. (13) we define the following partial specific inertial tensors Ja and Ba:

pa t)Ja (x, t) := na (x, t)JJa,

(23)

pa (x, t)Ba (x, t):= na (x, ta.

All field quantities in these two equations are identified by explicit arguments. In this context it was observed that the following partial average moment of inertial tensor, defined in complete analogy to Eq. (19)2:

Ja (x, t) :=

Na (x. t)„

E J

i_1 _ va Na

J a Na (x, t)

(24)

Na (x, t) Na (x, t) is not a field but a constant. The same remark holds for B a. Hence, because of Eqs. (23) and (20)2, we conclude that the specific moment of inertia and the specific coupling moment of inertia are no fields either. Rather they must be constants, too:

Ja = Ja/ma , Ba = Ba/ma . (25)

We now turn to the question how to relate macroscopic momenta to translational and angular velocities of particles belonging to the same species in a mixture. Traditional mixture theory does not consider angular velocities, hence we shall not consider them either. However, we shall incorporate the inertia tensor since its changes are of interest as outlined in the previous sections. Moreover, we want to show that the production of the inertia tensor will appear within our "extended mixture theory" and clarify different terms of this production. This goal is reached in terms of the expressions (22) relating the macroscopic and mixture inertia tensors and by putting ra t = 0 in Eq. (14) for the kinetic energy of one particle, which then simply reads

K, (t) = 1/2 m, v. (t) • V. (t). (26)

Then this relation serves as a potential for the usual linear momentum of one particle as follows: dKt

K1,i (t ):= —L = mi v. (t). (27)

dvi

We now define the average linear momentum within AV by

N ( x, t) N (x,t)

E KM(t) E mivi (t)

Ki(x, t ):=

(28)

N(x,t) N(x,t)

and conclude because of Eqs. (10) that n(x, t)K (x, t) = p(x, t)v(x, t),

N(x,t) In (x,t) (29)

v(x, t)= E mivi (t W E mi.

¿=1 / ¿=1

Consequently, in this case the velocity field of macroscopic theory v(x, t) is identified as the barycentric velocity. Now, in order to bring in mixture quantities we define the partial linear momentum analogously to Eq. (28) by

K a,1 (x t) ma

1

N ( x, t )

"E Va, (t ).

(30)

Na (x, t) i=1

A further simplification of the sum is not possible because the velocities va t of the a particles are not necessarily all the same. We conclude that

na (x, t)Ka,1(x, t) = pa (x, t )va (x, t),

1 Na (x, t) (31)

va (x'tE va,i (t).

Na (x, t) i=1

In other words, there is no difference between partial barycentric velocity and partial average velocity, because all the masses are the same.

Summarizing we conclude from the above that the following relations between macroscopic and partial field quantities hold:

Va (x>t)

n(x, t)J(x, t)_ E na (x, t)Ja ,

a_1

Va (x>t)

(32)

p(x, t)J(x,t)= E pa (x> t)Ja ,

a=1

Va (x>t)

p(x, t) v(x, t)= E pa (x> t )va (x, t).

a=1

For latter purpose we also note the following relation resulting from these:

J(x, t) =

Va(x> t)„

E Ja na (x> t)

a_1

Va (x>t) !

E ma na (x, t)

a_1

(33)

which relates the partial inertia tensors to the macroscopic specific tensor of inertia field.

4.2. Partial balances

In this section we will, first, briefly recall some well known relations on partial balances of mixtures and how they relate to the corresponding macroscopic balance equations of continuum theory in regular points. Second, we will present some new relations so far untouched by mixture theory. These relate to the inertia tensor and to its production term. We start with the partial balances for particle numbers, mass, and linear momentum, which read a = 1,..., Va(x, t) [10, p. 68]:

+ v )_T dPa +V-(P V )_m T

dt v a va) La, dt a a) ma La,

dva

at

P -a + P V •Vv _

^ a ^, ^a * a * a

(34)

= V • «a + Pafa + ma + maTa va >

Ta and ma being the productions of partial particle numbers and partial momenta. Following the principle that after summation with regard to all components a Eqs. (34) should lead to the macroscopic relations (1), (2), and (8), we find the following constraints:

Va (x>t)

T _ E (Ta + V •[na(v - Va)]),

a_1

Va (x>0 Va (x>0 Va (x>0

E maTa _0, E ma _ 0 Pf _ E Pafa' (35)

a_1 a_1 a_1

Va (x>t)

« _ E [«a — Pa( V -Va) ® (V - Va)].

a_1

Note that Eq. (34)2 is a consequence of Eq. (34)1 and obtained after multiplication by ma (see Eq. (20)2). Similarly, if we multiply it by J a we obtain

aJan«+V-(JarcaVa)_ JaV (36)

dt

We sum up with regard to a, observe Eqs. (32) and arrive at

dpJ V" (X'

dt

-+V-(pJv) ^

E J aTa +

a=1

+V

a (». t)„

E Ja

a=1

"a ( v -va )

(37)

If we now observe the mass balance (1) and the relations (11) and (21)3, we obtain

J

dt

+ vVJ = ■

Va (x>t)„

E Ja

a=1

(Ta+V- ["a (V - Va )])

Va (x>t)

E ma na

a=1

(38)

Hence in comparison with Eq. (3^ we must conclude that the production of specific moment of inertia tensor is given in context with a theory of mixture by the following equation:

X j (x. t ) =

Va (x>t)„

E Jc

a=1

(Ta + V - ["a (V - Va )])

Va (x>t)

(39)

E ma na a=1

At this point a few comments regarding the status and meaning of this equation are in order. Recall that a constitutive theory of materials on the macrolevel is a formal one. It should be based on "principles" and representation theorems leading to suitable reduction of constitutive equations in form and content. In other words, the introduction of "mesoscopic ingredients" through the backdoor are a no go for the purist. On the other hand, a mesoscopic, i.e. informal, physics-based line of reasoning can be very helpful during the reduction process of possible forms of the constitutive equations. For example, by looking at Eq. (39) we realize that the inertia production contains the excess velocity. This in mind one may want to propose viscosity-like terms in the production term for inertia on the macrolevel. In other words, it can give you a clue what the macroterm looks like, which variables have to be averaged, which processes or quantities it depends on, etc. But Eq. (39) cannot be used as an all-purpose equation in macrotheory.

In conclusion, it is only fair to say that such an approach has pros and cons. Indeed, on the one hand side, the last equation shows very clearly how mesoquantities are related to macro ones and that the newly introduced production results from particle productions due to chemical reactions or fragmentation Ta as well as from relative motion of particles with regard to the barycentric velocity v - va, the so-called excess or diffusion velocity term, which we also encounter in context with the stress tensor, Eq. (35). On the other hand side, the theory of mixture is limited to a discrete number of particle sorts va, whereas the /J in Eq. (3)j is not. For instance, a crushing process may lead to an essentially continuous distribution of particles of differ-

ent size. We may therefore conclude that Eq. (39) determines macrocharacteristics through mesoscopic ones, but only for a finite number of particle species. Also, we do not claim that Eq. (39) explains all possible microscopic effects. For example, the inertia production of the ensemble of pressurized particles discussed in [7] or [16] is not described by it. This may be considered as a con, but it was also never claimed that in this paper we present a theory for everything.

For completeness it should be mentioned that Eq. (39) can also be derived by combining the definition shown in Eq. (22)1 with

J(x, t ) =

Va(x> t)„

E J a "a (x. t)

a=1

Va (x>t) '

E ma "a (x, t)

a=1

(40)

then applying the substantial derivative (5), observing the macroscopic mass balance (1), the partial particle balance (34)1, and the relation for the barycentric velocity shown in (29).

One final remark concerns an intuitive interpretation of the two terms in the numerator of Eq. (39). The first one involving chemical productions is a volume based term as it should be. It cannot be written more suggestively. The second term is a flux term, which becomes even more evident if we integrate it over a volume and apply Gauss' law (the divergence theorem). Then it presents itself as a net flux of inertia across the volume surface, analogously to the stress tensor from Eq. (35)5 after the divergence has been applied to it, in order to make it a true balance quantity, the kind of which /J is from the very beginning on.

We shall now proceed and illustrate the macroscopic and mixture theories of Sect. 2 and 4, respectively, especially the production term Eq. (39), by an example. To this end we shall consider charged particles moving through a gravitational field. However, putting in a caveat seems necessary at this point. Note once more that Eq. (39) gives us clues of how the production of inertia in Eq. (3)1 may look like, but it cannot be used in macrotheory directly. In fact, so far we do not know how the macroproduction term (based on Eq. (39)) can be written. Hence we will try to eliminate all terms unknown to us in the example, and this includes besides the production of inertia the stress as well. The details of the elimination process, which involves Eqs. (39) and (35), are outlined in the Appendix A, but, in addition, a summary will be given as we move on in the main text.

5. Example: A binary mixture of electrically charged particles subjected to gravity

Consider the situation depicted in Fig. 2. A binary mix of matter is contained in a column of height H. The two constituents carry mass and, in addition, they are also electrically charged, i.e., they "feel" the presence of gravita-

Fig. 2. Electrically charged binary particle mix

tional acceleration g and of an electric field E both of which are assumed to be prescribed and constant. Later we shall idealize the situation in terms of one-dimensional continua. Then we shall assume that both accelerating fields point exclusively in direction z, such that g = -gez and E = Eez, respectively.

5.1. Macroscopic fields predicted by mixture theory

In Appendix B it is shown that according to the theory of mixtures the macroscopic fields of particle and mass density, velocity, and the specific moment of inertia are given by

n( z, t) = n0( z -1/2 ft2 ) + n0 (z -1/2 f t2 ), p( z, t) = m1n0 (z -1/2 f1t2) + m2 (z -1/2 /212),

(mf + m2f2)t_ (41)

v z ( Z, t )_

m^ z -1/2 fxt2) + m2 n0 (z -12 /212 ) '

J(z t) = Jn0 (z -1/2 /1 t2 ) + J2n20 (z - 12 /212 ) ' m1n0( z -1/2 /1t2) + m2 n20 (z -12 / t2 )' where n°2 (•) are the initial distributions of the partial particle densities and

fa := -g + ejma E, ae (1,2). (42)

These results hold under certain assumptions, namely reduction of the problem to one dimension, no chemical reactions, no momentum interaction between the species, and vanishing partial stress tensors, i.e., the species are assumed to behave like "dusts."

In order to put it quite bluntly. Based on detailed knowledge of the mesoscopic substructure we have predicted macroproperties. This, of course, is not the way a macroscopic theory is supposed to operate. Rather it is phenom-enologically based on the balance equations (1)-(3)1 and (8) in combination with suitable constitutive relations for the stress tensor and for the production of moment of inertia, which need to be established independently from a mesoscopic or mixture theory. Since we also want to study analytical solutions and compare with the results above, the constitutive equations must be "simple" enough. We proceed to explore this in more detail.

5.2. Macroscopic fields based on macroscopic theory

In Appendix C it is shown that according to macroscopic theory the macroscopic fields of particle and mass density, velocity, and the specific moment of inertia are given by

n(z, t)_ n0(Z-1/2/012),

P(z, t)_ Po(z-1/2f012),

V z ( z, t )

/it

(43)

m^jm2 n1 (z) + n2(z-12/21 )

J(z, t)_ Jo(z-1/2f0t2), ft :_-g + e2/m2 E, where, for comparison with Subsect. 5.1, the initial conditions for particle density, mass density, and specific moment of inertia must be chosen consistently to the requirements imposed before: n0(z)_ n0( z) + n0(z),

0, if z e (H, ),

n0( z)_

n0 _ const, if z e [0, H], 0, if z e (-^,0),

P0( z )_ p0( z ) + p0( z )_

(44)

= mn

m

m2

1 n0( z ) + n0( z )

- m2 n2 ( z ),

J0( z )

_ J2 J1

/J2 n0( z ) + n2°( z )

0, if z e (H, ), J [0, H ],

^,if zi m

m2 mjm2 n°(z) + n^(z)

0, if z e ,0). These equations hold under the following assumptions, which are explained in great detail in the Appendix B:

(a) The mass of species 1 is much smaller than that of species 2, m1 << m2, analogously we have for the inertial tensors J1 << J2 .

(b) Gravitational and electrostatic accelerations of species 1 are in balance; its net acceleration is zero and it does not move.

(c) The initial velocity of all species is zero; motion starts out of a state of rest.

(d ) There are no chemical reactions and no partial stress tensors (dust assumption).

(e) The particle density of species 1 is constant.

Realize that it can be argued as to whether the small mass approximation must be made when rewriting the algebraic Eq. (44)3. If it is made, then the specific moment of inertia assumes constant values throughout space, namely zero and J2/ m2. This is in contrast to the mesoscopic result (41)4, where we need to put f1 = 0 for consistency. If it is not made then both species travel as one at one speed, as indicated by the argument of the solution shown in Eq. (43)4, which is also in contrast to the solution from mixture theory, Eq. (41)4.

In the next section we shall compare predictions from Subsects. 5.1 and 5.2.

6. Numerical examples and discussion

In what follows we shall illustrate the spatial and temporal evolution of the solution for all fields compiled in Subsects. 5.1 and 5.2. This will be done in normalized form. The details of the rather extensive normalization process are outlined in Appendix D.

Specifically we choose for the initial conditions n0 = 0.8, n2(z) = L4z + 0.5, m1m 2 = 0.3, J J2 = 0.1. The latter we chose in order to show some effect, even if these ratios are not really small as it was required above. Recall that it was assumed that species 1 stays put, in other words, gravity and electric acceleration annihilate each other, f = = 0. However, species 2 is heaved upwards by f2 = 1.5.

Figure 3 visualizes the temporal development of the particle density predicted by macroscopic and mesoscopic (mixture theory) analysis, respectively. However, for completeness the development of the partial particle densities is also shown. The motions of the latter n° and n° are depicted in the solid and dashed lines, the motion of the prediction from macroscopic theory (based on Eq. (43)1 or rather (D.9) in the dotted line and the one from mesoscopic theory (Eq. (41)1 or rather (D.6)) in the dash and dotted

line. In the original domain z e [0,1] both predictions agree in so far that the macroscopic analysis "forgets" species 1 during the upward movement of the species. The remainder is equal zero. Of course, the differences were less pronounced if we had chosen a smaller mass ratio.

Similar remarks hold for the mass densities in Fig. 4 (based on Eqs. (43)2or (D.14), macro, and Eq. (41)2 or (D.13), meso), and finally for the inertia tensor in Fig. 5 (based on Eqs. (43)4 or (D.19), macro, and Eq. (43)4 or (D.18), meso). The results from macroscopic theory are always shown by the dashed line and the ones for mesoscopic theory by the solid line.

Summarizing we may say that in order to arrive at a closed form solution on the macrolevel we had to compromise quite extensively. For lack of better knowledge we made sure that the inertia production is equal to zero. This leads to differences in the solution from the macro- and from the mesoperspective, Eqs. (41) and (43). In conclusion, it is impossible to "force" a macroscopic theory to describe a binary mixture, which wants to segregate and not stay a single continuum, unless most severe measures are taken and one component is almost annihilated in its presence.

z/H t/T=0 ^

1.2- ij

IL-l-h-L-. / /

0.8- / /

/ /

- / /

/ /

0.4- / /

/ /

/ f /

n n / /

u.u 1 1 1 1 1

0 1 2 3

Normalized particle density

z/H 1.2

0.8

0.4

0.0

0 12 3

Normalized particle density

t/T= 0.494949 \b_

T /

/ / /

/___/

/ //

/ //

' /

/ /

/ //

. _ / /

-1-1— —i-1-1-1-

z/H 1.20.80.40.00 12 3 Normalized particle density

// / / * *

/ //___> * * i

/ / __j— / / f / / / /

! t/r= 0.656566

z/H 1.2

0.4-

0.0

/ / k

/

/•

/

/

/ /

__/

j t/T= 0.767677

0 12 3

Normalized particle density

z/H

1.2

0.8

0.4-

0.0

TU

t/T= 0.89899

0 12 3

Normalized particle density

z/H . / / f

1.2-

r---n

0.8" 1 '

0.4- 1

o.o-i '-1— t/T= 0.959596 —1-1-1-1-

0 12 3

Normalized particle density

Fig. 3. Spatial and temporal development of the particle density; n° (-), n° (---), n (macro) (........), n (meso) (----)

z/H

z/H

0.0 0.5 1.0 1.5 Normalized mass density

z/H

0.0 0.5 1.0 1.5 Normalized mass density

z/H

0.0 0.5 1.0 1.5 Normalized mass density

z/H

1.2

0.8-1

0.4-1

0.0

t/T= 0.89899

0.0 0.5 1.0 1.5 Normalized mass density

0.0 0.5 1.0 1.5 Normalized mass density

z/HV

1.2-

0.8

0.4-

0.0

t/T= 0.959596

0.0 0.5 1.0 1.5 Normalized mass density

Fig. 4. Spatial and temporal development of the mass density p(macro) (---), p(meso) (-)

7. Conclusions and outlook

This paper presented a comparison of results from a recent, extended formulation of macroscopic micropolar theory with relations expressing a mesoscopic point of view, specifically in terms of the theory of mixtures. The following tasks have been accomplished.

First, the extended set of balance equations of macroscopic micropolar theory was presented. In contrast to classical micropolar theory they are complemented by balances for the specific moment of inertia tensor J and, in principle, also for the second inertia tensor B, which is used for coupling translational and angular momentum, and appears if there is a difference between the center of mass and the geometric center. This new approach enables us to study the temporal development of rotational inertial characteristics. In this context it is particularly noteworthy that the additional balances contain production terms characteristic of structural transformations. In context with structural transformations a production of moment of inertia will arise. The concept of an indestructible material particle becomes obsolete and requires us to turn to a spatial (Eulerian) description when solving concrete problems.

Second, a mesoscopic framework was presented, which allows us to comprehend the meaning of all macroscopic fields, especially of the new ones for the inertia tensors, in terms of averages within a representative volume element. It is important to say that the particles within the elementary volume were replaced by an ensemble of identical particles each having an average mass and an average tensor of inertia. To this end the inertial characteristics of the representative volume element were assumed to coincide with those of the average particle. In addition, the inertia tensor fields were constrained by the requirement that the macroscopic linear and the angular momenta are equal to averages from the corresponding momenta of the particles. Moreover, the standard principle of continuum theory was observed according to which the number of particles within the representative volume element is very large so that it can be assumed that the mesoscopic characteristics of the particles, like particle mass, velocity, moment of inertia, etc., vary continuously.

Third, an alternative, yet mesoscopically particle-based viewpoint to this continuous approach was presented in terms of the theory of mixtures. Here it is assumed that

z/H

z/H

0.0 0.4 0.8 1.2 Normalized specific inertial tensor

z/H

1.2

0.8

0.4

0.0

t/T =0.767677

Jd

0.0 0.4 0.8 1.2 Normalized specific inertial tensor

z/H

0.0 0.4 0.8 1.2 Normalized specific inertial tensor

z/H

1.2-

0.4-

0.0

t/T = 0.89899 , / /

0.0 0.4 0.8 1.2 Normalized specific inertial tensor

0.0 0.4 0.8 1.2 Normalized specific inertial tensor

z/H

t/T = 0.959596 /

1.2

0.8

0.4

f

0.0

0.0 0.4 0.8 1.2 Normalized specific inertial tensor

Fig. 5. Spatial and temporal development of the moment of inertia J (macro) (---), J (meso) (-)

there exists a finite number of particle characteristics attributable to a finite number of particle species. The standpoint of the theory of mixtures is appealing because it allows to derive explicit relations for the newly introduced productions in the balances of the inertial tensors.

Fourth, the example of a binary mixture was studied. It shows very clearly a drawback of the newly presented macroscopic theory. It is not possible to reconcile results for the macroscopic fields from macroscopic theory with those obtained from the theory of mixtures completely. Only if extremely simplifying assumptions are made, which make one of the components of the mixture quasi non-existent, we come to a certain amount of agreement. Otherwise the binary mixture is simply "ripped apart" — a process that certainly cannot be described by a single continuum.

In future work we shall see if these simplifications can be ameliorated, for instance, by taking a much higher number of species into account.

Appendix

In this section we will compile some important formulae, based on the method of characteristics, which allow to obtain analytical solutions for the transport equations of interest to this paper. It is a hands-on approach far from mathematical stringency. For more details the reader is referred to the Appendix in [6] or to the pertinent literature, for example to [19].

A. Relating macroscopic body forces to mixture quantities

In complete analogy to Eq. (21)2 between the total and partial mass densities we write

Va (x' t)

q(x, t)= E qa (x, t) (A.1)

a=1

in order to connect partial and total charge density fields qa (x, t) and q(x, t), respectively (both in Coulomb per m3). If we assign the elementary charge ea to each species we

have just as in Eqs. (20) and (21):

Va (x>t)

q«(x 0 = eana (x, t), q(x, t) = ^ eana (x, t). (A.2)

a=1

The partial body force densities are given by Pa(X, t)fa = Pa(x, t)g + qa(x, t)E. (A.3)

Then because of Eqs. (A.2)1 and (20)2 we may write for the specific partial body forces:

fa = g + -

-E.

ma

(A.4)

According to Eqs. (35)4 and (21)2 the total body force density becomes

f (x, t)= g +

Va (x>t)

E eana (x, t)

a=1

Va (x>t)

E mana (x, t)

a=1

E.

(A.5)

B. Solution of the field equations for the mixture

We now focus on a one-dimensional binary mixture. In other words, all fields depend only on the z-direction. Nevertheless, in order to solve all equations in closed form we will need to make some further assumptions. Let us start with an examination of the partial balances. We turn to the partial balances of momenta first, Eq. (34)3. For a closed-form solution we shall assume that both species are "dusts," i.e., the partial stress tensors vanish o a = 0 and the interaction terms leading to exchange ofmomenta vanish ma = 0, ae (1, 2). Hence Eq. (34)3 reads for the two species:

% + Vx^ = -g + ^E =:fa, ae (1,2). (B.1)

dt

dz

As detailed in the Appendix in [16] these transport equations of the Burgers type can be integrated by using the method of characteristics:

+ fat, ae (1,2),(B.2)

va (z, t) = Oa z - J va (z, t')dt' _ t'-0 _ where ®a[-] are arbitrary functions. These solutions are obviously implicit. However, we now choose as initial conditions vanishing partial velocities

(z, t = 0) = 0, ae(1,2), (B.3)

and then conclude that Oa[-] must identically vanish, so that

(z,t) ^ (t) = fat, ae (1,2), (B.4)

i.e., the species move as a single block or rigid body and their velocities increase linearly, as they should when subjected to a constant acceleration. All of these results in mind, the relations for the partial number densities (34)1 reduce to

/• dna

- + fat—

- = °, ae (1,2),

(B.5)

dna

dt Ja' dz

which can also be solved in closed form by using the method of characteristics:

na (z, t) = r0

ae (1,2).

Z -J fat 'dt' t'=°

-Ta [ z -1/2 fa t2] (B.6)

The arbitrary functions ra [•] must be determined from initial conditions for the partial particle densities, na (z, t - 0) -: «a (z). By combining the initial conditions with the solution (B.6), we find that:

na (Z, t) = n°(Z - 1/2 fat2).

(B.7)

Hence, according to Eqs. (21)1 (20)2, (29), and (33) mesomechanics predicts for the macroscopic particle density, the mass density, and for the specific moment of inertia field:

n(z, t) = n°(z -1/2 fit2) + n°(z -1/2 f2t2), p(z, t) = mn0 (z -1/2 fit2) + m2n°° (z -1/2 f2t2),

(mifi + m2f2)t_ (B.8)

vz (Z, t) =

J(Z, t)

mn0 ( z - V2 ft2) + m2n°( z - V2 f2t2 ) ' _ Jn (z -1/2 ft2 ) + J2n°° (z -12 f2t2 )

m1n°( z - V2 fit2) + m2n°( z - V2 f>t2)' where the specific inertia tensor has also been adjusted to the one-dimensional case.

C. Determination offields from macroscopic theory

We now turn to the question what the macroscopic equations (1)—(3)1, and (8) have to say about these four quantities. These are no field equations yet and we have to complement them by constitutive relations. Because we wish to compare with results for the macroscopic fields from mixture theory we have to make sure that the constitutive equations are suited to do that and comply with the assumptions of Appendix B. Therefore in context with macroscopic theory some additional requirements using partial quantities must be made.

We start our examination with the balance of momentum shown in Eq. (2). A constitutive relation for the stress tensor is required. In view of Eq. (35)5, which relates partial quantities to the macroscopic stress we argue as follows. In order to obtain closed-form solutions we assume additionally that species 1 has much less mass than species 2, m1 << m2. We also recall that we considered a mixture of dusts so that the partial stress tensors vanish, aa = 0, ae (1,2). Moreover, the differences between the barycentric and the partial velocities can be rewritten, while taking into account that mx << m2. We find m,n,n2

a =---^i-2-(V2 - vt) ® (V2 - vt) =

mj m2 n + n2

~ —m1n1(v2 -v1) ® (v2 -v1). (C-1) If specialized to the one-dimensional case this equation reads:

a = -m1n1(v2 - v1)2. (C.2)

In the one-dimensional situation the partial velocities are only functions of time, Eq. (B.4). Thus we can make the contribution of the stress tensor in the balance of momentum vanish, if we require that n = const on z e [0, H] at all times t, since then V- o = 0 ^ da/dz = 0. Eq. (B.7), if evaluated for a = 1, tells us when this is the case. We

must make the argument time independent and guarantee that n (z, t) is constant throughout the interval [0, H] at all times t. This means that the gravitational and electrostatic accelerations are in balance:

f = -g + exjm E = 0. (C.3)

This is the first step toward solving the balance of momentum (2) in closed form. The second one concerns the simplification of the expression for the total body force. Therefore, if we combine the assumptions of vanishing partial body force on species 1 and smallness of its mass, then according to Eq. (35)4 the total specific body force (A.5) is given by

f (z, t)= n2 -g + ^ E =: f\ (C.4)

mdm2 n2 + n2 m2

In other words, it is also a constant and not a field. Hence, for the one-dimensional case, the balance of momentum shown in Eq. (2) simplifies to

dvz

-+ vz

dvz

fz fz

(C.5)

dt z dz

which can once more be solved in closed form by using the method of characteristics:

+A

(C.6)

vz(z, t) = O z- J v(z, t')dt' _ t'=0 _ where $[•] is an arbitrary function determinable from initial conditions. We choose a vanishing initial velocity

vz (z, t = 0) = 0, (C.7)

and conclude that $[•] = 0, so that

v z (z, t) = v z (t ) = fz0t. (C.8)

Thus macroscopic theory predicts that the dusts are moving as one block with a velocity increasing linearly in time. Note that this is in contrast to the theory of mixture, where both species move independently, even if this velocity is zero for species 1, because we have to put f = 0 for consistency in Eq. (B.8)3. Thus the macroscopic predic-

tion reads:

Vz (z, t)

ft

(C.9)

mjm2 n1)(z) + n°(z- V2 />t2)

We can now use Eq. (C.9) during integration of the particle balance (8), provided the particle production t is known. For this purpose we turn to Eq. (35)1. Recall that there are no chemical reactions, i.e., Ta _0, ae (1,2). The remaining terms containing differences between the bary-centric and the partial velocities can be rewritten similarly as in Eq. (C.1):

n^! - m^ m2)

t _ v'

m1/m2 n1 + n2 = V^[n1(v2 -V1)],

-(v2 - v1)

(C.10)

where, if restricted to the one-dimensional case, the differences between the partial velocities are functions of time only. Moreover, recall that we have assumed before that the particle density of species 1 is constant. Thus the pro-

duction term vanishes, t = 0, and the particle balance in Eq. (8) reads: 3n 0 dn

— + /0t— _0. dt Jz dz

(C.11)

As before this type of equation is solved by the method of characteristics in closed form:

n(z, t)_ n (z-1/2/012), (C.12)

where the initial condition for the particle density must be chosen consistently to Eq. (21)1:

n0 (z) _ n10 (z) + n° (z),

n0(z) =

(C.13)

0, if z e (H, +«>), nf = const, if z e [0, H], 0, if z e (-«>, 0).

Note the difference to the solution predicted by the theory of mixtures, Eq. (B.8)1. The part of the particle density related to species 1 will now move at speed ft and not stay put.

A similar line of reasoning leads to the following solution for the macroscopic mass balance (1):

p(z, t) = p0(z -1/2 f°t2), (C.14)

where the initial condition follows the rule

p0( z)= p0( z) + p2( z) = = m2(m1lm2 n°(z) + n°(z)) = m2n°(z). (C.15)

It can be argued as to whether the approximation in the last equation should be performed since, originally, the assumption m1 << m2 was made in order to arrive at closed forms of differential equations. It is not necessary in context with algebraic equations. Be that as it may, in contrast to the solution predicted by mesomechanics, Eq. (B.8)2, there is the same movement in both species, as indicated by the argument of the right hand side in Eq. (C.14).

We now turn to the differential equation (3)1 for the specific moment of inertia tensor. In this context we first study the production density %J according to Eq. (39). The situation is quite similar as in context with the particle production t, Eq. (C.10):

x j

_ J2 • J2 • J1 + 1

m2 m2 / m1 n1 + n2

= — V-MV2 - V1)], m2

-V •

n,n

1n2

m2 / m1 n1 + n2

(v2 - v1)

(C.16)

if we assume consistently that if m1 << m2 then JJ1 << J2, too. Now, if n1 is a constant, this production will vanish as well. Assuming one-dimensional conditions our field equation for the specific moment of inertia tensor therefore reduces to

-+f°t - _0, dt Jz dz

(C.17)

which can be solved by using the method of characteristics:

J(z, t) = J°(z -1/2 f°t2),

(C.18)

where the initial condition follows the rule (40), after specialization to one dimension and small mass approximation for species 1:

J n°( z) + n°°( z)

J°( z)= J h-

m2 m n°( z) + n°°( z)

0 if, z e (H, +«>),

^,if ze [0, H], (C.19)

m2

0, if z e (-«>, 0). It can be argued again as to whether the small mass approximation must be made when rewriting this algebraic equation. If it is made, then the specific moment of inertia assumes constant values throughout space, namely zero and J2jm2. This is in contrast to the mesoscopic result (B.8)4, where we need to put f = 0 again for consistency. If it is not made then both species travel as one at one speed, as indicated by the argument of the solution shown in Eq. (C.18), which is also in contrast to the solution from mixture theory, Eq. (B.8)4.

D. Normalizations

For a numerical evaluation and comparison of the solutions compiled in Subsects. 5.1 and 5.2 we need to find suitable normalizations. We will normalize space by height H and time by the duration T for a mass traversing H under the influence of gravitational acceleration g:

z _ t ^ \lH _ ,.

z := -, t := -, T := —. (D-1)

H T \ g

For the normalizations of the accelerations we write:

f2:= £ = -1 + e g m2 g

E = fz_=: f 0 : J z ,

(D.2)

which was chosen to be 1.5 during the simulations.

In the case of particle densities we will normalize with the average initial particle density nave, which is given by:

nave := ■-1 H n (z)dz = nf + nj(z = H/ 2), (D.3)

H z=0

if we assume that the first species is distributed equally throughout [0, H] and the second one varies linearly, such as AH z + B, A and B being two constants, which have to be chosen suitably, as we shall see shortly. We define the normalized particle densities accordingly:

n(Z, 7):= n(H Z, TJVnave,

n1,2 (z, T) n1,2 (Hz, TtVnave.

Similar conditions hold for the initial conditions. Because of Eqs. (21)1 and (C.13) we must conclude that

(D.4)

J n°(z ,°)dz = 1

z =°

°

«1° + - + b = 1,

(D.5)

: navea, B

naVeb.

—

In the simulations of Sect. 6 we put a = 1.4, b = 0.5, 0.8. The solutions for the particle density stemming from mesoscopic considerations, Eq. (B.8)1 reads in normalized form:

n(z, t ) = n°(z) + n°(z - J2t 2) with the initial conditions

°,if z e (1, + ^),

nVf z e [°,1], if z e (-«>, °)

n0_( z ) =

(D.6)

(D.7)

and

0, if z e (1, +

az + b,if z e [0,1], (D.8)

0, if z e (-«>, 0).

On the other hand, the solution for the particle density from macroscopic analysis, Eq. (C.12) is given by

n(z, 7)- n- fz°T2) (D.9)

with the initial condition 0, if z e (1,

n2°( z ) =

n °( z ) =

n1° + az + b, if z e[°,1], if z e (-«>, °).

(D.10)

Note that initially both solutions coincide. We now turn to the mass density solutions, which we will normalize by the average initial mass density

Pave := ■-1 Jfp° (Z)dz = p° + p°(z = J 2) =

H z=°

= m1nj° + m2n°(z = H/ 2). (D.11)

Hence the mass densities in dimensionless form are given by

P( z, 7 ):= P( HZ,TT ) =

Pave

_ m1/m2 n1(z, t ) + n2(z, t ) _ m1/m2 n1(z e [ö,1 ], ö) + n2 (V2, Ö)'

_ (_T) P1,2( Hz, T7 ) P1,2(z, t ):=^-=

(D.12)

m

„21m2 n1/2(z, t )

m1 /m2 n1(z e [°,1 ], °) + n2 (1/2, °)

The mesolevel solution (B.8)2 reads in dimensionless form:

P(z, t )

_ m^m2 n1°(z) + n2°(z - f2t 2)

mxlm2 n0 (z e [0,1]) + n0 (1/2) and the macrolevel one from Eq. (C.14):

p( z, 7 )- p0( z - fz°72) with the initial condition Pc(z )- p1°(z ) +P20(z )-_ mjm2 nf(z ) + «20(z )

(D.13)

(D.14)

(D.15)

mjm2 n0(z e [0,1]) + ^2)' where the expression "e [0, 1]" is supposed to indicate that any number from this interval can be chosen as argument to nf.

Finally the relations for the specific moment of inertia. We first define its average value

J ._ M0 (z e [0, H]) + J2n0 (z = 2H) ave._ m«0 (z e [0, H]) + m2«0 (z = 2H)

(D.16)

and define a dimensionless specific moment of inertia as follows:

j ( z, 7 ).==

j

(D.17)

Ji

J2

ni(z, t) + n2(z, t) mni(z e [0,1],0) + «2(1/2,0)

m, _ —N _ —„ J

—Ln1(z,t) + n2(z, T) Jn1(z e [0,1],0) + ^(V2,0) m2 J2

The result from mesoscopic theory, Eq. (B.8)4 now takes the dimensionless form:

J(z, t) = (D.18)

Jn0 (z) + n0 (z - f F2) mnf(z e [0,1]) + «2 (12)

J

m2

m n0 (z) + n0 (z - f r2) Jn0 (z e [0,1]) + n20 (12) m2 ./2

Finally the dimensionless form of the specific moment of inertia according to macroscopic theory from Eq. (C.18):

J(z, 7) = J0(z - f012), (D.19)

where the normalized initial condition reads:

J)( z) =

J «i0(z) + «0(z) m««(je [0,1]) + «0(1/2)

_ J

m«1 (z) + «20(z) Jnj0(z e [0,1]) + «0(1/2)

-. (D.20)

m2

J/

Acknowledgments

Support of her work by a grant from the Russian Foundation for Basic Research (16-01-00815) is gratefully acknowledged by E.N.Vilchevskaya. Both authors want to

thank Dr. A. Sokolov for most valuable help when converting the TeX file into MS-Word.

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nocTynaaa b peaaKUHro 23.01.2017 r.

Ceedenun 06 aemopax

Wolfgang H. Müller, Prof. Dr., Institute of Mechanics, Chair of Continuum Mechanics and Materials Theory, Technische Universität Berlin, [email protected]

Elena N. Vilchevskaya, Cand. Sci. (Phys.-Math.), Assoc. Prof., Senior Researcher, Sci. Secr., Institute for Problems in Mechanical Engineering RAS, Assoc. Prof., Peter the Great Saint-Petersburg Polytechnic University, [email protected]