AUTHOR COPY
Asymptotic Analysis 83 (2013) 331–353 DOI 10.3233/ASY-131162 IOS Press
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Asymptotic behavior of a suspension of oriented particles in a viscous incompressible fluid Maksym Berezhnyi a,∗ and Eugen Khruslov b a
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Darmstadt University of Technology, Darmstadt, Germany E-mails: [email protected], [email protected] b Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Kharkiv, Ukraine E-mail: [email protected]
Dedicated to the memory of my mother, T.M. Berezhna (22.09.1953–06.03.2012)
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Abstract. A viscous incompressible fluid with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of the fluid move translationally or rotate around a symmetry axis with the direction of their symmetry axes unchanged. The asymptotic behavior of oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a non-standard hydrodynamics. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor. Keywords: microstructure, suspension, anisotropic material, inhomogeneous material, viscous incompressible fluid, asymmetric hydrodynamics, asymptotic analysis
1. Introduction
Mechanics of suspensions is a part of a general physical–chemical sphere of knowledge about dispersions. Dispersion is a mixture of 2 phases one of which forms a continuum medium (we will call it a dispersive phase) and the other one is dispersed and distributed in the form of separate volume elements inside the first one (we will call it a disperse phase). In this work it is supposed that the dispersive phase is a viscous incompressible fluid and the disperse phase consists of a great number of small solid ferromagnetic particles suspended in the fluid. The sizes of particles are assumed to be of the same order as the distances between the nearest particles. *
Corresponding author: Maksym Berezhnyi, Darmstadt University of Technology, Schloßgartenstr. 7, Darmstadt 64289, Germany. Tel.: +49 6151 163495; E-mails: [email protected], [email protected]. 0921-7134/13/$27.50 © 2013 – IOS Press and the authors. All rights reserved
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When neglecting all physical–chemical processes, the study of the suspension motion can be considered as a problem of pure classical mechanics. In this case the motion of the dispersive phase is governed by the Navier–Stokes equations, and the motion of solid particles forming a dispersed phase is described by the equations of continuum mechanics. However, the study of the properties of the fluids in the framework of such a model by using both analytical and numerical methods appears to be an unsurmountable problem because of the great number of small particles. Therefore it is necessary to develop adequate macroscopic models that can help in studying such fluids. It is known that under the absence of external forces the motion of the compound is governed by the following homogenized equations: ρ
∂v + (v, ∇)v − div σ[v] = ρf , ∂t
div v = 0,
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where ρ = ρ(x) is the homogenized specific mass density of the mixture, v(x, t) is the homogenized velocity of the suspension, σ[v] = {σij [v]}3i,j=1 is the homogenized stress tensor and f is the external force acting on the suspension. Moreover, the stress tensor linearly depends on the strain tensor e[v] = ∂v ∂vi {eij [v] = 12 ( ∂x + ∂xji )}3i,j=1 : j
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σ[v] = Ae[v] − Ep,
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where A = {anpqr (x, t)}3n,p,q,r=1 is the effective viscosity tensor (it is symmetrical with respect to permutation of pairs of subscripts and of subscripts in pairs themselves), E = {δij }3i,j=1 is the unity matrix, and p(x, t) is the pressure. The result is qualitatively the same in the case of weak electric or magnetic forces affecting the suspension. If the suspension is subjected to the influence of a very strong electric or magnetic field then its behavior appears to be different. The study of such a behavior leads to the development of the socalled asymmetric hydrodynamics in which case the stress tensor appears to be non-symmetric (see, for example, the pioneer works [1] and [15] where this fact was established from physical considerations, and our previous work [4] where the same linear problem was considered):
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σ[v] = AD e[v] + AR ω[v] − Ep.
(1)
Here AD and AR are the deformative and rotational parts of the effective viscosity tensor, and ω[v] = ∂v ∂vi {ωij [v] = 12 ( ∂x − ∂xji )}3i,j=1 . j In this paper we suggest a non-linear mathematical model of a suspension which is a mixture of a viscous incompressible fluid with a large number of small perfectly rigid inclusions which are the prolate particles oriented along the fixed direction l. Under the influence of the surrounding fluid the particles can move translationally or rotate around symmetry axis but the direction of their symmetry axes does not change. Such a motion of the composite can be realized, for example, if the particles are strongly magnetizable and subjected to the influence of the strong magnetic field, so that they are oriented along the field direction B (see Fig. 1). We study the asymptotic behavior of such a mixture when the diameters of inclusions tend to zero and the inclusions are distributed in the whole volume. As a result, we obtain the homogenized equations corresponding to asymmetric hydrodynamics.
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M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
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2. Statement of the problem
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Fig. 1. The suspension with oriented particles.
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Consider a bounded domain Ω in R3 with smooth boundary ∂Ω. Suppose that this domain is filled with a mixture consisting of a viscous incompressible fluid with a large number Nε = O(ε−3 ) of small solids Qiε (t) bounded by smooth surfaces ∂Qiε (t) and suspended in the fluid. Further we will call them “the particles”. i Let Ωε (t) = Ω \ N i=1 Qε (t) be a domain filled with the fluid, ρf and ρs be the specific mass density of the fluid and of solid particles respectively, μ be the dynamic viscosity of the fluid, v ε = v ε (x, t) be the velocity of the fluid, pε = pε (x, t) be the pressure, f ε = f ε (x, t) be the external force acting on the ∂v
p 3 n suspension, e[v ε ] = {enp [v] = 12 ( ∂v ∂xp + ∂xn )}n,p=1 be the strain tensor in the fluid, σ[v ε ] = {σnp [v] = 2μenp [v] − pε δnp }3n,p=1 be the stress tensor in the fluid, xiε (t) be the position of the center of mass of Qiε (t), uiε (t) be the displacement of the center of mass of Qiε (t), θiε (t) be the rotation vector of Qiε (t), miε be the mass of Qiε (t), Iεi (t) be the inertia tensor of Qiε (t). Consider the following system of equations:
∂v ε + ρf (v ε , ∇)v ε − μΔv ε = ∇pε + ρf f ε , div v ε = 0, ∂t i i i v ε = u˙ iε + θ˙ ε × x − xiε , θ˙ ε = P d θ˙ ε , x ∈ ∂Qiε , ρf
x ∈ Ωε ,
(2) (3)
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miε u ¨ iε +
Sεi
σ[v ε ]ν ds =
d i ˙i P I θ + Pd dt ε ε
Qiε
d
∂Qiε
x−
ρs f ε dx, xiε
(4)
× σ[v ε ]ν ds = P
d Qiε
x − xiε × ρs f ε dx,
(5)
v ε (x, 0) = v ε0 (x),
x ∈ Ωε (0),
u˙ iε (0) = v iε ,
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uiε (0) = 0,
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where f ε = f ε (x, t) is the external force acting on the mixture, ν is the unit inner normal vector to the surface ∂Qiε (t), and P d is a projection operator onto some fixed d-dimensional subspace S d ⊂ R3 . Depending on d, such a system describes non-stationary motions of the mixture under various regimes of particles rotations. Namely, if d = 3 then the particles can rotate without any constraints. Such a situation was considered in [11] (for the case of an elastic medium filled with the particles) and in [2,12,17,26] (for the case of a viscous incompressible fluid filled with the particles). If d = 0 then the particles move translationally without any rotations. In this paper, we focus on the non-standard cases where d = 1 or d = 2 (similar linear problems for the case of elastic and fluid media were considered in our previous works [9] and [4]; see also [10]). The case d = 1 can be realized, for example, if we consider strongly magnetizable prolate ellipsoidal particles in the strong magnetic field directed along a constant vector B. Then all the particles are aligned along B [20], and under the influence of elastic forces they can move translationally or rotate only around their symmetry axis l = B, but the direction of their symmetry axis does not change (see Fig. 1). In this case, the subspace S 1 is a linear subspace spanned by the vector l. The case d = 2 can be realized, for example, if we consider strongly magnetizable oblate ellipsoidal particles in the strong magnetic field. Moreover, it is assumed that the particles are aligned in such a way that their symmetry axes are identically oriented along the direction l perpendicular to the field direction B and they can rotate both around their symmetry axis and around the field direction. In this case, subspace S 2 is a linear subspace spanned by vectors l and B. The result both in case d = 1 and in case d = 2 is qualitatively the same: the stress tensor in the homogenized model is expressed via the strain tensor and the rotation tensor in accordance with (1). The system of Eqs (2)–(5) is supplemented by the initial conditions
θεi (0) = 0,
(6) θ˙εi (0) = ωεi
(7)
(div v ε0 = 0 at x ∈ Ωε (0) and v ε0 (x) = v iε +ω iε ×(x−xiε (0)) at x ∈ ∂Qiε (0)) and the boundary condition on ∂Ω v ε (x, t) = 0,
x ∈ ∂Ω.
(8)
Theorem 1. There exists a unique solution of the problem (2)–(8) for t ∈ [0, T ] (local in time (0 < T < ∞) or even global (T = ∞) if the data are small enough and the particles do not collide with each other and with the boundary ∂Ω. The proof of the theorem is given in [28] (see also references therein). The main goal of the paper is to study the asymptotic behavior of the solution of problem (2)–(8) as ε → 0. At first, we get uniform (with respect to ε) bounds for the derivatives of that solution extended onto the particles Qiε (t) by equality (3).
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3. A priori estimations of the solution of the problem (2)–(8) Starting from the solution {v ε (x, t), uiε (t), θiε (t) = P d θiε (t), i = 1, Nε } of the problem (2)–(8) we construct the vector function v˜ε (x, t) = χε (x, t)v ε (x, t) +
Nε
i χiε (x, t) u˙ iε (t) + θ˙ ε (t) × x − xiε (t) ,
(9)
i=1
where χε (x, t) is the characteristic function of the domain Ωε (t), filled with the fluid, and χiε (x, t) is the characteristic function of a particle Qiε (t). We also denote by Nε
ρε (x, t) = ρf χε (x, t) + ρs
χiε (x, t)
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T ∂v ε dx dt + ρf v ε , ρf v ε , (v ε · ∇)v ε dx dt ∂t Ωε (t) 0 Ωε (t) T T 3 2 ekl [v ε ] dx dt − + 2μ σ[v ε ]ν, v ε dS dt
Ωε (t) k,l=1
0
T = ρf
(v ε , f ε ) dx dt.
∂Ωε (t)
(10)
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Ωε (t)
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the density of the suspension “the fluid-the particles”. (1) At the first step we estimate ∇˜ v ε (x, t)L2 (ΩT ) , where ΩT = Ω × [0, T ]. To do so, we multiply Eq. (2) by v ε (x, t) and integrate over the domain ΩεT = [0, T ] × Ωε (t). Using Green’s formula, we get
With the aid of the boundary conditions (3), Eqs (4)–(5) and Reynolds transport theorem (see, for example, [3]), the surface integral in (10) can be transformed as follows:
∂Ωε (t)
0
T
= 0
σ[v ε ]ν, v ε dS dt
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−
T
Nε i ρs d u˙ ε (t) + θ˙ iε (t) × x − xiε (t) 2 dx dt 2 dt Qiε (t) i=1
Nε T
− ρs 0
i=1
i u˙ iε (t) + θ˙ ε (t) × x − xiε (t) , f ε (x, t) dx dt.
Qiε (t)
(11)
Consider now the second term in the LHS of (10). Taking into account the boundary condition (8) and the divergence-free condition for the velocity v ε (x, t), with integrating by parts we get T 0
Ω (t)
ρf v ε , (v ε · ∇)v ε dx dt = 0,
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whence it follows that T ρf v ε , (v ε · ∇)v ε dx dt = −ρf Ωε (t)
0
Nε T 0
Qiε (t)
i=1
v ε , (v ε · ∇)v ε dx dt.
With the help of equality (3) one can easily check that v ε , (v ε · ∇)v ε dx = 0, i = 1, Nε .
(12)
(13)
Qiε (t)
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Combining now (12) and (13), we conclude that the second term in the LHS of (10) is equal to 0. Consider now the first term in the LHS of (10). It is easy to see that the following identity holds:
T ∂v ε dx dt ρf v ε , ∂t 0 Ωε (t) T ρf d |v |2 dx dt = 2 dt Ωε (t) ε 0 T T Nε Nε ρf d ρf d|v ε |2 2 dx dt. |v ε | dx dt − + 2 dt 2 0 Qiε (t) 0 Qiε (t) dt i=1
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Using Reynolds transport theorem and recalling that v ε = u˙ iε + θ˙ ε ×(x−xiε ) for x ∈ Qiε (t) (i = 1, Nε ), one can be convinced that the second and the third terms in the RHS of that equality coincide with each other. Hence,
T T ρf d ∂v ε dx dt = ρf v ε , |v |2 dx dt. (14) ∂t 2 dt Ωε (t) ε 0 Ωε (t) 0 Combining now equalities (10)–(14), we obtain T 0
ρf d 2 dt
Ωε (t)
− ρs
Nε T
0
i=1
T + 2μ 0
T
|v ε | dx dt + 2
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0
Nε i ρs d u˙ ε (t) + θ˙ iε (t) × x − xiε (t) 2 dx dt 2 dt Qiε (t) i=1
i u˙ iε (t) + θ˙ ε (t) × x − xiε (t) , f ε (x, t) dx dt
Qiε (t) 3
Ωε (t) k,l=1
T e2kl [v ε ] dx dt = ρf 0
Ωε (t)
(v ε , f ε ) dx dt.
It is easy to see that we can rewrite this equality as follows:
T 3 2 1 T d ρε (x, t) v˜ε (x, t) dx dt + 2μ e2kl [˜ v ε ] dx dt 2 0 dt Ω 0 Ω k,l=1 T ρε (x, t)(˜ v ε , f ε ) dx dt. = 0
Ω
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Due to the first Korn’s inequality (see [25]) ˜ v ε 2◦ 1
H (Ω )
3
2
Ω k,l=1
e2kl [˜ v ε ] dx,
(15)
the last identity gives us the required bound ∇˜ v ε 2L2 (ΩT ) C
(16)
provided the external force f ε is bounded. (2) At the second step we prove the following estimation:
k,l=1
(17)
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3 ∂v ε 2 + 2μ ekl v ε (x, T ) dx C. ρf ∂t Ωε (T ) L2 (ΩεT )
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To prove this bound we assume that the following conditions hold.
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(3.1) Let diε be the diameter of the ellipsoidal particle Qiε (t), B(Qiε (t)) be a minimal ball containing Qiε (t), and Rεi (t) be a distance from the ball B(Qiε (t)) to other minimal balls and to the boundary ∂Ω. We suppose that both diε and Rεi (t) (for any fixed t ∈ [0, T ]) satisfy the following inequalities: (18)
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C1 ε diε , Rεi (t) C2 ε,
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where constants C1 and C2 do not depend on ε (0 < C1 < C2 < ∞). (3.2) The velocities u˙ iε (t) of the centers of mass of Qiε (t) for any fixed t ∈ [0, T ] change in a smooth way when passing from one particle to another, i.e. there exist smooth vector functions V ε (x, t) ∂V such that u˙ iε (t) = V ε (xiε , t), V ε (x, t) = 0 for (x, t) ∈ ∂Ω × [0, T ], | ∂tε | < C and |∇x V ε | < C. i (3.3) The instant angular velocities and accelerations of the particles Qε (t) are bounded, so that the ε ¨i 2 i 3 i following estimates hold: |θ˙ ε (t)| < C, N i=1 |θ ε | (dε ) < C. (3.4) max(x,t)∈ΩεT |v ε (x, t)| < C. (3.5) The external force f ε (x, t) and the initial velocity v ε0 (x) are bounded: f ε L2 (ΩT ) C, v ε0 H 1 (Ω ) C. Here, all constants C do not depend on ε. Let ϕiε (x, t) ∈ C ∞ (Ω × [0, T ]) be the functions satisfying the following conditions for any t ∈ [0, T ]: = 1 for x ∈ B(Qiε (t)), ϕiε = 0 for x ∈ / B(1+α)diε (t), 0 ϕiε 1 for x ∈ B(1+α)diε (t) and |∇x ϕiε (x, t)| dCi , where B(1+α)diε (t) denotes a ball of diameter (1 + α)diε concentric with the minimal ε ball B(Qiε (t)), and constants α > 0, C > 0 do not depend on ε. Introduce the following vector field:
ϕiε
lε (x, t) = V ε (x, t) +
Nε i=1
i u˙ iε (t) + θ˙ ε (t) × x − xiε (t) − V ε (x, t) ϕiε (x, t),
(x, t) ∈ ΩT .
(19)
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From the properties of V iε (x, t) and ϕiε (x, t) it follows that i lε (x, t) = u˙ iε (t) + θ˙ ε (t) × x − xiε (t) for x ∈ Qiε (t) (i = 1, Nε ); ∂lε (x, t) ∇x lε (x, t) < C; lε (x, t) = 0 for (x, t) ∈ ∂Ω × [0, T ], ∂t < C;
(20)
where C > 0 does not depend on ε. Consider in ΩT the following vector function: ∂˜ 1/2 ∂˜ ∂˜ v v vε lεi ε = 1 + |lε |2 , q ε (x, t) = ε + ∂t ∂xi ∂ ˆlε 3
(21)
i=1
∂ v˜ε ∂ˆ lε
denotes the directional derivative of u ˜ε along the vector field
1 l ε1 l ε2 l ε3 , , , . (1 + |lε |2 )1/2 (1 + |lε |2 )1/2 (1 + |lε |2 )1/2 (1 + |lε |2 )1/2
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ˆl = ε
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where
∂ v˜ (x,t)
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Since v˜ε (x, t) is continuous everywhere in ΩT , ∇x v˜ε (x, t) and ε∂t are continuous both in ΩεT and in QTε = ΩT \ ΩεT , v˜ε (x, t) = 0 for (x, t) ∈ ∂Ω × [0, T ], and the vector field ˆlε is tangent to the lateral surface of ∂ΩεT ([22], pp. 225–226), then q ε (x, t) is continuous everywhere in ΩT and q ε (x, t) = 0 for ◦
3 vεj ∂lεi ∂˜ . ∂xj ∂xi
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div q ε =
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(x, t) ∈ ∂Ω × [0, T ]. Therefore, from (19) and (21), it follows that for any t ∈ [0, T ] q ε (x, t) ∈H 1 (Ω). Moreover, since div v˜ε = 0, we get (22)
For further considerations we need a technical lemma which can be proved analogously to [12]. ◦
◦
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Lemma 1. For any vector function q ε (x) ∈H 1 (Ω) there exists a vector function z ε (x) ∈H 1 (Ω) such that div z ε (x) = div q ε (x) for x ∈ Ω, z ε (x) = q ε (x) + aiε for x ∈ Qiε (i = 1, Nε ) and
z ε H 1 (Ω ) C div q ε L2 (Ω ) + 2
2
Nε
∇q ε L2 (Qiε ) , 2
(23)
i=1
where aiε are constant vectors and C does not depend on ε. Applying this lemma to the function q ε (x) defined by equality (21) for every t ∈ [0, T ], we construct a vector function z ε (x, t). From estimate (23), using equalities (21) and (22) and taking into account the properties of the vector field lε (x, t), we get Nε 2 2 i 2 i 4 i 3 z ε (t) 1 C ∇˜ ¨θε (t) + θ˙ ε (t) dε v ε (t) + . H (Ω )
L2 (Ω )
i=1
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Hence, due to the estimate (16) and conditions (3.1) and (3.3), the following inequality follows
z ε (t)2
T
H 1 (Ω )
0
dt C,
(24)
where C does not depend on ε.
◦
Ωε (t) k,l=1
Nε T
− 0
i=1
∂Qiε (t)
σ[v ε ] dS, aiε (t)
T dt = ρf 0
Ωε (t)
(wε , f ε ) dx dt.
(25)
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Set now wε = q ε − z ε . It is clear that wε (x, t) ∈ L2 (0, T ; H1 (Ω)), div wε = 0 and wε (x, t) = aiε (t) for x ∈ Qiε (t) (i = 1, Nε ). Multiply Eq. (2) by wε and integrate over domain ΩεT . Using Green’s formula, we get
T T ∂v ε dx dt + ρf w ε , ρf wε , (v ε · ∇)v ε dx dt ∂t 0 Ωε (t) 0 Ωε (t) T 3 ekl [wε ]ekl [v ε ] dx dt + 2μ
Ωε (t)
T = ρf
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C
Estimate now each of the terms in (25). Taking into account the form of the vector function q ε (x, t) and using inequality (20), Cauchy–Schwarz and Young’s inequalities (with arbitrary δ > 0), estimate from above the RHS of equality (25): T ρf (wε , f ε ) dx dt
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(q ε − z ε , f ε ) dx dt 0 Ωε (t)
∂v ε + C∇v ε L2 (ΩεT ) + z ε L2 (ΩεT ) ρf f ε L2 (ΩεT ) ∂t L2 (ΩεT )
∂v ε 2 ρf 2 2 2 2 f + 3ρf δ + C ∇v ε L2 (Ω T ) + z ε L2 (Ω T ) . ε ε 4δ ε L2 (ΩT ) ∂t L2 (ΩεT )
(26)
Consider now the first term on the LHS of equality (25). With the help of the same arguments as before, we obtain the following lower bound:
T T ∂v ε ∂v ε dx dt = dx dt ρf w ε , ρf q ε − z ε , ∂t ∂t 0 Ωε (t) 0 Ωε (t)
T 3 T ∂v ε 2 ∂v ε ∂v ε = dx dt ρf dx dt + ρf lεi , ∂t ∂xi ∂t 0 Ωε (t) i=1 0 Ωε (t)
T ∂v ε − dx dt ρf zε , ∂t 0 Ωε (t)
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∂v ε 2 ∂v ε 2 1 2 ∇v ε L2 (Ω T ) + δ ρf − 3Cρf ε ∂t L2 (ΩεT ) 4δ ∂t L2 (ΩεT ) ∂v ε 2 ρf 2 − z ε L2 (Ω T ) − ρf δ . (27) ∂t ε 4δ L2 (ΩεT ) Next, analogously to (12)–(13), it can be shown that T Ωε (t)
0
ρf wε , (v ε · ∇)v ε dx dt = 0.
(28)
Estimate now the fourth term in the LHS of (25):
∂Qiε (t)
i=1
Nε T
¨ iε (t), aiε (t) miε u
0
Nε T
dt − ρs 0
i=1
Qiε (t)
i=1
f ε (x, t) dx, aiε (t)
dt
C
=
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0
σ[v ε ] dS, aiε (t) dt
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Nε T
−
= Jε1 − Jε2 .
Nε T
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1/2
T
1/2
2 miε aiε dt
0
2 miε u ¨ iε dt
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0
2 miε u ¨ iε dt
0
i=1
1 2
T
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Nε 1 Jε
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Using Cauchy–Schwarz inequality and identity aiε (t) = q ε (x, t) − z ε (x, t) for x ∈ Qiε (t) (i = 1, Nε ), we get:
i=1
ρs + 2
Nε T
0
i=1
Qiε (t)
q (x, t) 2 + z ε (x, t) 2 dx dt, ε
whence, taking into account the form of the vector function q ε (x, t) on the particles Qiε (t) and conditions (3.2)–(3.3), the following upper bound follows: N Nε Nε ε 1 ρs i i 2 5 2 3 2 3 ¨θε diε + u θ˙ ε (t) diε Jε z ε 2 T + C max ¨ iε diε + L2 (Qε ) 0tT 2 i=1
ρs z ε 2L2 (QT ) + C. ε 2 The term Jε2 can be estimated analogously: 2 ρs Jε z ε 2 T + ρs f 2 T + C. L2 (Qε ) ε L2 (Qε ) 2
i=1
i=1
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Thus, N
ε T σ[v ε ] dS, aiε (t) dt ρs z ε 2L2 (QT ) + ρs f ε 2L2 (QT ) + 2C. ε ε 0 ∂Qiε (t)
(29)
i=1
It remains to consider only the third term in the LHS of (25). Taking into account that wε = q ε − z ε , we have: 3
Ωε (t) k,l=1
0
T
ekl [wε ]ekl [v ε ] dx dt 3
= 2μ
Ωε (t) k,l=1
0
ekl [q ε ]ekl [v ε ] dx dt − 2μ
T 0
3
Ωε (t) k,l=1
ekl [z ε ]ekl [v ε ] dx dt.
(30)
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T 2μ
The second term in the RHS of (30) we can estimate as follows:
C
O
3 T ekl [z ε ]ekl [v ε ] dx dt ∇z ε 2L2 (Ω T ) + ∇v ε 2L2 (Ω T ) . 2μ ε ε 0 Ωε (t) k,l=1
(31)
3
2μ
Ωε (t) k,l=1
0
=μ 0
−μ
ekl [q ε ]ekl [v ε ] dx dt
3 3 ∂ ˆ 1/2 e2kl [v ε ] dx dt lεi 1 + |lε |2 ∂x i Ωε (t)
AU
T
TH
T
O
R
Recalling the definition of q ε and ˆlε , the first term in the RHS of (30) can be written as follows ∂ ∂ ≡ ∂t ): ( ∂x 0
i=0
T 0
3 3 ∂lεi
Ωε (t) i=1
T +μ 0
k,l=1
∂xi
e2kl [v ε ] dx dt
k,l=1
3 ∂lεi ∂vεl ∂lεi ∂vεk + ekl [v ε ] dx dt ∂xl ∂xi ∂xk ∂xi
Ωε (t) i,k,l=1
= Jε1 + Jε2 + Jε3 . Due to (20), we have: 2 3 Jε + Jε C∇v ε 2
L2 (ΩεT ) .
342
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
Applying the divergence theorem to the integral Jε1 and taking into account that the vector field 2 T ˆl (x, t)(1 + |l |2 )1/2 3 ε ε k,l=1 ekl [v ε ] is tangent to the lateral surface of the domain Ωε , we get: 1
Jε = μ
3
Ωε (T ) k,l=1
ekl v ε (x, T ) dx − μ 2
3
Ωε (0) k,l=1
e2kl v ε (x, 0) dx.
Thus, 3
0
Ωε (t) k,l=1
μ
ekl [wε ]ekl [v ε ] dx dt
3
Ωε (T ) k,l=1
ekl v ε (x, T ) dx − μ 2
3
Ωε (0) k,l=1
e2kl v ε0 (x) dx − C∇v ε 2L2 (Ω T ) . ε
(32)
PY
T 2μ
Combining now (25)–(32), we obtain:
C
k,l=1
O
3 ∂v ε 2 2 ρf 1 − (3C + 4)δ + μ e (x, T ) dx v ε kl ∂t L2 (ΩεT ) Ωε (T )
O
R
ρf fε 2L2 (ΩT ) + ρs fε 2L2 (QT ) ε 4δ
C 1 2 2 + C δ + 2C + 1 ∇v ε L2 (Ω T ) + ρf + 3δ z ε 2L2 (Ω T ) + ∇z ε 2L2 (Ω T ) + 3ρf ε ε ε 4δ 4δ 3 + ρs z ε 2L2 (QT ) + 2C + e2kl v ε0 (x) dx. Ωε (0) k,l=1
TH
ε
AU
1 Taking here δ = 2(3C+ 4) , using the previously obtained bounds (16) and (24), and taking into consideration condition (3.5), we get the required bound (17):
3 ∂v ε 2 + 2μ ekl v ε (x, T ) dx C, ρf ∂t L2 (ΩεT ) Ωε (T ) k,l=1
where constant C does not depend on ε. Before formulating the main result we introduce some definitions and assumptions. 4. Additional assumptions and the main result Consider now the following auxiliary linear stationary problem in the domain Ωε (t) (t is a parameter): −μΔv ε = ∇pε + Φε , div v ε = 0, i i i v ε = aε + bε × x − xε ,
x ∈ Ωε , biε
=
P d biε ,
(33) x∈
∂Qiε ,
(34)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
343
∂Qiε
σ[v ε ]ν ds =
P
d ∂Qiε
x−
xiε
Qiε
Φε dx,
(35)
× σ[v ε ]ν ds = P
d Qiε
x − xiε × Φε dx,
(36)
x ∈ ∂Ω.
v ε (x) = 0,
(37)
It will be shown in the next section that for any fixed t ∈ [0, T ] the solution v ε (x, t) of the problem (2)–(8) can be represented in the form v ε (x, t) = Rεt Φtε [x], where Rεt is the resolving operator of the problem (33)–(37) in the domain Ωε (t) (t is a parameter), and Φtε [x] = ρf χε (x, t) + ρs 1 − χε (x, t) f ε (x, t)
Nε ∂v ε ∂V iε i + (v ε · ∇)v ε χε (x, t) − ρs + V ε · ∇ V iε χiε (x, t), ∂t ∂t
PY
− ρf
i=1
min ◦
v ε ∈J ε (Ω )
Fε v ε ,
C
Fε (v ε ) =
O
i where V iε (x, t) = u˙ iε (t) + θ˙ ε (t) × (x − xiε (t)). Note that the problem (33)–(37) is equivalent to the variational problem
◦
(38) ◦
Fε (v ε ) =
2μ Ω
3
enp [v ε ] − 2 Φε , v ε dx. 2
TH
O
R
where J ε (Ω) is the class of divergence free vector-functions from H1 (Ω) which are equal to aiε + biε × (x − xiε ) on the particles Qiε (aiε and biε = P d biε are arbitrary vectors) and (39)
n,p=1
AU
Let Khy be a cube with the side length h (ε h 1) centered at y ∈ Ω. We assume that the edges of
this cube are parallel to the coordinate axes. Let Jεθ [Khy ] be the following class of vector-functions: Jεθ Khy = wε ∈ H 1 Khy ; div wε = 0; θ × x − xiε , x ∈ Qiε ∩ Khy , wε (x) = wiε + P d θiε + 1 − P d where wiε and θiε are arbitrary vectors, and θ is a given vector. Consider a minimization problem in this class for the following functional (mesocharacteristic): Aγεh (wε , y, T ) = EKhy [wε , wε ] εhγ + PK wε (x) − y h
3
n,p=1
Tnp ϕnp (x − y), wε (x) −
3
q ,r=1
Tqr ϕqr (x − y) ,
(40)
344
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
where EG [uε , v ε ] = 2μ
3
G n,p=1
PGεhγ uε (x), v ε (x)
=h
enp [uε ]enp [v ε ] dx,
−2−γ
G
(41)
uε (x), v ε (x) dx,
(42)
3 δqr 1 q r ϕ (x) = xr e + xq e − xn en , 2 3 n=1
qr
(43)
min
wε ∈Jεθ [Khy ]
AU
TH
O
R
C
O
PY
∂vl n 3 k ekl [v] = 12 ( ∂v ∂xl + ∂xk ), T = {Tqr } is an arbitrary symmetric second rank tensor, {e }n=1 is an orthonormal basis in R3 , and 0 < γ < 2 is a penalty parameter. This mesocharacteristic plays the crucial role in our consideration. Roughly speaking, it allows us to compute the energy of the suspension in some mesoscopic cube of size h (ε h 1), which is a so-called representative volume element. In other words, if a suspension can be described within the effective single medium approach, then the rheological properties of the suspension can be determined by calculation or measurements in some representative volume element of an intermediate mesoscale h, which is why we choose the cube Khy . Next, observe that the first term (41) in (40) represents the energy of the suspension. The minimizer wε of (40) is “close”, up to an additive constant, to the true global minimizer v ε of the variational problem (38) corresponding to (33)–(37), if the tensor T is chosen appropriately. Now one should choose T . If the single medium homogenized description is possible, then v ε (x) is “close” to some smooth (homogenized) vector-function v(x), which depends only on the macroscopic variable x and does not depend on ε, so that it does not vary on the microscale ε. We then minimize the energy of the suspension, adding the constraint that the minimizer wε is “close” to the linear part (differential) of the global minimizer v, so that |wε − v| = o(h) ∼ h1+γ/2 for some γ > 0. This condition is imposed by introducing the penalty term (42). It can be proved (see [4]) that there exists the unique vector-function which minimizes the functional (40); the minimal value of this functional is given by
Aγεh (wε , y, T ) =
3
γ d a0, npqr y, S , ε, h Tnp Tqr
n,p,q ,r=1
+2
3 3
n,p=1 q=1
3 bγnpq y, S d , ε, h Tnp θq + cγqr y, S d , ε, h θq θr ,
γ γ γ d d d where a0, npqr (y, S , ε, h), bnpq (y, S , ε, h) and cqr (y, S , ε, h) are the components of the fourth-, and second-rank tensors respectively, defined as follows np qr εhγ np γ d + PK (x) − ϕnp (x − y), wqr (x) − ϕqr (x − y) , a0, y w npqr y, S , ε, h = EKhy w , w h εhγ bγnpq y, S d , ε, h = EKhy wnp , v q + PK y wnp (x) − ϕnp (x − y), v q (x) , h q r εhγ q γ d cqr y, S , ε, h = EKhy v , v + PK y v (x), v r (x) . h
(44)
q ,r=1
third(45) (46) (47)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
345
Here wnp (x) is the vector-function that minimizes the functional (40) in Jε [Khy ] as T = T np = eq y 1 n p p n q 2 (e ⊗ e + e ⊗ e ), v (x) is the vector-function minimizing the functional (40) in Jε [Kh ] as T = 0, n 3 and e (n = 1, 2, 3) form an orthonormal basis in R . Starting from the solution {v ε (x), aiε , biε (t) = P d biε (t), i = 1, Nε } of the problem (33)–(37) we construct the vector function 0
v˜ε (x) = χε (x)v ε (x) +
Nε
χiε (x) aiε + biε × x − xiε ,
(48)
i=1
where χε (x) is the characteristic function of the domain Ωε , filled with the fluid, and χiε (x) is the characteristic function of a particle Qiε . We assume that the following conditions hold:
PY
(4.1) for some real number γ > 0 the following limits exist heterogeneously at x ∈ Ω: (a)
γ γ d d a0, a0, npqr (x, S , ε, h) npqr (x, S , ε, h) = lim lim = a0npqr x, S d , 3 3 h→0 ε→0 h→0 ε→0 h h
(b)
bγnpq (x, S d , ε, h) bγnpq (x, S d , ε, h) = lim lim = bnpq x, S d , 3 3 ε→ 0 h→0 h→0 ε→0 h h
(c)
cγqr (x, S d , ε, h) cγqr (x, S d , ε, h) = lim lim = cqr x, S d , 3 3 h→0 ε→0 h→0 ε→0 h h
O
lim lim
C
lim lim
R
lim lim
O
where {a0npqr (x, S d )}, {bnpq (x, S d )}, {cqr (x, S d )} are continuous tensors (at x ∈ Ω). (4.2) the sequence Φε (x) converges weakly in L2 (Ω) to a vector function Φ(x), as ε → 0.
TH
Note, that the existence of the limits (4.1) is a general restriction on the spatial distributions of the particles. Since we do not require any spatial periodicity, we have to impose some conditions on these distributions.
AU
Remark. If the limits in (4.1) exist for some γ > 0, then they exist for any γ > 0 and the limiting tensors do not depend on γ; moreover, {a0npqr (x, S d )} and {cqr (x, S d )} are positive definite tensors (these facts can be proved analogously to [22]). In our previous work [4] the following theorem was proved. Theorem 2. Let conditions (4.1)–(4.2) hold. Then the sequence of vector-functions v˜ε (x), defined by (48), converges strongly in L2 (Ω) to a vector-function v(x, t), which is a solution of the following homogenized linear stationary problem: −
3
n ∂ D anpqr (x)eqr [v] + aR npqr (x)ωqr [v] e = ∇p + Φ, ∂xp n,p,q ,r=1
x ∈ Ω,
(49)
div v = 0,
x ∈ Ω,
(50)
v(x) = 0,
x ∈ ∂Ω.
(51)
346
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
Here 1 = anpqr + bqrl lnp , 2 l=1
1 ∂vq ∂vr , ωqr [v] = − 2 ∂xr ∂xq 3
aD npqr
aR npqr
0
3 3 1 1 = clm lnp mqr + bnpl lqr , 4 2
l,m=1
(52)
l=1
(53)
where {lnp } is Levi–Civita permutation tensor. The problem (49)–(51) has the unique solution.
ε→0
2
PY
Let Rεt and Rt (t is a parameter) be the resolving operators of the problems (33)–(37) and (49)–(51), respectively (v ε = Rεt F ε and v = Rt F ). Analogously to [8] and [22], one can show that Rεt and Rt are compact and self-adjoint in L2 (Ω). Moreover, with the help of Theorem 2, it can be proved that for any t ∈ [0, T ] and any f ∈ L2 (Ω) lim Rεt f − Rt f L (Ω ) = 0, Rεt f L (Ω ) C, (54) 2
Nε
χiε (x, t) C(x, t)
C
O
where constant C depends neitheron ε nor on t (for more details see [22]). ε i ∞ Assume now that the sequence N i=1 χε (x, t) = 1 − χε (x, t) converges as ε → 0 ∗-weakly in L (ΩT ) to the function 0 < C(x, t) < 1: *-weakly in L∞ (ΩT ) ,
(55)
R
i=1
TH
lim f ε − f L2 (ΩT ) = 0,
O
where Ct (x, t) ∈ L4 (0, T ; L2 (Ω)), ∇C(x, t) ∈ L2 (ΩT ), and the sequences f ε (x, t) and v˜ε0 (x) converge as ε → 0 strongly in L2 (ΩT ) to f (x, t) and strongly in L2 (Ω) to v 0 (x), respectively: ε→0
v ε0 − v 0 L2 (Ω ) = 0. lim ˜
(57)
AU
ε→0
(56)
Theorem 3. Let conditions (3.1)–(3.5) are satisfied, the limits (55)–(57) exist, the limits in (4.1) exist for every t ∈ [0, T ] and the limiting tensors {a0npqr (x, t, S d )}, {bnpq (x, t, S d )}, {cnp (x, t, S d )} are continuous in ΩT . Then the sequence of vector-functions v˜ε (x, t), defined by (9), converges strongly in L2 (ΩT ) (and in L2 (Ω) uniformly with respect to t) to a vector-function v(x, t), which is a generalized solution of the following homogenized problem: 3 n ∂ D ∂(ρv) + (v · ∇)(ρv) − anpqr (x, t)eqr [v] + aR npqr (x, t)ωqr [v] e ∂t ∂xp n,p,q ,r=1
= ∇p + F ,
x ∈ ΩT ,
(58)
div v(x, t) = 0,
x ∈ ΩT ,
(59)
v(x, t) = 0,
(x, t) ∈ ∂Ω × [0, T ],
(60)
v(x, 0) = v 0 (x),
(61)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
347
where ρ(x, t) = ρf 1 − C(x, t) + ρs C(x, t),
F (x, t) = ρ(x, t)f (x, t).
(62)
◦
◦
Remark. A function v(x, t) ∈ L(ΩT ) = H 1 (ΩT ) ∩ L2 (0, T ; J (Ω)) ∩ L∞ (ΩT ), where J (Ω) is the ◦ class of divergence free vector functions from H1 (Ω), is said to be a generalized solution to the problem (58)–(61) if it satisfies the following integral identity
−ρv, Φt + (v · ∇)Φ +
Ω
0
3
n,p,q ,r=1
ρv, Φ(x, 0) =
ρv, Φ(x, τ ) −
+ Ω
aD npqr (x, t)
+
aR npqr (x, t)
∂vn ∂Φq ∂xp ∂xr
dx dt
τ (F , Φ) dx dt
Ω
0
Ω
(63)
PY
τ
for any Φ(x, t) ∈ L(ΩT ) and τ (0 < τ T ).
O
Lemma 2. Problem (58)–(61) can have at most one generalized solution from L(ΩT ).
C
Proof. Assume that there exist two solutions v (x, t) ∈ L(ΩT ) and v (x, t) ∈ L(ΩT ). Then the function v = v − v ∈ L(ΩT ) satisfies the following identity for any Φ ∈ L(ΩT ):
R
Ω
(ρv, Φt ) + ρv, v · ∇ Φ − ρv , (v · ∇)Φ dx dt
τ
3
+
Ω n,p,q ,r=1
0
O
0
R aD npqr (x, t) + anpqr (x, t)
TH
−
τ
∂vn ∂Φq dx dt ∂xp ∂xr
ρ(x, τ )v(x, τ ), Φ(x, τ ) dx = 0.
+
AU
Ω
Choosing here Φ(x, t) = v(x, t), after obvious transformations we get that 1 2
2 ρ(x, τ ) v(x, τ ) dx +
τ
Ω
Ω n,p,q ,r=1
0
1 2
τ 0
|ρt ||v|2 dx dt + Ω
3
τ 0
R aD npqr (x, t) + anpqr (x, t)
∂vn ∂vq dx dt ∂xp ∂xr
ρ v, v · ∇ v + v , (v · ∇)v dx dt. Ω ◦
Since for any fixed t ∈ [0, τ ] v(x, t) ∈J (Ω), the following inequality holds (see [4]):
Ω
R aD npqr + anpqr
∂vn ∂vq dx v2H 1 (Ω ) , ∂xp ∂xr
(64)
348
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
R where aD npqr and anpqr are defined by (52). Using now inequality (64) and the fact that both the function ρ(x, t) and the vector functions v (x, t) and v (x, t) are bounded, we obtain:
2 ρ(x, τ ) v(x, τ ) dx +
max
0tτ
Ω
C
τ
τ
|ρt ||v| dx dt +
|∇v|2 dx dt Ω
0
τ
Ω
0
|v||∇v| dx dt ,
2
(65)
Ω
0
where constant C does not depend on τ . Using Hölder’s inequality, we can write |ρt ||v|2 dx dt
τ
Ω
0
|v|q dx
r/q 2/r τ
r /q 1/r dt |ρt |q dx dt
Ω
0
0
= vLr (0,τ ;Lq (Ω )) ρt Lr (0,τ ;Lq (Ω )) ,
(Ωτ ) = v(x, t): vΩτ = ess max
C
◦ V2
v(x, t) 2 dx +
2
0tτ
Ω
τ 0
∇v(x, t) 2 dx dt < ∞; Ω
◦
O
R
v(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ] .
(66)
O
2
where q = q−q 2 , r = r−r 2 (q 2, r 2). Introduce now the following space of vector functions:
Ω
PY
τ
TH
For any vector function v(x, t) ∈V2 (Ωτ ) the following inequality holds (see [19]): vLr (0,τ ;Lq (Ω )) βvΩτ ,
AU
(67)
where q and r are arbitrary constants such that
1 r
+
3 2q
◦
= 34 , r 2, 2 q 6 and β = 42/r . Note that
due to the Embedding theorems (see [19]) L(ΩT ) ⊂V2 (ΩT ). Choosing in (66) q = 4 and r = 83 and taking into account (67), we get τ 0
Ω
|ρt ||v|2 dx dt β 2 v2Ωτ ρt L4 (0,τ ;L2 (Ω )) .
Next, using Hölder’s inequality, we have τ 0
|v||∇v| dx dt vΩτ Ω
τ 0
1/2 |v| dx dt 2
Ω
(τ )1/r (meas Ω)1/q vΩτ vLr (0,τ ;Lq (Ω )) ,
(68)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
where r = τ 0
Ω
2r r−2
and q =
2q q−2 .
Choosing, as before, q = 4 and r =
8 3
349
and using inequality (67), we get
|v||∇v| dx dt β(τ )1/8 (meas Ω)1/4 v2Ωτ .
(69)
From (65), (68) and (69) it follows that min{ρf , ρs , 1}v2Ωτ δ(τ )v2Ωτ ,
(70)
PY
where δ(τ ) = β 2 ρt L4 (0,τ ;L2 (Ω )) + β(τ )1/8 (meas Ω)1/4 . Taking now τ so that δ(τ ) < min{ρf , ρs , 1}, from (70) we conclude that v(x, t) = 0. If τ < T , then partitioning interval [0, T ] into subintervals of length τk (for which v(x, t) = 0) and repeating the above arguments, after a final number of steps one can show that v(x, t) = v (x, t) − v (x, t) ≡ 0 in ΩT . The lemma is proved. 2
O
5. Proof of Theorem 3
In accordance with the bounds (16)–(17) and conditions (3.1)–(3.3) the sequence of vector functions ◦
Nε ∂v ε ∂V iε i + (v ε · ∇)v ε χε (x, t) + ρs + V ε · ∇ V iε χiε (x, t), ∂t ∂t
AU
TH
O
R
C
{v ε (x, t), ε > 0} is bounded in H 1 (ΩT ) and, thus, it is weakly compact in H 1 (ΩT ) (and in H 1 (Ω) for any t ∈ [0, T ]). Due to the Embedding theorem, this sequence is compact in L2 (ΩT ) (and in L2 (Ω) for any t ∈ [0, T ]). Hence, there exists a subsequence {v εk (x, t), εk > 0} which converges weakly in H 1 (ΩT ) (and in H 1 (Ω) for any t ∈ [0, T ]) and strongly in L2 (ΩT ) (and in L2 (Ω) for any t ∈ [0, T ]) to some vector function v(x, t). Using conditions (3.3)–(3.4), we conclude that v(x, t) ∈ L(ΩT ). As it is shown below, the limiting vector function v(x, t) is a solution of the problem (58)–(61). Since this problem, due to Lemma 2, has a unique generalized solution from L(ΩT ), the entire sequence {v ε (x, t), ε > 0} also converges to v(x, t). Introduce now the following vector function
W ε (x, t) = ρf
(71)
i=1
i
where V iε (x, t) = u˙ iε (t) + θ˙ ε (t) × (x − xiε (t)). Lemma 3. The sequence of vector functions {W ε (x, t), ε > 0} is weakly compact in L2 (ΩT ). So, there exists a subsequence {W εk (x, t), εk > 0} which converges as εk → 0 weakly in L2 (ΩT ) to the limiting vector function W (x, t) which is given by W (x, t) =
∂(ρv) + (v · ∇)(ρv), ∂t
(72)
where ρ(x, t) is determined by Eq. (62) and v(x, t) is a limit of the subsequence {v εk (x, t), εk → 0}.
350
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
Proof. From the bounds (16)–(17) and conditions (3.3)–(3.4) it follows that the sequence {W ε (x, t), ε > 0} is bounded in L2 (ΩT ) and, hence, it is weakly compact. Let ϕ(x, t) ∈ C01 (ΩT ) be an arbitrary smooth vector function with a compact support. Then
0
∂v εk + (v εk · ∇)v εk , ϕ dx dt (W εk , ϕ) dx dt = ρf ∂t Ω 0 Ωε (t)
Nε T i ∂V iεk i + V εk · ∇ V εk , ϕ dx dt + ρs ∂t i i=1 0 Qε (t)
T ∂v εk + (v εk · ∇)v εk , ϕ dx dt = ρf ∂t 0 Ω
Nε T i ∂V iεk i + (ρs − ρf ) + V εk · ∇ V εk , ϕ dx dt ∂t 0 Qiε (t) T
PY
T
i=1
1
2
(73)
O
= J εk + J εk .
T 1
lim Jεk = ρf
0
Ω
∂v + (v · ∇)v, ϕ dx dt. ∂t
(74)
R
εk →0
C
Since v εk (x, t) converges weakly in H 1 (ΩT ) and strongly in L2 (ΩT ) to v(x, t),
Nε T i=1
0
j=0
Nε T
AU
− (ρs − ρf )
3 ∂ i V εk , ϕ Vˆεik ,j dx dt ∂xj Qiε (t)
TH
Jεk = (ρs − ρf ) 2
O
The second integral in (73) we can write in the form
i=1
0
3
Qiε (t) j=0
∂ϕ i i ˆ Vεk ,j V εk , dx dt ∂xj
= Jε2,1 + Jε2,2 , k k
i where x0 ≡ t, Vˆεik ,0 ≡ 1, Vˆεik ,j = Vεik ,j (x, t) (j = 1, 3). Since the vector Vˆ εk = {Vˆεik ,j }3j=0 is tangent to the lateral surface of [0, T ] × Qiεk (t) (i = 1, Nεk ) and the vector function ϕ(x, t) has a compact support in ΩT , with the help of the divergence theorem we conclude that Jε2,1 = 0. k i Using (55) and taking into account that V εk (x, t) = v εk (x, t) at (x, t) ∈ Qiεk (t) × [0, T ] and v εk (x, t) converges strongly in L2 (ΩT ) to v(x, t), we obtain
lim Jε2,2 = −(ρs − ρf ) k
εk →0
T C(x, t) 0
Ω
∂ϕ v, ∂t
+
3
j=1
vj
∂ϕ v, ∂xj
dx dt.
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
351
Thus, lim Jεk = −(ρs − ρf )
T
2
εk →0
= (ρs − ρf )
C(x, t) Ω
0
T 0
∂ϕ v, ∂t
Ω
+
3
vj
j=1
∂ϕ v, ∂xj
dx dt
∂(Cv) + (v · ∇)(Cv), ϕ dx dt. ∂t
(75)
Here, the integration by parts is justified, because v(x, t) ∈ H 1 (ΩT ) ∩ L∞ (ΩT ), ρ(x, t) ∈ H 1 (ΩT ) ∩ L∞ (ΩT ) and ϕ(x, t) ∈ C01 (ΩT ). Combining now (73)–(75) and taking into account (62), we finally get T εk →0 0
Ω
T = 0
Ω
Ω
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0
W εk (x, t), ϕ(x, t) dx dt
W (x, t), ϕ(x, t) dx dt = lim
∂(ρv) + (v · ∇)(ρv), ϕ dx dt, ∂t
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2
whence the statement of the lemma follows. Denote by Fε the RHS of problem (2)–(5):
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T
Fε (x, t) = ρf χε (x, t) + ρs 1 − χε (x, t) f ε (x, t).
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(76)
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Recalling now (71) and taking into account that i ∂V iε i i i + V ε · ∇ V iε = u ¨ iε (t) + ¨θε (t) × x − xiε (t) + θ˙ ε (t) × θ˙ ε (t) × x − xiε (t) , ∂t
(77)
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we can represent the solution v ε (x, t) of the problem (2)–(8) in the form v ε (x, t) = Rεt Φtε [x], where Rεt is the resolving operator of the problem (33)–(37) in the domain Ωε (t) (t is a parameter), and Φtε [x] = F ε (x, t) − W ε (x, t). Since for every t the operator Rεt is self-adjoint in L2 (Ω), for any continuous vector function ϕ(x, t) ≡ t ϕ [x] we have: T
T 0
Ω
(v ε , ϕ) dx dt = 0
T
Ω
= 0
Rεt Φtε , ϕt
Ω
T
dx dt =
Φtε , Rt ϕt dx dt +
0
T 0
Ω
Φtε , Rεt ϕt dx dt
Ω
Φtε , Rεt ϕt − Rt ϕt dx dt.
(78)
Using Lemma 3 and conditions (55)–(56), we conclude that the subsequence {Φtεk [x] ≡ Φεk (x, t), εk > 0} converges weakly in L2 (ΩT ) to the vector function Φ(x, t) = F (x, t) − W (x, t) ≡ Φt [x], where
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the vector functions F and W are defined in (62) and (72), respectively. Thus, T
lim
εk →0 0
Φtεk , Rt ϕt
Ω
T εk →0 0
T
Φεk (x, t), Rt ϕ(x, t) dx dt
dx dt = lim
Ω
=
T
t
Φt , Rt ϕt dx dt
Φ(x, t), R ϕ(x, t) dx dt =
0
T
Ω
Rt Φt , ϕt dx dt =
= 0
Ω
0
T 0
Ω
Rt Φt [x], ϕ(x, t) dx dt.
(79)
Ω
Here we took into consideration that for almost all t ∈ [0, T ] Φ(x, t) ∈ L2 (Ω) and the operator Rt is self-adjoint in L2 (Ω). Next,
0
Ω
T
Rεt ϕt − Rt ϕt 2
L2 (Ω )
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T t t t t t Φε , Rε ϕ − R ϕ dx dt Φε L2 (ΩT )
0
1/2 dt
.
ε→0
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Since the sequence {Φε , ε > 0} is bounded in L2 (ΩT ) uniformly with respect to ε, with the help of (54) we get: T t t t t t lim Φε , Rε ϕ − R ϕ dx dt = 0. (80) Ω
0
0
Ω
0
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εk →0
R
Combining now (78)–(80), we obtain: T T t t (v ε , ϕ) dx dt = R Φ [x], ϕt dx dt. lim Ω
εk →0 0
Ω
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On the other hand, since the subsequence {v εk (x, t) ≡ v tεk [x], εk > 0} converges as εk → 0 strongly in L2 (ΩT ) (and in L2 (Ω) uniformly with respect to t ∈ [0, T ]) to the vector function v(x, t) ≡ v t [x], we have: T T t t lim v ε (x, t), ϕ(x, t) dx dt = v , ϕ dx dt. 0
Ω
So, for all t ∈ [0, T ] v(x, t) ≡ v t [x] = Rt Φt [x] = Rt [F (x, t) − W (x, t)]. Moreover, with the aid of condition (57), we conclude that v(x, 0) = v 0 . Recalling the definition of the resolving operator Rt and the form of the vector function W (x, t) (see (72)), we conclude that v(x, t) is the generalized solution of the problem (58)–(61). Theorem 3 is proved. References [1] E.L. Aero, A.N. Bulygin and E.V. Kuvshinsky, Asymmetric hydrodynamics, J. Appl. Math. Mech. (PMM) 29 (1965), 297–308 (in Russian). [2] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. II. Noncritical sizes of the holes for a volume distribution and surface distribution of holes, Arch. Rational Mech. Anal. 113 (1991), 209–259, 261–298.
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[3] T. Belytschko, W.K. Liu and B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley, New York, 2000. [4] M. Berezhnyi and E. Khruslov, Asymmetric hydrodynamics of suspensions subjected to the influence of strong external magnetic fields, Journal of Multiscale Modeling 4 (2012), 24–25. [5] M.A. Berezhnyy and L.V. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn’s inequality, Journal of the Mechanics and Physics of Solids 54 (2006), 635–669. [6] M.A. Berezhnyi, The asymptotic behavior of viscous incompressible fluid small oscillations with solid interacting particles, Journal of Mathematical Physics, Analysis, Geometry 3 (2007), 135–156. [7] M. Berezhnyi, L. Berlyand and E. Khruslov, The homogenized model of small oscillations of complex fluids, Networks and Heterogeneous Media 3 (2008), 835–869. [8] M. Berezhnyi, Homogenized Models of Complex Fluids, PhD thesis, ILTPE, 2009 (in Ukrainian), available at: http://www.dlib.com.ua/useredneni-modeli-strukturovanykh-ridyn.html. [9] M. Berezhnyi and E. Khruslov, Non-standard dynamics of elastic composites, Networks and Heterogeneous Media 6 (2011), 89–109. [10] M. Berezhnyi, Discrete model of the nonsymmetric theory of elasticity, Ukrainian Mathematical Journal 63 (2011), 891–913. [11] L.V. Berlyand and A.D. Okhotsimskii, Averaged description of an elastic medium with a large number of small absolutely rigid inclusions, Dokl. Akad. Nauk SSSR 268 (1983), 317–320 (in Russian). [12] L. Berlyand and E. Khruslov, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid, SIAM, Journal of Applied Mathematics 64 (2004), 1002–1034. [13] E. Cosserat and F. Cosserat, Théorie des corps deformables, Hermann, Paris, 1909. [14] V.A. Ditkin and A.P. Prudnikov, Integral Transforms and Operational Calculus, Oxford, New York, 1965. [15] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16.1 (1966), 1–18. [16] G. Grioli, Ellasticá asymmetrica, Annali di Matematica Pura ed Applicata 4 (1960), 389–418. [17] U. Hornung, Homogenization and Porous Media, Interdisciplinary Applied Mathematics, Springer, New York, 1996. [18] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995. [19] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1967. [20] L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics. Quantum Mechanics. Non-relativistic Theory, Pergamon, London, 1958. [21] A.I. Leonov, Algebraic theory of linear viscoelastic nematodynamics, Mathematical Physics, Analysis and Geometry 11 (2008), 87–116. [22] V. Marchenko and E. Khruslov, Homogenization of Partial Differential Equations, Birkhäuser, Boston, 2006. [23] A.I. Marcushevich, Theory of Analytic Functions: Brief Course, Mir, Moscow, 1983. [24] R.D. Mindlin and H.F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11 (1962), 415–448. [25] O.A. Oleinic, A.S. Shamaev and G.A. Iosif’yan, Mathematical problems in elasticity and homogenization, in: Studies in Mathematics and its Applications, Vol. 26, North-Holland, Amsterdam, 1992. [26] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer, New York, 1980. [27] I.Y. Smolin, P.V. Makarov, D.V. Shmick and I.V. Savlevich, A micropolar model of plastic deformation of polycrystals at the mesolevel, Computational Materials Science 19 (2000), 133–142. [28] T. Takahashi, Existence of strong solutions for the problem of a rigid-fluid system, C. R. Acad. Sci. Paris, Ser. I 336 (2003), 453–458. [29] X. Zhang and P. Sharma, Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems, International Journal of Solids and Structures 42 (2005), 3833–3851.
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Asymptotic behavior of a suspension of oriented particles in a viscous incompressible fluid Maksym Berezhnyi a,∗ and Eugen Khruslov b a
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Darmstadt University of Technology, Darmstadt, Germany E-mails: [email protected], [email protected] b Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Kharkiv, Ukraine E-mail: [email protected]
Dedicated to the memory of my mother, T.M. Berezhna (22.09.1953–06.03.2012)
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Abstract. A viscous incompressible fluid with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of the fluid move translationally or rotate around a symmetry axis with the direction of their symmetry axes unchanged. The asymptotic behavior of oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a non-standard hydrodynamics. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor. Keywords: microstructure, suspension, anisotropic material, inhomogeneous material, viscous incompressible fluid, asymmetric hydrodynamics, asymptotic analysis
1. Introduction
Mechanics of suspensions is a part of a general physical–chemical sphere of knowledge about dispersions. Dispersion is a mixture of 2 phases one of which forms a continuum medium (we will call it a dispersive phase) and the other one is dispersed and distributed in the form of separate volume elements inside the first one (we will call it a disperse phase). In this work it is supposed that the dispersive phase is a viscous incompressible fluid and the disperse phase consists of a great number of small solid ferromagnetic particles suspended in the fluid. The sizes of particles are assumed to be of the same order as the distances between the nearest particles. *
Corresponding author: Maksym Berezhnyi, Darmstadt University of Technology, Schloßgartenstr. 7, Darmstadt 64289, Germany. Tel.: +49 6151 163495; E-mails: [email protected], [email protected]. 0921-7134/13/$27.50 © 2013 – IOS Press and the authors. All rights reserved
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M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
When neglecting all physical–chemical processes, the study of the suspension motion can be considered as a problem of pure classical mechanics. In this case the motion of the dispersive phase is governed by the Navier–Stokes equations, and the motion of solid particles forming a dispersed phase is described by the equations of continuum mechanics. However, the study of the properties of the fluids in the framework of such a model by using both analytical and numerical methods appears to be an unsurmountable problem because of the great number of small particles. Therefore it is necessary to develop adequate macroscopic models that can help in studying such fluids. It is known that under the absence of external forces the motion of the compound is governed by the following homogenized equations: ρ
∂v + (v, ∇)v − div σ[v] = ρf , ∂t
div v = 0,
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where ρ = ρ(x) is the homogenized specific mass density of the mixture, v(x, t) is the homogenized velocity of the suspension, σ[v] = {σij [v]}3i,j=1 is the homogenized stress tensor and f is the external force acting on the suspension. Moreover, the stress tensor linearly depends on the strain tensor e[v] = ∂v ∂vi {eij [v] = 12 ( ∂x + ∂xji )}3i,j=1 : j
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σ[v] = Ae[v] − Ep,
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where A = {anpqr (x, t)}3n,p,q,r=1 is the effective viscosity tensor (it is symmetrical with respect to permutation of pairs of subscripts and of subscripts in pairs themselves), E = {δij }3i,j=1 is the unity matrix, and p(x, t) is the pressure. The result is qualitatively the same in the case of weak electric or magnetic forces affecting the suspension. If the suspension is subjected to the influence of a very strong electric or magnetic field then its behavior appears to be different. The study of such a behavior leads to the development of the socalled asymmetric hydrodynamics in which case the stress tensor appears to be non-symmetric (see, for example, the pioneer works [1] and [15] where this fact was established from physical considerations, and our previous work [4] where the same linear problem was considered):
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σ[v] = AD e[v] + AR ω[v] − Ep.
(1)
Here AD and AR are the deformative and rotational parts of the effective viscosity tensor, and ω[v] = ∂v ∂vi {ωij [v] = 12 ( ∂x − ∂xji )}3i,j=1 . j In this paper we suggest a non-linear mathematical model of a suspension which is a mixture of a viscous incompressible fluid with a large number of small perfectly rigid inclusions which are the prolate particles oriented along the fixed direction l. Under the influence of the surrounding fluid the particles can move translationally or rotate around symmetry axis but the direction of their symmetry axes does not change. Such a motion of the composite can be realized, for example, if the particles are strongly magnetizable and subjected to the influence of the strong magnetic field, so that they are oriented along the field direction B (see Fig. 1). We study the asymptotic behavior of such a mixture when the diameters of inclusions tend to zero and the inclusions are distributed in the whole volume. As a result, we obtain the homogenized equations corresponding to asymmetric hydrodynamics.
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M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
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2. Statement of the problem
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Fig. 1. The suspension with oriented particles.
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Consider a bounded domain Ω in R3 with smooth boundary ∂Ω. Suppose that this domain is filled with a mixture consisting of a viscous incompressible fluid with a large number Nε = O(ε−3 ) of small solids Qiε (t) bounded by smooth surfaces ∂Qiε (t) and suspended in the fluid. Further we will call them “the particles”. i Let Ωε (t) = Ω \ N i=1 Qε (t) be a domain filled with the fluid, ρf and ρs be the specific mass density of the fluid and of solid particles respectively, μ be the dynamic viscosity of the fluid, v ε = v ε (x, t) be the velocity of the fluid, pε = pε (x, t) be the pressure, f ε = f ε (x, t) be the external force acting on the ∂v
p 3 n suspension, e[v ε ] = {enp [v] = 12 ( ∂v ∂xp + ∂xn )}n,p=1 be the strain tensor in the fluid, σ[v ε ] = {σnp [v] = 2μenp [v] − pε δnp }3n,p=1 be the stress tensor in the fluid, xiε (t) be the position of the center of mass of Qiε (t), uiε (t) be the displacement of the center of mass of Qiε (t), θiε (t) be the rotation vector of Qiε (t), miε be the mass of Qiε (t), Iεi (t) be the inertia tensor of Qiε (t). Consider the following system of equations:
∂v ε + ρf (v ε , ∇)v ε − μΔv ε = ∇pε + ρf f ε , div v ε = 0, ∂t i i i v ε = u˙ iε + θ˙ ε × x − xiε , θ˙ ε = P d θ˙ ε , x ∈ ∂Qiε , ρf
x ∈ Ωε ,
(2) (3)
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M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
miε u ¨ iε +
Sεi
σ[v ε ]ν ds =
d i ˙i P I θ + Pd dt ε ε
Qiε
d
∂Qiε
x−
ρs f ε dx, xiε
(4)
× σ[v ε ]ν ds = P
d Qiε
x − xiε × ρs f ε dx,
(5)
v ε (x, 0) = v ε0 (x),
x ∈ Ωε (0),
u˙ iε (0) = v iε ,
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uiε (0) = 0,
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where f ε = f ε (x, t) is the external force acting on the mixture, ν is the unit inner normal vector to the surface ∂Qiε (t), and P d is a projection operator onto some fixed d-dimensional subspace S d ⊂ R3 . Depending on d, such a system describes non-stationary motions of the mixture under various regimes of particles rotations. Namely, if d = 3 then the particles can rotate without any constraints. Such a situation was considered in [11] (for the case of an elastic medium filled with the particles) and in [2,12,17,26] (for the case of a viscous incompressible fluid filled with the particles). If d = 0 then the particles move translationally without any rotations. In this paper, we focus on the non-standard cases where d = 1 or d = 2 (similar linear problems for the case of elastic and fluid media were considered in our previous works [9] and [4]; see also [10]). The case d = 1 can be realized, for example, if we consider strongly magnetizable prolate ellipsoidal particles in the strong magnetic field directed along a constant vector B. Then all the particles are aligned along B [20], and under the influence of elastic forces they can move translationally or rotate only around their symmetry axis l = B, but the direction of their symmetry axis does not change (see Fig. 1). In this case, the subspace S 1 is a linear subspace spanned by the vector l. The case d = 2 can be realized, for example, if we consider strongly magnetizable oblate ellipsoidal particles in the strong magnetic field. Moreover, it is assumed that the particles are aligned in such a way that their symmetry axes are identically oriented along the direction l perpendicular to the field direction B and they can rotate both around their symmetry axis and around the field direction. In this case, subspace S 2 is a linear subspace spanned by vectors l and B. The result both in case d = 1 and in case d = 2 is qualitatively the same: the stress tensor in the homogenized model is expressed via the strain tensor and the rotation tensor in accordance with (1). The system of Eqs (2)–(5) is supplemented by the initial conditions
θεi (0) = 0,
(6) θ˙εi (0) = ωεi
(7)
(div v ε0 = 0 at x ∈ Ωε (0) and v ε0 (x) = v iε +ω iε ×(x−xiε (0)) at x ∈ ∂Qiε (0)) and the boundary condition on ∂Ω v ε (x, t) = 0,
x ∈ ∂Ω.
(8)
Theorem 1. There exists a unique solution of the problem (2)–(8) for t ∈ [0, T ] (local in time (0 < T < ∞) or even global (T = ∞) if the data are small enough and the particles do not collide with each other and with the boundary ∂Ω. The proof of the theorem is given in [28] (see also references therein). The main goal of the paper is to study the asymptotic behavior of the solution of problem (2)–(8) as ε → 0. At first, we get uniform (with respect to ε) bounds for the derivatives of that solution extended onto the particles Qiε (t) by equality (3).
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
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3. A priori estimations of the solution of the problem (2)–(8) Starting from the solution {v ε (x, t), uiε (t), θiε (t) = P d θiε (t), i = 1, Nε } of the problem (2)–(8) we construct the vector function v˜ε (x, t) = χε (x, t)v ε (x, t) +
Nε
i χiε (x, t) u˙ iε (t) + θ˙ ε (t) × x − xiε (t) ,
(9)
i=1
where χε (x, t) is the characteristic function of the domain Ωε (t), filled with the fluid, and χiε (x, t) is the characteristic function of a particle Qiε (t). We also denote by Nε
ρε (x, t) = ρf χε (x, t) + ρs
χiε (x, t)
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i=1
T ∂v ε dx dt + ρf v ε , ρf v ε , (v ε · ∇)v ε dx dt ∂t Ωε (t) 0 Ωε (t) T T 3 2 ekl [v ε ] dx dt − + 2μ σ[v ε ]ν, v ε dS dt
Ωε (t) k,l=1
0
T = ρf
(v ε , f ε ) dx dt.
∂Ωε (t)
(10)
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Ωε (t)
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0
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the density of the suspension “the fluid-the particles”. (1) At the first step we estimate ∇˜ v ε (x, t)L2 (ΩT ) , where ΩT = Ω × [0, T ]. To do so, we multiply Eq. (2) by v ε (x, t) and integrate over the domain ΩεT = [0, T ] × Ωε (t). Using Green’s formula, we get
With the aid of the boundary conditions (3), Eqs (4)–(5) and Reynolds transport theorem (see, for example, [3]), the surface integral in (10) can be transformed as follows:
∂Ωε (t)
0
T
= 0
σ[v ε ]ν, v ε dS dt
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−
T
Nε i ρs d u˙ ε (t) + θ˙ iε (t) × x − xiε (t) 2 dx dt 2 dt Qiε (t) i=1
Nε T
− ρs 0
i=1
i u˙ iε (t) + θ˙ ε (t) × x − xiε (t) , f ε (x, t) dx dt.
Qiε (t)
(11)
Consider now the second term in the LHS of (10). Taking into account the boundary condition (8) and the divergence-free condition for the velocity v ε (x, t), with integrating by parts we get T 0
Ω (t)
ρf v ε , (v ε · ∇)v ε dx dt = 0,
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whence it follows that T ρf v ε , (v ε · ∇)v ε dx dt = −ρf Ωε (t)
0
Nε T 0
Qiε (t)
i=1
v ε , (v ε · ∇)v ε dx dt.
With the help of equality (3) one can easily check that v ε , (v ε · ∇)v ε dx = 0, i = 1, Nε .
(12)
(13)
Qiε (t)
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Combining now (12) and (13), we conclude that the second term in the LHS of (10) is equal to 0. Consider now the first term in the LHS of (10). It is easy to see that the following identity holds:
T ∂v ε dx dt ρf v ε , ∂t 0 Ωε (t) T ρf d |v |2 dx dt = 2 dt Ωε (t) ε 0 T T Nε Nε ρf d ρf d|v ε |2 2 dx dt. |v ε | dx dt − + 2 dt 2 0 Qiε (t) 0 Qiε (t) dt i=1
C
i=1
i
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Using Reynolds transport theorem and recalling that v ε = u˙ iε + θ˙ ε ×(x−xiε ) for x ∈ Qiε (t) (i = 1, Nε ), one can be convinced that the second and the third terms in the RHS of that equality coincide with each other. Hence,
T T ρf d ∂v ε dx dt = ρf v ε , |v |2 dx dt. (14) ∂t 2 dt Ωε (t) ε 0 Ωε (t) 0 Combining now equalities (10)–(14), we obtain T 0
ρf d 2 dt
Ωε (t)
− ρs
Nε T
0
i=1
T + 2μ 0
T
|v ε | dx dt + 2
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0
Nε i ρs d u˙ ε (t) + θ˙ iε (t) × x − xiε (t) 2 dx dt 2 dt Qiε (t) i=1
i u˙ iε (t) + θ˙ ε (t) × x − xiε (t) , f ε (x, t) dx dt
Qiε (t) 3
Ωε (t) k,l=1
T e2kl [v ε ] dx dt = ρf 0
Ωε (t)
(v ε , f ε ) dx dt.
It is easy to see that we can rewrite this equality as follows:
T 3 2 1 T d ρε (x, t) v˜ε (x, t) dx dt + 2μ e2kl [˜ v ε ] dx dt 2 0 dt Ω 0 Ω k,l=1 T ρε (x, t)(˜ v ε , f ε ) dx dt. = 0
Ω
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
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Due to the first Korn’s inequality (see [25]) ˜ v ε 2◦ 1
H (Ω )
3
2
Ω k,l=1
e2kl [˜ v ε ] dx,
(15)
the last identity gives us the required bound ∇˜ v ε 2L2 (ΩT ) C
(16)
provided the external force f ε is bounded. (2) At the second step we prove the following estimation:
k,l=1
(17)
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3 ∂v ε 2 + 2μ ekl v ε (x, T ) dx C. ρf ∂t Ωε (T ) L2 (ΩεT )
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To prove this bound we assume that the following conditions hold.
C
(3.1) Let diε be the diameter of the ellipsoidal particle Qiε (t), B(Qiε (t)) be a minimal ball containing Qiε (t), and Rεi (t) be a distance from the ball B(Qiε (t)) to other minimal balls and to the boundary ∂Ω. We suppose that both diε and Rεi (t) (for any fixed t ∈ [0, T ]) satisfy the following inequalities: (18)
R
C1 ε diε , Rεi (t) C2 ε,
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TH
O
where constants C1 and C2 do not depend on ε (0 < C1 < C2 < ∞). (3.2) The velocities u˙ iε (t) of the centers of mass of Qiε (t) for any fixed t ∈ [0, T ] change in a smooth way when passing from one particle to another, i.e. there exist smooth vector functions V ε (x, t) ∂V such that u˙ iε (t) = V ε (xiε , t), V ε (x, t) = 0 for (x, t) ∈ ∂Ω × [0, T ], | ∂tε | < C and |∇x V ε | < C. i (3.3) The instant angular velocities and accelerations of the particles Qε (t) are bounded, so that the ε ¨i 2 i 3 i following estimates hold: |θ˙ ε (t)| < C, N i=1 |θ ε | (dε ) < C. (3.4) max(x,t)∈ΩεT |v ε (x, t)| < C. (3.5) The external force f ε (x, t) and the initial velocity v ε0 (x) are bounded: f ε L2 (ΩT ) C, v ε0 H 1 (Ω ) C. Here, all constants C do not depend on ε. Let ϕiε (x, t) ∈ C ∞ (Ω × [0, T ]) be the functions satisfying the following conditions for any t ∈ [0, T ]: = 1 for x ∈ B(Qiε (t)), ϕiε = 0 for x ∈ / B(1+α)diε (t), 0 ϕiε 1 for x ∈ B(1+α)diε (t) and |∇x ϕiε (x, t)| dCi , where B(1+α)diε (t) denotes a ball of diameter (1 + α)diε concentric with the minimal ε ball B(Qiε (t)), and constants α > 0, C > 0 do not depend on ε. Introduce the following vector field:
ϕiε
lε (x, t) = V ε (x, t) +
Nε i=1
i u˙ iε (t) + θ˙ ε (t) × x − xiε (t) − V ε (x, t) ϕiε (x, t),
(x, t) ∈ ΩT .
(19)
338
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
From the properties of V iε (x, t) and ϕiε (x, t) it follows that i lε (x, t) = u˙ iε (t) + θ˙ ε (t) × x − xiε (t) for x ∈ Qiε (t) (i = 1, Nε ); ∂lε (x, t) ∇x lε (x, t) < C; lε (x, t) = 0 for (x, t) ∈ ∂Ω × [0, T ], ∂t < C;
(20)
where C > 0 does not depend on ε. Consider in ΩT the following vector function: ∂˜ 1/2 ∂˜ ∂˜ v v vε lεi ε = 1 + |lε |2 , q ε (x, t) = ε + ∂t ∂xi ∂ ˆlε 3
(21)
i=1
∂ v˜ε ∂ˆ lε
denotes the directional derivative of u ˜ε along the vector field
1 l ε1 l ε2 l ε3 , , , . (1 + |lε |2 )1/2 (1 + |lε |2 )1/2 (1 + |lε |2 )1/2 (1 + |lε |2 )1/2
O
ˆl = ε
PY
where
∂ v˜ (x,t)
C
Since v˜ε (x, t) is continuous everywhere in ΩT , ∇x v˜ε (x, t) and ε∂t are continuous both in ΩεT and in QTε = ΩT \ ΩεT , v˜ε (x, t) = 0 for (x, t) ∈ ∂Ω × [0, T ], and the vector field ˆlε is tangent to the lateral surface of ∂ΩεT ([22], pp. 225–226), then q ε (x, t) is continuous everywhere in ΩT and q ε (x, t) = 0 for ◦
3 vεj ∂lεi ∂˜ . ∂xj ∂xi
TH
i,j=1
O
div q ε =
R
(x, t) ∈ ∂Ω × [0, T ]. Therefore, from (19) and (21), it follows that for any t ∈ [0, T ] q ε (x, t) ∈H 1 (Ω). Moreover, since div v˜ε = 0, we get (22)
For further considerations we need a technical lemma which can be proved analogously to [12]. ◦
◦
AU
Lemma 1. For any vector function q ε (x) ∈H 1 (Ω) there exists a vector function z ε (x) ∈H 1 (Ω) such that div z ε (x) = div q ε (x) for x ∈ Ω, z ε (x) = q ε (x) + aiε for x ∈ Qiε (i = 1, Nε ) and
z ε H 1 (Ω ) C div q ε L2 (Ω ) + 2
2
Nε
∇q ε L2 (Qiε ) , 2
(23)
i=1
where aiε are constant vectors and C does not depend on ε. Applying this lemma to the function q ε (x) defined by equality (21) for every t ∈ [0, T ], we construct a vector function z ε (x, t). From estimate (23), using equalities (21) and (22) and taking into account the properties of the vector field lε (x, t), we get Nε 2 2 i 2 i 4 i 3 z ε (t) 1 C ∇˜ ¨θε (t) + θ˙ ε (t) dε v ε (t) + . H (Ω )
L2 (Ω )
i=1
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
339
Hence, due to the estimate (16) and conditions (3.1) and (3.3), the following inequality follows
z ε (t)2
T
H 1 (Ω )
0
dt C,
(24)
where C does not depend on ε.
◦
Ωε (t) k,l=1
Nε T
− 0
i=1
∂Qiε (t)
σ[v ε ] dS, aiε (t)
T dt = ρf 0
Ωε (t)
(wε , f ε ) dx dt.
(25)
O
0
PY
Set now wε = q ε − z ε . It is clear that wε (x, t) ∈ L2 (0, T ; H1 (Ω)), div wε = 0 and wε (x, t) = aiε (t) for x ∈ Qiε (t) (i = 1, Nε ). Multiply Eq. (2) by wε and integrate over domain ΩεT . Using Green’s formula, we get
T T ∂v ε dx dt + ρf w ε , ρf wε , (v ε · ∇)v ε dx dt ∂t 0 Ωε (t) 0 Ωε (t) T 3 ekl [wε ]ekl [v ε ] dx dt + 2μ
Ωε (t)
T = ρf
O
0
R
C
Estimate now each of the terms in (25). Taking into account the form of the vector function q ε (x, t) and using inequality (20), Cauchy–Schwarz and Young’s inequalities (with arbitrary δ > 0), estimate from above the RHS of equality (25): T ρf (wε , f ε ) dx dt
AU
TH
(q ε − z ε , f ε ) dx dt 0 Ωε (t)
∂v ε + C∇v ε L2 (ΩεT ) + z ε L2 (ΩεT ) ρf f ε L2 (ΩεT ) ∂t L2 (ΩεT )
∂v ε 2 ρf 2 2 2 2 f + 3ρf δ + C ∇v ε L2 (Ω T ) + z ε L2 (Ω T ) . ε ε 4δ ε L2 (ΩT ) ∂t L2 (ΩεT )
(26)
Consider now the first term on the LHS of equality (25). With the help of the same arguments as before, we obtain the following lower bound:
T T ∂v ε ∂v ε dx dt = dx dt ρf w ε , ρf q ε − z ε , ∂t ∂t 0 Ωε (t) 0 Ωε (t)
T 3 T ∂v ε 2 ∂v ε ∂v ε = dx dt ρf dx dt + ρf lεi , ∂t ∂xi ∂t 0 Ωε (t) i=1 0 Ωε (t)
T ∂v ε − dx dt ρf zε , ∂t 0 Ωε (t)
340
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
∂v ε 2 ∂v ε 2 1 2 ∇v ε L2 (Ω T ) + δ ρf − 3Cρf ε ∂t L2 (ΩεT ) 4δ ∂t L2 (ΩεT ) ∂v ε 2 ρf 2 − z ε L2 (Ω T ) − ρf δ . (27) ∂t ε 4δ L2 (ΩεT ) Next, analogously to (12)–(13), it can be shown that T Ωε (t)
0
ρf wε , (v ε · ∇)v ε dx dt = 0.
(28)
Estimate now the fourth term in the LHS of (25):
∂Qiε (t)
i=1
Nε T
¨ iε (t), aiε (t) miε u
0
Nε T
dt − ρs 0
i=1
Qiε (t)
i=1
f ε (x, t) dx, aiε (t)
dt
C
=
PY
0
σ[v ε ] dS, aiε (t) dt
O
Nε T
−
= Jε1 − Jε2 .
Nε T
O
1/2
T
1/2
2 miε aiε dt
0
2 miε u ¨ iε dt
AU
0
2 miε u ¨ iε dt
0
i=1
1 2
T
TH
Nε 1 Jε
R
Using Cauchy–Schwarz inequality and identity aiε (t) = q ε (x, t) − z ε (x, t) for x ∈ Qiε (t) (i = 1, Nε ), we get:
i=1
ρs + 2
Nε T
0
i=1
Qiε (t)
q (x, t) 2 + z ε (x, t) 2 dx dt, ε
whence, taking into account the form of the vector function q ε (x, t) on the particles Qiε (t) and conditions (3.2)–(3.3), the following upper bound follows: N Nε Nε ε 1 ρs i i 2 5 2 3 2 3 ¨θε diε + u θ˙ ε (t) diε Jε z ε 2 T + C max ¨ iε diε + L2 (Qε ) 0tT 2 i=1
ρs z ε 2L2 (QT ) + C. ε 2 The term Jε2 can be estimated analogously: 2 ρs Jε z ε 2 T + ρs f 2 T + C. L2 (Qε ) ε L2 (Qε ) 2
i=1
i=1
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
341
Thus, N
ε T σ[v ε ] dS, aiε (t) dt ρs z ε 2L2 (QT ) + ρs f ε 2L2 (QT ) + 2C. ε ε 0 ∂Qiε (t)
(29)
i=1
It remains to consider only the third term in the LHS of (25). Taking into account that wε = q ε − z ε , we have: 3
Ωε (t) k,l=1
0
T
ekl [wε ]ekl [v ε ] dx dt 3
= 2μ
Ωε (t) k,l=1
0
ekl [q ε ]ekl [v ε ] dx dt − 2μ
T 0
3
Ωε (t) k,l=1
ekl [z ε ]ekl [v ε ] dx dt.
(30)
PY
T 2μ
The second term in the RHS of (30) we can estimate as follows:
C
O
3 T ekl [z ε ]ekl [v ε ] dx dt ∇z ε 2L2 (Ω T ) + ∇v ε 2L2 (Ω T ) . 2μ ε ε 0 Ωε (t) k,l=1
(31)
3
2μ
Ωε (t) k,l=1
0
=μ 0
−μ
ekl [q ε ]ekl [v ε ] dx dt
3 3 ∂ ˆ 1/2 e2kl [v ε ] dx dt lεi 1 + |lε |2 ∂x i Ωε (t)
AU
T
TH
T
O
R
Recalling the definition of q ε and ˆlε , the first term in the RHS of (30) can be written as follows ∂ ∂ ≡ ∂t ): ( ∂x 0
i=0
T 0
3 3 ∂lεi
Ωε (t) i=1
T +μ 0
k,l=1
∂xi
e2kl [v ε ] dx dt
k,l=1
3 ∂lεi ∂vεl ∂lεi ∂vεk + ekl [v ε ] dx dt ∂xl ∂xi ∂xk ∂xi
Ωε (t) i,k,l=1
= Jε1 + Jε2 + Jε3 . Due to (20), we have: 2 3 Jε + Jε C∇v ε 2
L2 (ΩεT ) .
342
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
Applying the divergence theorem to the integral Jε1 and taking into account that the vector field 2 T ˆl (x, t)(1 + |l |2 )1/2 3 ε ε k,l=1 ekl [v ε ] is tangent to the lateral surface of the domain Ωε , we get: 1
Jε = μ
3
Ωε (T ) k,l=1
ekl v ε (x, T ) dx − μ 2
3
Ωε (0) k,l=1
e2kl v ε (x, 0) dx.
Thus, 3
0
Ωε (t) k,l=1
μ
ekl [wε ]ekl [v ε ] dx dt
3
Ωε (T ) k,l=1
ekl v ε (x, T ) dx − μ 2
3
Ωε (0) k,l=1
e2kl v ε0 (x) dx − C∇v ε 2L2 (Ω T ) . ε
(32)
PY
T 2μ
Combining now (25)–(32), we obtain:
C
k,l=1
O
3 ∂v ε 2 2 ρf 1 − (3C + 4)δ + μ e (x, T ) dx v ε kl ∂t L2 (ΩεT ) Ωε (T )
O
R
ρf fε 2L2 (ΩT ) + ρs fε 2L2 (QT ) ε 4δ
C 1 2 2 + C δ + 2C + 1 ∇v ε L2 (Ω T ) + ρf + 3δ z ε 2L2 (Ω T ) + ∇z ε 2L2 (Ω T ) + 3ρf ε ε ε 4δ 4δ 3 + ρs z ε 2L2 (QT ) + 2C + e2kl v ε0 (x) dx. Ωε (0) k,l=1
TH
ε
AU
1 Taking here δ = 2(3C+ 4) , using the previously obtained bounds (16) and (24), and taking into consideration condition (3.5), we get the required bound (17):
3 ∂v ε 2 + 2μ ekl v ε (x, T ) dx C, ρf ∂t L2 (ΩεT ) Ωε (T ) k,l=1
where constant C does not depend on ε. Before formulating the main result we introduce some definitions and assumptions. 4. Additional assumptions and the main result Consider now the following auxiliary linear stationary problem in the domain Ωε (t) (t is a parameter): −μΔv ε = ∇pε + Φε , div v ε = 0, i i i v ε = aε + bε × x − xε ,
x ∈ Ωε , biε
=
P d biε ,
(33) x∈
∂Qiε ,
(34)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
343
∂Qiε
σ[v ε ]ν ds =
P
d ∂Qiε
x−
xiε
Qiε
Φε dx,
(35)
× σ[v ε ]ν ds = P
d Qiε
x − xiε × Φε dx,
(36)
x ∈ ∂Ω.
v ε (x) = 0,
(37)
It will be shown in the next section that for any fixed t ∈ [0, T ] the solution v ε (x, t) of the problem (2)–(8) can be represented in the form v ε (x, t) = Rεt Φtε [x], where Rεt is the resolving operator of the problem (33)–(37) in the domain Ωε (t) (t is a parameter), and Φtε [x] = ρf χε (x, t) + ρs 1 − χε (x, t) f ε (x, t)
Nε ∂v ε ∂V iε i + (v ε · ∇)v ε χε (x, t) − ρs + V ε · ∇ V iε χiε (x, t), ∂t ∂t
PY
− ρf
i=1
min ◦
v ε ∈J ε (Ω )
Fε v ε ,
C
Fε (v ε ) =
O
i where V iε (x, t) = u˙ iε (t) + θ˙ ε (t) × (x − xiε (t)). Note that the problem (33)–(37) is equivalent to the variational problem
◦
(38) ◦
Fε (v ε ) =
2μ Ω
3
enp [v ε ] − 2 Φε , v ε dx. 2
TH
O
R
where J ε (Ω) is the class of divergence free vector-functions from H1 (Ω) which are equal to aiε + biε × (x − xiε ) on the particles Qiε (aiε and biε = P d biε are arbitrary vectors) and (39)
n,p=1
AU
Let Khy be a cube with the side length h (ε h 1) centered at y ∈ Ω. We assume that the edges of
this cube are parallel to the coordinate axes. Let Jεθ [Khy ] be the following class of vector-functions: Jεθ Khy = wε ∈ H 1 Khy ; div wε = 0; θ × x − xiε , x ∈ Qiε ∩ Khy , wε (x) = wiε + P d θiε + 1 − P d where wiε and θiε are arbitrary vectors, and θ is a given vector. Consider a minimization problem in this class for the following functional (mesocharacteristic): Aγεh (wε , y, T ) = EKhy [wε , wε ] εhγ + PK wε (x) − y h
3
n,p=1
Tnp ϕnp (x − y), wε (x) −
3
q ,r=1
Tqr ϕqr (x − y) ,
(40)
344
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
where EG [uε , v ε ] = 2μ
3
G n,p=1
PGεhγ uε (x), v ε (x)
=h
enp [uε ]enp [v ε ] dx,
−2−γ
G
(41)
uε (x), v ε (x) dx,
(42)
3 δqr 1 q r ϕ (x) = xr e + xq e − xn en , 2 3 n=1
qr
(43)
min
wε ∈Jεθ [Khy ]
AU
TH
O
R
C
O
PY
∂vl n 3 k ekl [v] = 12 ( ∂v ∂xl + ∂xk ), T = {Tqr } is an arbitrary symmetric second rank tensor, {e }n=1 is an orthonormal basis in R3 , and 0 < γ < 2 is a penalty parameter. This mesocharacteristic plays the crucial role in our consideration. Roughly speaking, it allows us to compute the energy of the suspension in some mesoscopic cube of size h (ε h 1), which is a so-called representative volume element. In other words, if a suspension can be described within the effective single medium approach, then the rheological properties of the suspension can be determined by calculation or measurements in some representative volume element of an intermediate mesoscale h, which is why we choose the cube Khy . Next, observe that the first term (41) in (40) represents the energy of the suspension. The minimizer wε of (40) is “close”, up to an additive constant, to the true global minimizer v ε of the variational problem (38) corresponding to (33)–(37), if the tensor T is chosen appropriately. Now one should choose T . If the single medium homogenized description is possible, then v ε (x) is “close” to some smooth (homogenized) vector-function v(x), which depends only on the macroscopic variable x and does not depend on ε, so that it does not vary on the microscale ε. We then minimize the energy of the suspension, adding the constraint that the minimizer wε is “close” to the linear part (differential) of the global minimizer v, so that |wε − v| = o(h) ∼ h1+γ/2 for some γ > 0. This condition is imposed by introducing the penalty term (42). It can be proved (see [4]) that there exists the unique vector-function which minimizes the functional (40); the minimal value of this functional is given by
Aγεh (wε , y, T ) =
3
γ d a0, npqr y, S , ε, h Tnp Tqr
n,p,q ,r=1
+2
3 3
n,p=1 q=1
3 bγnpq y, S d , ε, h Tnp θq + cγqr y, S d , ε, h θq θr ,
γ γ γ d d d where a0, npqr (y, S , ε, h), bnpq (y, S , ε, h) and cqr (y, S , ε, h) are the components of the fourth-, and second-rank tensors respectively, defined as follows np qr εhγ np γ d + PK (x) − ϕnp (x − y), wqr (x) − ϕqr (x − y) , a0, y w npqr y, S , ε, h = EKhy w , w h εhγ bγnpq y, S d , ε, h = EKhy wnp , v q + PK y wnp (x) − ϕnp (x − y), v q (x) , h q r εhγ q γ d cqr y, S , ε, h = EKhy v , v + PK y v (x), v r (x) . h
(44)
q ,r=1
third(45) (46) (47)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
345
Here wnp (x) is the vector-function that minimizes the functional (40) in Jε [Khy ] as T = T np = eq y 1 n p p n q 2 (e ⊗ e + e ⊗ e ), v (x) is the vector-function minimizing the functional (40) in Jε [Kh ] as T = 0, n 3 and e (n = 1, 2, 3) form an orthonormal basis in R . Starting from the solution {v ε (x), aiε , biε (t) = P d biε (t), i = 1, Nε } of the problem (33)–(37) we construct the vector function 0
v˜ε (x) = χε (x)v ε (x) +
Nε
χiε (x) aiε + biε × x − xiε ,
(48)
i=1
where χε (x) is the characteristic function of the domain Ωε , filled with the fluid, and χiε (x) is the characteristic function of a particle Qiε . We assume that the following conditions hold:
PY
(4.1) for some real number γ > 0 the following limits exist heterogeneously at x ∈ Ω: (a)
γ γ d d a0, a0, npqr (x, S , ε, h) npqr (x, S , ε, h) = lim lim = a0npqr x, S d , 3 3 h→0 ε→0 h→0 ε→0 h h
(b)
bγnpq (x, S d , ε, h) bγnpq (x, S d , ε, h) = lim lim = bnpq x, S d , 3 3 ε→ 0 h→0 h→0 ε→0 h h
(c)
cγqr (x, S d , ε, h) cγqr (x, S d , ε, h) = lim lim = cqr x, S d , 3 3 h→0 ε→0 h→0 ε→0 h h
O
lim lim
C
lim lim
R
lim lim
O
where {a0npqr (x, S d )}, {bnpq (x, S d )}, {cqr (x, S d )} are continuous tensors (at x ∈ Ω). (4.2) the sequence Φε (x) converges weakly in L2 (Ω) to a vector function Φ(x), as ε → 0.
TH
Note, that the existence of the limits (4.1) is a general restriction on the spatial distributions of the particles. Since we do not require any spatial periodicity, we have to impose some conditions on these distributions.
AU
Remark. If the limits in (4.1) exist for some γ > 0, then they exist for any γ > 0 and the limiting tensors do not depend on γ; moreover, {a0npqr (x, S d )} and {cqr (x, S d )} are positive definite tensors (these facts can be proved analogously to [22]). In our previous work [4] the following theorem was proved. Theorem 2. Let conditions (4.1)–(4.2) hold. Then the sequence of vector-functions v˜ε (x), defined by (48), converges strongly in L2 (Ω) to a vector-function v(x, t), which is a solution of the following homogenized linear stationary problem: −
3
n ∂ D anpqr (x)eqr [v] + aR npqr (x)ωqr [v] e = ∇p + Φ, ∂xp n,p,q ,r=1
x ∈ Ω,
(49)
div v = 0,
x ∈ Ω,
(50)
v(x) = 0,
x ∈ ∂Ω.
(51)
346
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
Here 1 = anpqr + bqrl lnp , 2 l=1
1 ∂vq ∂vr , ωqr [v] = − 2 ∂xr ∂xq 3
aD npqr
aR npqr
0
3 3 1 1 = clm lnp mqr + bnpl lqr , 4 2
l,m=1
(52)
l=1
(53)
where {lnp } is Levi–Civita permutation tensor. The problem (49)–(51) has the unique solution.
ε→0
2
PY
Let Rεt and Rt (t is a parameter) be the resolving operators of the problems (33)–(37) and (49)–(51), respectively (v ε = Rεt F ε and v = Rt F ). Analogously to [8] and [22], one can show that Rεt and Rt are compact and self-adjoint in L2 (Ω). Moreover, with the help of Theorem 2, it can be proved that for any t ∈ [0, T ] and any f ∈ L2 (Ω) lim Rεt f − Rt f L (Ω ) = 0, Rεt f L (Ω ) C, (54) 2
Nε
χiε (x, t) C(x, t)
C
O
where constant C depends neitheron ε nor on t (for more details see [22]). ε i ∞ Assume now that the sequence N i=1 χε (x, t) = 1 − χε (x, t) converges as ε → 0 ∗-weakly in L (ΩT ) to the function 0 < C(x, t) < 1: *-weakly in L∞ (ΩT ) ,
(55)
R
i=1
TH
lim f ε − f L2 (ΩT ) = 0,
O
where Ct (x, t) ∈ L4 (0, T ; L2 (Ω)), ∇C(x, t) ∈ L2 (ΩT ), and the sequences f ε (x, t) and v˜ε0 (x) converge as ε → 0 strongly in L2 (ΩT ) to f (x, t) and strongly in L2 (Ω) to v 0 (x), respectively: ε→0
v ε0 − v 0 L2 (Ω ) = 0. lim ˜
(57)
AU
ε→0
(56)
Theorem 3. Let conditions (3.1)–(3.5) are satisfied, the limits (55)–(57) exist, the limits in (4.1) exist for every t ∈ [0, T ] and the limiting tensors {a0npqr (x, t, S d )}, {bnpq (x, t, S d )}, {cnp (x, t, S d )} are continuous in ΩT . Then the sequence of vector-functions v˜ε (x, t), defined by (9), converges strongly in L2 (ΩT ) (and in L2 (Ω) uniformly with respect to t) to a vector-function v(x, t), which is a generalized solution of the following homogenized problem: 3 n ∂ D ∂(ρv) + (v · ∇)(ρv) − anpqr (x, t)eqr [v] + aR npqr (x, t)ωqr [v] e ∂t ∂xp n,p,q ,r=1
= ∇p + F ,
x ∈ ΩT ,
(58)
div v(x, t) = 0,
x ∈ ΩT ,
(59)
v(x, t) = 0,
(x, t) ∈ ∂Ω × [0, T ],
(60)
v(x, 0) = v 0 (x),
(61)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
347
where ρ(x, t) = ρf 1 − C(x, t) + ρs C(x, t),
F (x, t) = ρ(x, t)f (x, t).
(62)
◦
◦
Remark. A function v(x, t) ∈ L(ΩT ) = H 1 (ΩT ) ∩ L2 (0, T ; J (Ω)) ∩ L∞ (ΩT ), where J (Ω) is the ◦ class of divergence free vector functions from H1 (Ω), is said to be a generalized solution to the problem (58)–(61) if it satisfies the following integral identity
−ρv, Φt + (v · ∇)Φ +
Ω
0
3
n,p,q ,r=1
ρv, Φ(x, 0) =
ρv, Φ(x, τ ) −
+ Ω
aD npqr (x, t)
+
aR npqr (x, t)
∂vn ∂Φq ∂xp ∂xr
dx dt
τ (F , Φ) dx dt
Ω
0
Ω
(63)
PY
τ
for any Φ(x, t) ∈ L(ΩT ) and τ (0 < τ T ).
O
Lemma 2. Problem (58)–(61) can have at most one generalized solution from L(ΩT ).
C
Proof. Assume that there exist two solutions v (x, t) ∈ L(ΩT ) and v (x, t) ∈ L(ΩT ). Then the function v = v − v ∈ L(ΩT ) satisfies the following identity for any Φ ∈ L(ΩT ):
R
Ω
(ρv, Φt ) + ρv, v · ∇ Φ − ρv , (v · ∇)Φ dx dt
τ
3
+
Ω n,p,q ,r=1
0
O
0
R aD npqr (x, t) + anpqr (x, t)
TH
−
τ
∂vn ∂Φq dx dt ∂xp ∂xr
ρ(x, τ )v(x, τ ), Φ(x, τ ) dx = 0.
+
AU
Ω
Choosing here Φ(x, t) = v(x, t), after obvious transformations we get that 1 2
2 ρ(x, τ ) v(x, τ ) dx +
τ
Ω
Ω n,p,q ,r=1
0
1 2
τ 0
|ρt ||v|2 dx dt + Ω
3
τ 0
R aD npqr (x, t) + anpqr (x, t)
∂vn ∂vq dx dt ∂xp ∂xr
ρ v, v · ∇ v + v , (v · ∇)v dx dt. Ω ◦
Since for any fixed t ∈ [0, τ ] v(x, t) ∈J (Ω), the following inequality holds (see [4]):
Ω
R aD npqr + anpqr
∂vn ∂vq dx v2H 1 (Ω ) , ∂xp ∂xr
(64)
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M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
R where aD npqr and anpqr are defined by (52). Using now inequality (64) and the fact that both the function ρ(x, t) and the vector functions v (x, t) and v (x, t) are bounded, we obtain:
2 ρ(x, τ ) v(x, τ ) dx +
max
0tτ
Ω
C
τ
τ
|ρt ||v| dx dt +
|∇v|2 dx dt Ω
0
τ
Ω
0
|v||∇v| dx dt ,
2
(65)
Ω
0
where constant C does not depend on τ . Using Hölder’s inequality, we can write |ρt ||v|2 dx dt
τ
Ω
0
|v|q dx
r/q 2/r τ
r /q 1/r dt |ρt |q dx dt
Ω
0
0
= vLr (0,τ ;Lq (Ω )) ρt Lr (0,τ ;Lq (Ω )) ,
(Ωτ ) = v(x, t): vΩτ = ess max
C
◦ V2
v(x, t) 2 dx +
2
0tτ
Ω
τ 0
∇v(x, t) 2 dx dt < ∞; Ω
◦
O
R
v(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ] .
(66)
O
2
where q = q−q 2 , r = r−r 2 (q 2, r 2). Introduce now the following space of vector functions:
Ω
PY
τ
TH
For any vector function v(x, t) ∈V2 (Ωτ ) the following inequality holds (see [19]): vLr (0,τ ;Lq (Ω )) βvΩτ ,
AU
(67)
where q and r are arbitrary constants such that
1 r
+
3 2q
◦
= 34 , r 2, 2 q 6 and β = 42/r . Note that
due to the Embedding theorems (see [19]) L(ΩT ) ⊂V2 (ΩT ). Choosing in (66) q = 4 and r = 83 and taking into account (67), we get τ 0
Ω
|ρt ||v|2 dx dt β 2 v2Ωτ ρt L4 (0,τ ;L2 (Ω )) .
Next, using Hölder’s inequality, we have τ 0
|v||∇v| dx dt vΩτ Ω
τ 0
1/2 |v| dx dt 2
Ω
(τ )1/r (meas Ω)1/q vΩτ vLr (0,τ ;Lq (Ω )) ,
(68)
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
where r = τ 0
Ω
2r r−2
and q =
2q q−2 .
Choosing, as before, q = 4 and r =
8 3
349
and using inequality (67), we get
|v||∇v| dx dt β(τ )1/8 (meas Ω)1/4 v2Ωτ .
(69)
From (65), (68) and (69) it follows that min{ρf , ρs , 1}v2Ωτ δ(τ )v2Ωτ ,
(70)
PY
where δ(τ ) = β 2 ρt L4 (0,τ ;L2 (Ω )) + β(τ )1/8 (meas Ω)1/4 . Taking now τ so that δ(τ ) < min{ρf , ρs , 1}, from (70) we conclude that v(x, t) = 0. If τ < T , then partitioning interval [0, T ] into subintervals of length τk (for which v(x, t) = 0) and repeating the above arguments, after a final number of steps one can show that v(x, t) = v (x, t) − v (x, t) ≡ 0 in ΩT . The lemma is proved. 2
O
5. Proof of Theorem 3
In accordance with the bounds (16)–(17) and conditions (3.1)–(3.3) the sequence of vector functions ◦
Nε ∂v ε ∂V iε i + (v ε · ∇)v ε χε (x, t) + ρs + V ε · ∇ V iε χiε (x, t), ∂t ∂t
AU
TH
O
R
C
{v ε (x, t), ε > 0} is bounded in H 1 (ΩT ) and, thus, it is weakly compact in H 1 (ΩT ) (and in H 1 (Ω) for any t ∈ [0, T ]). Due to the Embedding theorem, this sequence is compact in L2 (ΩT ) (and in L2 (Ω) for any t ∈ [0, T ]). Hence, there exists a subsequence {v εk (x, t), εk > 0} which converges weakly in H 1 (ΩT ) (and in H 1 (Ω) for any t ∈ [0, T ]) and strongly in L2 (ΩT ) (and in L2 (Ω) for any t ∈ [0, T ]) to some vector function v(x, t). Using conditions (3.3)–(3.4), we conclude that v(x, t) ∈ L(ΩT ). As it is shown below, the limiting vector function v(x, t) is a solution of the problem (58)–(61). Since this problem, due to Lemma 2, has a unique generalized solution from L(ΩT ), the entire sequence {v ε (x, t), ε > 0} also converges to v(x, t). Introduce now the following vector function
W ε (x, t) = ρf
(71)
i=1
i
where V iε (x, t) = u˙ iε (t) + θ˙ ε (t) × (x − xiε (t)). Lemma 3. The sequence of vector functions {W ε (x, t), ε > 0} is weakly compact in L2 (ΩT ). So, there exists a subsequence {W εk (x, t), εk > 0} which converges as εk → 0 weakly in L2 (ΩT ) to the limiting vector function W (x, t) which is given by W (x, t) =
∂(ρv) + (v · ∇)(ρv), ∂t
(72)
where ρ(x, t) is determined by Eq. (62) and v(x, t) is a limit of the subsequence {v εk (x, t), εk → 0}.
350
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
Proof. From the bounds (16)–(17) and conditions (3.3)–(3.4) it follows that the sequence {W ε (x, t), ε > 0} is bounded in L2 (ΩT ) and, hence, it is weakly compact. Let ϕ(x, t) ∈ C01 (ΩT ) be an arbitrary smooth vector function with a compact support. Then
0
∂v εk + (v εk · ∇)v εk , ϕ dx dt (W εk , ϕ) dx dt = ρf ∂t Ω 0 Ωε (t)
Nε T i ∂V iεk i + V εk · ∇ V εk , ϕ dx dt + ρs ∂t i i=1 0 Qε (t)
T ∂v εk + (v εk · ∇)v εk , ϕ dx dt = ρf ∂t 0 Ω
Nε T i ∂V iεk i + (ρs − ρf ) + V εk · ∇ V εk , ϕ dx dt ∂t 0 Qiε (t) T
PY
T
i=1
1
2
(73)
O
= J εk + J εk .
T 1
lim Jεk = ρf
0
Ω
∂v + (v · ∇)v, ϕ dx dt. ∂t
(74)
R
εk →0
C
Since v εk (x, t) converges weakly in H 1 (ΩT ) and strongly in L2 (ΩT ) to v(x, t),
Nε T i=1
0
j=0
Nε T
AU
− (ρs − ρf )
3 ∂ i V εk , ϕ Vˆεik ,j dx dt ∂xj Qiε (t)
TH
Jεk = (ρs − ρf ) 2
O
The second integral in (73) we can write in the form
i=1
0
3
Qiε (t) j=0
∂ϕ i i ˆ Vεk ,j V εk , dx dt ∂xj
= Jε2,1 + Jε2,2 , k k
i where x0 ≡ t, Vˆεik ,0 ≡ 1, Vˆεik ,j = Vεik ,j (x, t) (j = 1, 3). Since the vector Vˆ εk = {Vˆεik ,j }3j=0 is tangent to the lateral surface of [0, T ] × Qiεk (t) (i = 1, Nεk ) and the vector function ϕ(x, t) has a compact support in ΩT , with the help of the divergence theorem we conclude that Jε2,1 = 0. k i Using (55) and taking into account that V εk (x, t) = v εk (x, t) at (x, t) ∈ Qiεk (t) × [0, T ] and v εk (x, t) converges strongly in L2 (ΩT ) to v(x, t), we obtain
lim Jε2,2 = −(ρs − ρf ) k
εk →0
T C(x, t) 0
Ω
∂ϕ v, ∂t
+
3
j=1
vj
∂ϕ v, ∂xj
dx dt.
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
351
Thus, lim Jεk = −(ρs − ρf )
T
2
εk →0
= (ρs − ρf )
C(x, t) Ω
0
T 0
∂ϕ v, ∂t
Ω
+
3
vj
j=1
∂ϕ v, ∂xj
dx dt
∂(Cv) + (v · ∇)(Cv), ϕ dx dt. ∂t
(75)
Here, the integration by parts is justified, because v(x, t) ∈ H 1 (ΩT ) ∩ L∞ (ΩT ), ρ(x, t) ∈ H 1 (ΩT ) ∩ L∞ (ΩT ) and ϕ(x, t) ∈ C01 (ΩT ). Combining now (73)–(75) and taking into account (62), we finally get T εk →0 0
Ω
T = 0
Ω
Ω
PY
0
W εk (x, t), ϕ(x, t) dx dt
W (x, t), ϕ(x, t) dx dt = lim
∂(ρv) + (v · ∇)(ρv), ϕ dx dt, ∂t
O
2
whence the statement of the lemma follows. Denote by Fε the RHS of problem (2)–(5):
C
T
Fε (x, t) = ρf χε (x, t) + ρs 1 − χε (x, t) f ε (x, t).
O
R
(76)
TH
Recalling now (71) and taking into account that i ∂V iε i i i + V ε · ∇ V iε = u ¨ iε (t) + ¨θε (t) × x − xiε (t) + θ˙ ε (t) × θ˙ ε (t) × x − xiε (t) , ∂t
(77)
AU
we can represent the solution v ε (x, t) of the problem (2)–(8) in the form v ε (x, t) = Rεt Φtε [x], where Rεt is the resolving operator of the problem (33)–(37) in the domain Ωε (t) (t is a parameter), and Φtε [x] = F ε (x, t) − W ε (x, t). Since for every t the operator Rεt is self-adjoint in L2 (Ω), for any continuous vector function ϕ(x, t) ≡ t ϕ [x] we have: T
T 0
Ω
(v ε , ϕ) dx dt = 0
T
Ω
= 0
Rεt Φtε , ϕt
Ω
T
dx dt =
Φtε , Rt ϕt dx dt +
0
T 0
Ω
Φtε , Rεt ϕt dx dt
Ω
Φtε , Rεt ϕt − Rt ϕt dx dt.
(78)
Using Lemma 3 and conditions (55)–(56), we conclude that the subsequence {Φtεk [x] ≡ Φεk (x, t), εk > 0} converges weakly in L2 (ΩT ) to the vector function Φ(x, t) = F (x, t) − W (x, t) ≡ Φt [x], where
352
M. Berezhnyi and E. Khruslov / Asymptotic behavior of a suspension of oriented particles
the vector functions F and W are defined in (62) and (72), respectively. Thus, T
lim
εk →0 0
Φtεk , Rt ϕt
Ω
T εk →0 0
T
Φεk (x, t), Rt ϕ(x, t) dx dt
dx dt = lim
Ω
=
T
t
Φt , Rt ϕt dx dt
Φ(x, t), R ϕ(x, t) dx dt =
0
T
Ω
Rt Φt , ϕt dx dt =
= 0
Ω
0
T 0
Ω
Rt Φt [x], ϕ(x, t) dx dt.
(79)
Ω
Here we took into consideration that for almost all t ∈ [0, T ] Φ(x, t) ∈ L2 (Ω) and the operator Rt is self-adjoint in L2 (Ω). Next,
0
Ω
T
Rεt ϕt − Rt ϕt 2
L2 (Ω )
PY
T t t t t t Φε , Rε ϕ − R ϕ dx dt Φε L2 (ΩT )
0
1/2 dt
.
ε→0
C
O
Since the sequence {Φε , ε > 0} is bounded in L2 (ΩT ) uniformly with respect to ε, with the help of (54) we get: T t t t t t lim Φε , Rε ϕ − R ϕ dx dt = 0. (80) Ω
0
0
Ω
0
O
εk →0
R
Combining now (78)–(80), we obtain: T T t t (v ε , ϕ) dx dt = R Φ [x], ϕt dx dt. lim Ω
εk →0 0
Ω
AU
TH
On the other hand, since the subsequence {v εk (x, t) ≡ v tεk [x], εk > 0} converges as εk → 0 strongly in L2 (ΩT ) (and in L2 (Ω) uniformly with respect to t ∈ [0, T ]) to the vector function v(x, t) ≡ v t [x], we have: T T t t lim v ε (x, t), ϕ(x, t) dx dt = v , ϕ dx dt. 0
Ω
So, for all t ∈ [0, T ] v(x, t) ≡ v t [x] = Rt Φt [x] = Rt [F (x, t) − W (x, t)]. Moreover, with the aid of condition (57), we conclude that v(x, 0) = v 0 . Recalling the definition of the resolving operator Rt and the form of the vector function W (x, t) (see (72)), we conclude that v(x, t) is the generalized solution of the problem (58)–(61). Theorem 3 is proved. References [1] E.L. Aero, A.N. Bulygin and E.V. Kuvshinsky, Asymmetric hydrodynamics, J. Appl. Math. Mech. (PMM) 29 (1965), 297–308 (in Russian). [2] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. II. Noncritical sizes of the holes for a volume distribution and surface distribution of holes, Arch. Rational Mech. Anal. 113 (1991), 209–259, 261–298.
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353
AU
TH
O
R
C
O
PY
[3] T. Belytschko, W.K. Liu and B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley, New York, 2000. [4] M. Berezhnyi and E. Khruslov, Asymmetric hydrodynamics of suspensions subjected to the influence of strong external magnetic fields, Journal of Multiscale Modeling 4 (2012), 24–25. [5] M.A. Berezhnyy and L.V. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn’s inequality, Journal of the Mechanics and Physics of Solids 54 (2006), 635–669. [6] M.A. Berezhnyi, The asymptotic behavior of viscous incompressible fluid small oscillations with solid interacting particles, Journal of Mathematical Physics, Analysis, Geometry 3 (2007), 135–156. [7] M. Berezhnyi, L. Berlyand and E. Khruslov, The homogenized model of small oscillations of complex fluids, Networks and Heterogeneous Media 3 (2008), 835–869. [8] M. Berezhnyi, Homogenized Models of Complex Fluids, PhD thesis, ILTPE, 2009 (in Ukrainian), available at: http://www.dlib.com.ua/useredneni-modeli-strukturovanykh-ridyn.html. [9] M. Berezhnyi and E. Khruslov, Non-standard dynamics of elastic composites, Networks and Heterogeneous Media 6 (2011), 89–109. [10] M. Berezhnyi, Discrete model of the nonsymmetric theory of elasticity, Ukrainian Mathematical Journal 63 (2011), 891–913. [11] L.V. Berlyand and A.D. Okhotsimskii, Averaged description of an elastic medium with a large number of small absolutely rigid inclusions, Dokl. Akad. Nauk SSSR 268 (1983), 317–320 (in Russian). [12] L. Berlyand and E. Khruslov, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid, SIAM, Journal of Applied Mathematics 64 (2004), 1002–1034. [13] E. Cosserat and F. Cosserat, Théorie des corps deformables, Hermann, Paris, 1909. [14] V.A. Ditkin and A.P. Prudnikov, Integral Transforms and Operational Calculus, Oxford, New York, 1965. [15] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16.1 (1966), 1–18. [16] G. Grioli, Ellasticá asymmetrica, Annali di Matematica Pura ed Applicata 4 (1960), 389–418. [17] U. Hornung, Homogenization and Porous Media, Interdisciplinary Applied Mathematics, Springer, New York, 1996. [18] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995. [19] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1967. [20] L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics. Quantum Mechanics. Non-relativistic Theory, Pergamon, London, 1958. [21] A.I. Leonov, Algebraic theory of linear viscoelastic nematodynamics, Mathematical Physics, Analysis and Geometry 11 (2008), 87–116. [22] V. Marchenko and E. Khruslov, Homogenization of Partial Differential Equations, Birkhäuser, Boston, 2006. [23] A.I. Marcushevich, Theory of Analytic Functions: Brief Course, Mir, Moscow, 1983. [24] R.D. Mindlin and H.F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11 (1962), 415–448. [25] O.A. Oleinic, A.S. Shamaev and G.A. Iosif’yan, Mathematical problems in elasticity and homogenization, in: Studies in Mathematics and its Applications, Vol. 26, North-Holland, Amsterdam, 1992. [26] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer, New York, 1980. [27] I.Y. Smolin, P.V. Makarov, D.V. Shmick and I.V. Savlevich, A micropolar model of plastic deformation of polycrystals at the mesolevel, Computational Materials Science 19 (2000), 133–142. [28] T. Takahashi, Existence of strong solutions for the problem of a rigid-fluid system, C. R. Acad. Sci. Paris, Ser. I 336 (2003), 453–458. [29] X. Zhang and P. Sharma, Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems, International Journal of Solids and Structures 42 (2005), 3833–3851.
