INTRODUCTION. One of the basic problems of rheology is the study of macroscopic properties of anisotropic fluids. Anisotropic properties are inherent in liquid ...
ISSN 1061933X, Colloid Journal, 2011, Vol. 73, No. 5, pp. 614–620. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.S. Volkov, V.G. Kulichikhin, 2011, published in Kolloidnyi Zhurnal, 2011, Vol. 73, No. 5, pp. 608–614.
Rheology of Complex Anisotropic Fluids V. S. Volkov and V. G. Kulichikhin Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Leninskii pr. 29, Moscow, 119991 Russia Received February 25, 2011
Abstract—A structurecontinual theory is developed for viscous fluids with arbitrary anisotropy based on the dissipation function. Its canonical form determined by the principal viscosities is established. A transversely isotropic incompressible fluid with a single preferable direction is considered as a particular case. The pecu liarities of the rheological behavior of this anisotropic fluid are investigated upon directed shear and elonga tional flows. DOI: 10.1134/S1061933X11050176
INTRODUCTION One of the basic problems of rheology is the study of macroscopic properties of anisotropic fluids. Anisotropic properties are inherent in liquid crystals, suspensions, colloidal systems, and many polymeric and biological fluids. The dynamics of anisotropic fluids is described with the help of the Euler continuum mechanics. In this case, it is important to take into account internal rota tion, which is not reduced to the translational motion. Constitutive equations describing continuous media with internal rotation are needed to close the two equations of the Euler dynamics. The idea to assign anisotropic structure to particles of continuous media was proposed by Duhem [1]. To allow for these structural properties, it is necessary to construct anisotropic constitutive equations relating variables entering into the basic laws of the Euler dynamics for continuous media. The formulation of the constitutive equations is an entirely theoretical problem, which, in principle, cannot be solved by the experimental methods of rheology. Anzelius was the first to study colloidal solutions as anisotropic fluids [2]. Developing his ideas, Ericksen formulated an invariant theory of a simple anisotropic fluid [3]. Each particle of the fluid may be represented as an ellipsoid of revolution. A director determining the orientation of the particles in the fluid is intro duced as a parameter. The dynamics of orientational processes in a fluid with a single preferable direction is described by the director equation postulated by Ericksen; it may be derived directly from the dynamic equation for internal rotation [4]. The effects of rheo logical anisotropy, which arise in flowing solutions of flexiblechain polymers are considered based on the Ericksen theory [5–10]. Moreover, the Ericksen theory is the foundation for investigating the dynamic properties of suspensions and liquid crystals. Director description of the behav ior of microstructures is used in the continuous theory of suspensions [11]. Leslie developed the director the
ory for liquid crystals [12]. This theory describes the main effects that are observed for flows of lowmolec ularmass liquid crystals [13, 14]. It should be stressed that liquid crystals are distinguished by an enhanced response to the action of external electric and mag netic fields. In this case, diverse phenomena relevant to asymmetric stresses are observed. These effects are most pronounced in colloidal magnetic fluids. The foundations of the modern theory of anisotro pic fluids, which is based on the spectral (algebraic) approach and makes it possible to judge the structure of anisotropic viscosity for diverse anisotropic fluids, were developed in [15–18]. Moreover, this approach opens up novel possibilities for the classification of anisotropic fluids. In this work, we developed a ther modynamic approach to the construction of rheologi cal equations describing the state of anisotropic vis cous fluids with complex microstructure. Being based on the spectral decomposition of the viscosity tensor, this approach enables us to reveal the general structure of the anisotropic dissipation function. CANONICAL FORM OF DISSIPATION FUNCTION Let us base the definition of anisotropic viscous flu ids on the concept of dissipation function D. In this case, the stress tensor may be represented as follows:
σ ij = − pδ ij + σ ij (eks ).
(1)
Here, eij = (v i, j + v j,i ) 2 is the symmetric part of the velocity gradient, p is the pressure independent on the velocity gradient, and σ ij ( eks ) is the dissipative part of the stress tensor. Fluids are distinguished by the absence of shear stresses at rest. Therefore, the part of stress tensor that is determined by pressure p, is isotropic.
614
RHEOLOGY OF COMPLEX ANISOTROPIC FLUIDS
Here and below, we shall consider isothermal pro cesses. In this case, dissipation function D for a viscous fluid with an arbitrary anisotropy is equal to
D ≡ TPs = σ ij (eks )eij ,
(2)
where T is the temperature and Ps is the entropy pro duction. The scalar function of tensor argument D(e) is equal to the product of thermodynamic flux σ ij and thermodynamic force eij . According to Eq. (2), the dissipation processes that lead to an increase in the entropy due to the stresses can only take place at movement. The development of specific models for viscous fluids is associated with the assessment of dis sipative stresses. In anisotropic fluids, dissipative stress tensor σ ij is an anisotropic function of strain rate ten sor eij . In this case, the principal directions of these tensors do not coincide with one another. Let us consider a simple anisotropic relation
σ ij = ηijkeeke.
(3)
This constitutive equation is the generalization of the Newtonian law for an anisotropic case. The viscosity of anisotropic fluids is characterized by the fourth rank tensor ηijke. If anisotropic relation (3) is fulfilled, the dissipation in the system is determined by the follow ing expression:
D = eij ηijkeeke.
(4)
In order to completely define the dissipation func tion, it is necessary to give a concrete expression for the viscosity tensor, which satisfies the common equa tions of symmetry
ηijke = η jike, ηijke = ηijek .
(5a)
They are related to the symmetry of the strain rate and stress tensors. Moreover, it possesses higherorder symmetry
ηijke = ηkeij .
(5б)
ηiike = 0, ηijkk = 0.
(6)
These restrictions result from Eq. (4) for the dissipa tion function with regard to the condition of incom pressibility ekk = 0 and with the application of the decomposition of the general fourth rank tensor into simpler tensors, i.e., the isotropic, deviator and nonor components [19]. Due to the properties (5), viscosity tensor may be represented in the following form: n
∑η a
α α ijke ,
n ≤ 6.
(7)
α =1
COLLOID JOURNAL
Vol. 73
This expression is called the spectral decomposition of the viscosity tensor [18]. It describes the tensor via α eigenvalues of ηα and corresponding eigentensors aijke . The set of all eigenvalues of the viscosity tensor is referred to as its spectrum. In the general case, it con sists of six real numbers. Some of them may coincide with each other. Note that the components of the vis cosity tensor ηijke depend on a selected coordinate sys tem and are not invariant characteristics of the prop erties of anisotropic fluids. Their material parameters are the eigenvalues of viscosity tensor ηα. These dissi pative coefficients may be considered to be coordi nates of the viscosity tensor relative to orthogonal ten α sor basis aijke , α = 1,…, n. An important role in the rheology of anisotropic α fluids is played by eigentensors of viscosity aijke . They determine the principal directions of the anisotropy in a fluid and possess properties similar to the conditions of the orthonormalization of unit vectors as shown below: n
I ijke
α = ∑ aijke ,
α α α aijke akemn = aijmn ,
α =1
α β aijkeakemn
(8)
= 0, для α ≠ β,
where I ijke = ( δ ik δ je + δ ieδ jk ) 2 is a fouth rank unit tensor. Eigentensors are idempotent and mutually orthogonal. Being multiplied by themselves, idempo α tent tensors aijke remain unchanged. Therefore, the use of structural formulas (7) noticeably simplifies the analysis of rheological effects relevant to the viscosity anisotropy. The construction of the spectral decompo sitions for specific anisotropic fluids implies the deter mination of eigentensors, which is a rather difficult problem. However, when multiple eigenvalues are absent, simple explicit formulas can be obtained for them as follows: 6
These important expressions are related to the exist ence of dissipation function (4). In the case of incom pressible fluids, the viscosity tensor possesses the fol lowing additional properties:
ηijke =
615
No. 5
2011
α aijke
=
ηijke − ηβ I ijke , β ≠ α. η α − ηβ β=1
∏
(9)
Substituting spectral decomposition (7) into expression (4), we derive the following canonical form of the dissipation function: n
D=
∑η e ,
n ≤ 6.
2 α α
(10)
α=1
Here, the following intensities were introduced:
(
eα = eijαeijα
)
12
,
(11)
α where eijα = aijke eke are the principal strain rates. According to the second law of thermodynamics, the dissipation function describes the entropy increase due to internal processes and, therefore, cannot be
616
VOLKOV, KULICHIKHIN
negative D ≥ 0. The necessary and sufficient condition for the nonnegativity of dissipation function (10) is the nonnegativity of the principal viscosities: η1 ≥ 0,…, ηn ≥ 0, n ≤ 6.
(12)
The extremal simplicity of these conditions should be noted. Thus the canonical form, to which the dissipa tion function for anisotropic viscous fluids is reduced, demonstrates the fundamental significance of princi pal values (eigenvalues) of viscosities ηα. An important feature of these dissipative coefficients is that they cannot be negative. The number of viscosity coefficients markedly diminishes in the important particular case, when a fluid is incompressible and transversely isotropic. Its properties remain unchanged upon rotation by an arbitrary angle relative to the axis and upon any reflec tion relative to the plane that comprises this axis. Con sequently, director n is oriented along this axis. The dissipation function of this anisotropic fluid is described applying three viscosity coefficients as fol lows: 3
D=
∑η e e . α α α ij ij
(13)
α=1
Eigentensors for the viscosity of an incompressible transversely isotropic fluid are determined by the fol lowing relations:
(
)
aijke = 1 δik δ je + δ jk δie − δij δ ke , aijke = mij mke, 2 (14) 3 ⊥ ⊥ ⊥ ⊥ 1 aijke = δik n je + nikδ je + δien jk + nieδ jk . 2 1
⊥ ⊥
⊥
⊥
⊥ ⊥
(
2
)
(
)
Here, mij = 2 3 nij − δij⊥ 2 , while nij = ni n j and
δij⊥
= δij − ni n j are the orthogonal generating tensors. In this situation, principal strain rate tensors eijα have the following form:
eij = eij − eij − eij , eij = 1 nkeeke(3nij − δij ), 2 3 eij = nikekj + eik nkj − 2nijkseks. 1
2
3
2
(15)
In this case, strain rate tensor eij is expressed as the sum of three pairwise orthogonal tensors eijα. A degenerate case is the classical model of an iso tropic viscous fluid in which any direction is the prin cipal one. In the case of an isotropic viscous incom pressible fluid, for which η1 = η2 = η3, dissipation function (13) is reduced to the known Rayleigh func tion
D = 2η eij eij ,
(16)
into which the denotation η1 = 2η. has been intro duced. It is the quadratic form of its arguments. The
positivity of the Rayleigh dissipation function is pro vided by the condition η > 0. SPECTRAL DECOMPOSITION OF STRESSES The concept of stress potentiality may be extended to anisotropic fluids. The canonical representation of dissipation function (10) supports the validity of the constitutive equations
σijα = 1 ∂Dα . 2 ∂eij
(17)
Hence, the dissipation function plays the role of the α potential of principal stresses σijα = aijke σ ke . Expres sions (17) establish the following linear relations between generalized thermodynamic forces and flows: σ1ij = η1eij1 ,…, σijn = ηneijn, n ≤ 6,
(18)
where n is the number of viscous modes. Principal vis cosities ηα define the energy of these modes. Thus, for an arbitrary anisotropic fluid, the gener alized Newtonian law may be presented in the form of a set of n ≤ 6 independent linear equations. Constitu tive relations (18) are identical to the decomposition of the stress tensor into mutually orthogonal tensors; i.e., n
σij + pδij =
∑η e , α α ij
n ≤ 6.
(19)
α=1
At the same time, these relations have a clearer physi cal meaning. The set of constitutive relations (18) characterizes the viscous drags in the principal direc tions of rheological anisotropy. It generalizes the known representation of the stress tensor for an isotro pic viscous compressible fluid in the form of two equa tions for the proportionality of spherical and deviator components of the stress and strain rate tensors as fol lows:
(
)
(20) σ1ij = 2η eij − 1 δij ekk , σij2 = 3ζ ekk δij , 3 where η1 = 2η and η2 = 3ζ. The first relation defines the linear law of the shear drag. Therefore, η is referred to as the shear viscosity coefficient. The second rela tion expresses the linear law of the volume drag; there fore, ζ is the volume viscosity coefficient. In the case of incompressible isotropic and aniso tropic fluids, without a loss of generality, we may believe that the dissipative stress tensor has a zero trace. σ ii = 0.
(21)
This result follows from Eq. (2) for the dissipation function with regard to the condition of incompress COLLOID JOURNAL
Vol. 73
No. 5
2011
RHEOLOGY OF COMPLEX ANISOTROPIC FLUIDS
ibility ekk = 0 and the representation of the general dissipative stress tensor σ ij in the following form:
σ ij = aδij + bij + cij ,
(22)
where a = 1 σ ii , bij = 1 (σ ij + σ ji ) − 1 σ kk δij , cij = 1 ( σ ij − σ ji ) . 3 2 3 2 Here, bij is the symmetric stress deviator. Thus, if ther modynamic force eij is a traceless symmetric tensor, the relevant thermodynamic flow σ ij is also symmetric and traceless. Given this, the pressure in incompress ible (isotropic and anisotropic) fluids is evaluated as follows:
σ11 + σ 22 + σ33 (23) . 3 For an isotropic viscous incompressible fluid, the validity of this equation was experimentally confirmed [20]. According to Eqs. (15) and (18), for a transversely isotropic incompressible fluid, the rheological equa tion is simplified and reduced to the following three independent relations: p=−
(
)
σ1ij = η1 eij − eij2 − eij3 , σij2 = η2I 0 (3nij − δij ) 2, σ3ij = η3 ( nik ekj + eik nkj − 2I 0nij ) ,
(24)
where I 0 = nkeeke is the combined invariant of the two tensors. These relations are equivalent to the following more compact rheological equation:
σ ij = 2η⊥eij + 2 (μ|| − η⊥ ) eij2 + 2 ( η|| − η⊥ ) eij3,
(25)
where η⊥ = η1 2, μ || = η2 2, η|| = η3 2,and σ ij is the deviator dynamic stresses. In the general case, a trans versely isotropic incompressible fluid is characterized by one transverse η⊥ and two longitudinal (relative to the director) η||, µ || viscosities. It is of importance that these viscosities have a certain physical meaning; therefore, they can be measured. Viscosities η|| and η⊥ determine the shear drags along and normal to the director (see figure). The second longitudinal viscosity µ || characterizes the normal stress applied to a plane normal to the director; therefore, it is called normal viscosity. The concept of the normal viscosity, which is determined by rheological equation (25), was first introduced in [18]. The values of the principal viscos ities depend on the physical nature of a fluid. They can be obtained experimentally or based on microscopic theory. Constitutive equation (25) is of principle signifi cance. Using this equation, we may formulate the sim plest rheological law for anisotropic fluids. Its signifi cance for the continuum mechanics is comparable with the significance of the Newtonian law. It should be noted that constitutive relation (25) is reversible COLLOID JOURNAL
Vol. 73
No. 5
2011
617
with respect to the strain rate tensor and contains a term proportional to δij . It has the same pattern as pδij , and ensures the fulfillment of the condition for the incompressibility of a fluid −1 (26) ekk = ηkkmn σ mn = 0. In this case, p is the pressure in the ordinary sense (23). −1 Inverse viscosity tensor ηijke is described by the rela −1 tion ηijkl ηklmn = I ijmn. In the general case, this relation has the following form:
n
−1 ηijke
=
∑ η1 a
α ijke,
α=1
n ≤ 6.
(27)
α
The spectral theory of anisotropic fluids is advanta geous in the simplicity of the calculation of tensor −1 . ηijke The stressed state of a fluid may vary and the prin cipal axes of the rheological anisotropy may rotate. Therefore, equations that describe the dynamics of these axes must be involved into the consideration. The orientation of simplest anisotropic fluids is char acterized by single vector n . A formula describing this vector can be derived from the equation for the inter nal rotation, which is referred to as the moment equa tion [4]. In the linear approximation with respect to the velocity gradient, this equation is as follows:
Dni (28) = λ ( eis ns − nimnemn ) . Dt where Dni Dt = ni − ωij n j is the corotation time deriv ative and nimn = ni nmnn. The orientation of director n is governed by flow and depends on the velocity gradient. According to Eq. (28), it depends significantly on dimensionless constant λ. At λ < 1 the director changes periodically with time. For stationary shear flows of anisotropic fluid with λ ≥ 1, the director acquires a stationary orientation, which is indepen dent of the shear rate. It is oriented at some angle to the flow rather than along it. When the geometry of the flow changes, the orientation of the director changes as well. Calculations performed based on orientation equation (28) demonstrate that simple extension causes a stronger orientation effect than simple shear flow. DIRECTED FLOWS The introduction of the principal directions of vis cosity tensor makes it possible to distinguish preferable flows for different types of anisotropic fluids. In this case, the directed shear flow plays a key role. The kine matics of such flow is described by the velocity gradi ent having the following form:
v ij = γ bi g j , bi g i = 0,
(29)
618
VOLKOV, KULICHIKHIN
n
η||
η⊥ Schematic representation of shear flows directed along and normal to director n.
where γ is the shear rate. Directions of the shear and the velocity gradient are characterized by orthogonal unit vectors b and g, respectively. Transversely isotropic fluids have a single preferable direction, which is determined by director n. For this fluid, two principal shear flows may be distinguished (figure). The first of these is a longitudinal shear flow that is parallel to the director of the fluid. The second principal flow is the transverse shear flow, which is characterized by the conditions b ⊥ n and g ⊥ n. and is perpendicular to the director of the fluid. The longitudinal and transverse shear flows divide the properties of the fluid with respect to the direction that has a particular physical significance. Two shear viscosities may be determined for these stationary shear flows as follows: ηL =
⊥
σ12 σ , ηT = 12 . γ || γ ⊥ ||
(30)
For a transversally isotropic incompressible fluid (25), we obtain the following: (31) ηL = η||, ηT = η⊥. The longitudinal and transverse shear flows are real ized at shear rates γ || and γ ⊥ respectively. The values of || ⊥ and σ12 determine the shear stresses under these σ12 directions. Transversally isotropic fluids are distin guished by the fact that the shear viscosities are equal in all directions normal to the director. This result indicates that these fluids are isotropic in the trans verse direction.
Now, let us consider a directed uniaxial elonga tional flow. It can be applied to measure normal vis cosity µ ||. Elongational flows represent flows with sym metric velocity gradient v ij which is determined by the spectral decomposition of the following form:
v ij = ε1ki k j + ε 2l il j + ε3mi m j .
(32)
Here, ε α are eigenvalues, while k, l , and m are mutu ally orthogonal eigenvectors. Given this, the following relation is fulfilled: (33) δij = ki k j + lil j + mi m j . It represents the unit tensor as the sum of dyads con structed from three mutually orthogonal unit vectors. For incompressible fluids, the eigenvalues satisfy the condition ε1 + ε 2 + ε 3 = 0. The uniaxial elongational flow is described by the following relations: (34) ε1 = ε, ε 2 = ε 3 = − ε . 2 Independent scalar parameter ε > 0 is called the elon gation rate. Thus, the kinematics of the uniaxial elon gational flow is determined by the following tensor:
(35) v ij = ε ki k j − ε l il j − ε mi m j . 2 2 Using relation (33), it may be represented in a more convenient and compact form as follows: v ij = ε (3ki k j − δij ) . 2 COLLOID JOURNAL
Vol. 73
(36) No. 5
2011
RHEOLOGY OF COMPLEX ANISOTROPIC FLUIDS
In this representation, elongation direction k, is explicitly specified, with the directions normal to it being equivalent. The longitudinal and transverse elongational flows are characterized by conditions k n and k ⊥ n,, respectively. In these cases, the follow ing simple expressions are derived for dissipative stresses from Eq. (25):
σ ij = 2μ eij , σ ij = 2μ ⊥eij .
(37)
An analysis of these equations with regard to the rela tion e11 = = − 2e22 = − 2e33 for uniaxial elongation yields the following expressions for the longitudinal and transverse elongational viscosities:
λ L = 3μ, λT = 3η⊥.
(38)
These equations are the generalizations of the Trouton law for transversally isotropic fluids. Upon uniaxial elongation, the viscosity is estimated as ηe = ( σ11 − σ33 ) e11 . Expressions (38) show that, at a fixed orientation of a fluid, the elongational viscosity is determined ambiguously and that it depends on the elongation direction. The shear and elongational viscosities in the longi tudinal and transverse directions (with respect to the director) are related as follows:
λ L = 3αηL, λT = 3ηT ,
(39)
where α = μ η|| is a dimensionless constant. Hence, upon elongation in these orthogonal directions, the rheological properties of the important subclass of transversally isotropic fluids characterized by the rela tion μ = η|| ≠ η⊥, are described by the Trouton for mula. CONCLUSIONS The rheology of anisotropic fluids is of extremal importance for understanding the transport properties of colloidal and polymeric systems. The study of the anisotropy of viscosity deserves especial attention. In this work, the theory of anisotropic viscous fluids is described, which is based on the spectral (canonical) representation of the dissipation function. The spec tral approach enables us to derive rheological equa tions for the state of anisotropic fluids with complex microstructures. It establishes the relation of principal viscosities and the principal basis tensors with eigen modes of anisotropic fluids. These modes are orthog onal for symmetric fluids. It should also be noted that the spectral theory makes it possible to solve a rather complex problem concerning the comparison of the tensor properties of fluids. Invariant characteristics of the viscous properties of anisotropic fluids are the eigenvalues of the viscosity tensor. These dissipative coefficients are referred to as principal viscosities. An important feature of these coefficients is their nonnegativity. As was established in this work, the principal viscosities are the coeffi COLLOID JOURNAL
Vol. 73
No. 5
2011
619
cients of the spectral decomposition of the dissipation function. This fact indicates their fundamental signif icance. The principal viscosities are independent of the orientation of a fluid and the direction of flow. Thus, in the case of anisotropic fluids, they may be considered to be the measures of material properties invariant with respect to measurement methods. The method of eigenmodes appears to be very fruit ful when studying flows of anisotropic fluids. It enables us to distinguish the principal flows for fluids with dif ferent symmetry. Within the framework of the devel oped approach, the peculiarities of the rheological behavior were investigated for flows of transversally isotropic fluids. In this case, there is one specific direc tion, along which the director is oriented. In this most important case of anisotropy, the number of macro scopic parameters is substantially decreased. The rheological properties of a transversally isotropic incompressible fluid are characterized by three princi pal viscosities. When studying fluids with a single pref erable direction, an important role is played by the shear and elongational flows directed along and nor mal to the director. Note that the rheological anisot ropy results in qualitative changes in the patterns of stress distributions based on isotropic concepts. For example, the Trouton law becomes invalid. The available experimental data confirm the gen eral conclusion that the viscosity anisotropy must be taken into account when investigating the flows of flu ids with orientable microstructure. Anisotropic viscos ity of colloidal solutions with micellar structure was experimentally measured in [21]. The anisotropy of the rheological properties in colloidal systems results from orientation of colloidal particles. Anisotropy of viscosity was experimentally found for other fluids. Special attention should be focused on the rheological anisotropy arising in flowing polymer solutions. In a flow, polymer molecules acquire the shape of elon gated ellipsoids. In flowing dilute polymer solutions, even low velocity gradients can markedly affect the orientation of macromolecules. The anisotropy of vis cosity in flexiblechain polymer solutions was mea sured in [22], where noticeably different shear viscos ities were observed for different directions. Effects of viscosity anisotropy are most pronounced in meso morphic polymer solutions [23]. The rheological study of flexiblechain polymer melts demonstrated that flow disturbs the isotropy in the macroscopic properties of the media [24]. REFERENCES 1. 2. 3. 4.
Duhem, P., Ann. Ecole Norm., 1893, vol. 10, p. 187. Anzelius, A., Ann. Univ. Uppsala, 1931, no. 1, p. 1. Ericksen, J.L., Kolloid Z., 1960, vol. 173, p. 117. Volkov, V.S. and Kulichikhin, V.G., Rheol. Acta, 2000, vol. 39, p. 360. 5. Ericksen, J.L., Trans. Soc. Rheol., 1960, vol. 4, p. 29.
620
VOLKOV, KULICHIKHIN
6. Abhiraman, A.S. and George, W., J. Polym. Sci., 1980, vol. 18, p. 127. 7. Allen, S.J. and De Silva, C.N., J. Fluid Mech., 1966, vol. 24, p. 801. 8. Shmakov, Yu.I. and Taran, E.Yu., Inzh.Fiz. Zh., 1970, vol. 18, p. 1019. 9. Shmakov, Y.I. and Begoulev, P.B., Rheol. Acta, 1974, vol. 13, p. 424. 10. Hinch, E.J. and Leal, L.G., J. Fluid Mech., 1975, vol. 71, p. 481. 11. Hand, G.L., Arch. Ration. Mech. Anal., 1961, vol. 7, p. 81. 12. Leslie, F.M., Arch. Ration. Mech. Anal., 1968, vol. 28, p. 265. 13. De Gennes, P.G. and Prost, J., The Physics of Liquid crystals, New York: Oxford Univ. Press, 1993. 14. Stewart, I.W., The Static and Dynamic Continuum The ory of Liquid crystals: A Mathematical Introduction, London: Taylor and Francis, 2004. 15. Volkov, V.S. and Kulichikhin, V.G., J. Rheol. (N. Y.), 1990, vol. 34, p. 281.
16. Volkov, V.S. and Kulichikhin, V.G., Rheol. Acta, 2007, vol. 46, p. 1131. 17. Volkov, V.S., Spectral Decomposition in Anisotropic Flu ids, LC: Communications, 2008. 18. Volkov, V.S., Vysokomol. Soedin., Ser. A, 2010, vol. 52, p. 1903. 19. Skhouten, Ya.A., Tenzornyi analiz dlya fizikov (Tensor Analysis for Physicists), Moscow: Nauka, 1965. 20. Loitsyanskii, L.G., Mekhanika zhidkosti i gaza (Fluid and Gas Mechanics), Moscow: Nauka, 1978. 21. Nikulin, V.A., Anisotropy of viscosity and its experi mental measurement, Preprint of Kamsk. Inst. of Humanitarian and Engineering Technologies, Izhevsk, 2003, no. 203, p. 22. 22. Highgate, D.J. and Whorlow, R.W., Br. J. Appl. Phys., 1967, vol. 18, p. 1019. 23. Malkin, A.Ya., Vasil’eva, R.V., Belousova, T.A., and Kulichikhin, V.G., Kolloidn. Zh., 1979, vol. 61, p. 200. 24. Boiko, B.B., Insarova, N.I., and Lugina, A.S., Mekh. Polim., 1965, no. 5, p. 13.
COLLOID JOURNAL
Vol. 73
No. 5
2011
