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Reciprocal and related theorems for nematic liquid crystals
International Journal of Engineering Science 43 (2005) 1185–1205 www.elsevier.com/locate/ijengsci
Reciprocal and related theorems for nematic liquid crystals C. Atkinson
a,*
, P.J.S. Pereira
a,b
a
b
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK ´ dio A´rea Cientı´fica de Matema´tica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emı Navarro, 1949-014 Lisboa, Portugal Received 4 March 2005; accepted 4 March 2005 Available online 21 September 2005
Abstract Reciprocal and related theorems are considered for uniaxial and biaxial nematic liquid crystals. In the general case, where the directors of real and reciprocal materials are assumed to coincide, integral relations between velocity and stress fields of the real and reciprocal materials are obtained. Simplified expressions are also obtained for the high viscosity and elasticity limits. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Reciprocal theorems for the classical equations of elasticity and fluid mechanics are well known. For example, for a Newtonian viscous fluid contained in the region Q bounded by a closed surface S with orientation given by an outward unit vector u normal to S, one has the result ZZ ZZ b bv i tij dS j vi t ij dS j ¼ ð1:1Þ S
*
S
Corresponding author. E-mail address: [email protected] (C. Atkinson).
0020-7225/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.03.008
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where both sets of velocity fields and stress tensors of the real and reciprocal materials (vi, tij) and ðbv i ; bt ij Þ (i, j = 1, 2, 3) satisfy the same constitutive equations and equations of motion (dSj = uj dS). The result is easily confirmed by means of the divergence theorem (see, e.g., [1,2]). A similar result for elasticity is ZZ ZZ b d ibt ij dS j ¼ ð1:2Þ d i tij dS j S
S
where tij is the stress tensor associated with the displacement field di (i, j = 1, 2, 3) in anisotropic bodies with the equilibrium equations tij,j = 0 and bt ij;j ¼ 0 also being satisfied. In this case, the elastic constants of both materials are the same. In all of the above equations the summation convention over repeated indices is used and a comma denotes partial differentiation with respect to an appropriate Cartesian coordinate xi (i = 1, 2, 3). Reciprocal theorems of this kind are the basis of the boundary element method and can be used to obtain information concerning solutions to boundary value problems and can occasionally be used to find the full solution, e.g., it provides a way to compute fields for arbitrarily shaped dislocation loops in an infinite solid (see [3]) or to find the singular stress fields in the neighbourhood of notches and corners in anisotropic media (see d i field of the reciprocal material can be selected for some [4]). In the above formulations the bv i or b auxiliary purpose and it does not have to be a physically attainable velocity or displacement field. It is the purpose of this paper to try to extend the above results to uniaxial and biaxial nematic liquid crystals. The hydrodynamic behaviour of a uniaxial nematic liquid crystal depends on both a velocity vector field v and a unit director vector field n with the spatial orientation defined by the twist and tilt angles. The director field is also considered to be non-polar such that the states n and n are indistinguishable. The velocity field is constrained by the incompressibility condition vi,i = 0 and the director field by nini = 1. For biaxial nematic liquid crystals there is one pair of orthonormal director vector fields n and m. The most general result derived (see Section 3) is of the form ZZ h ZZ i b b1 Þvj dS j et ij ðb e bv i t ij ðp þ W 1Þbv j dS j ¼ vi b pþW S ZSZ Z oni onj bv i;j l2 nj þ l3 ni þ ot ot QðSÞ ob ni ob nj onk ob nk b b b l3 vi;j b l2 n k;i k 1 nj þ b n i þ bv i nk;i k1 vi b dV ot ot ot ot ð1:3Þ where the director fields of both real and reciprocal materials are assumed to coincide at the time at which the relation above holds. In (1.3) et is the second order viscous stress tensor, p(xi, t) the hydrostatic pressure, W(ni, ni,j) the elastic energy of the director, and the scalar energy density function 1(xi, ni) represents some external body forces (i, j = 1, 2, 3). The constants lm (m = 1, 2, . . ., 6) are viscosity coefficients, k1 is the rotational viscosity coefficient, Kp (p = 1, 2, 3, 4) are elastic constants, xi (i = 1, 2, 3) Cartesian coordinates and t the time. The components of the reciprocal fluid are identified by a circumflex accent. Throughout the paper, unless otherwise stated, indices take the values 1, 2, and 3. For this theorem to hold various relations must hold between the viscosities of the real and reciprocal materials (see Section 3).
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In all cases we consider that the fluid inertial terms of both flows in the linear momentum equations are neglected when compared with the pressure, viscous stress tensor and elastic terms. In fact, the order of magnitude of a typical term of the force per unit volume due to the viscous stress is lU/L2, where U is the characteristic velocity, L the length, which is of the order of the thickness of the sample (10–100 lm) down to 1 lm, and l the viscosity, which is of the order of 101 poise. Therefore, in most cases the fluid inertial term qU_ ¼ wqU can be neglected up to angular frequencies, w, of the order of 106 Hz and similarly for the reciprocal flow (see [5–7]). In this last expression q is the mass density and U_ is the material time derivative of U. In addition to the main result (1.3) simplified results are obtained in the high viscosity and elasticity limits. The result (1.3) is also generalized to biaxial nematic liquid crystals (see Section 5). The plan of this paper is as follows. We begin with the Ericksen–Leslie partial differential equations in Cartesian coordinates (see Section 2). In Section 3, we investigate a reciprocal material associated with an anisotropic nematic liquid crystal described by the full asymmetric viscous stress tensor. A key feature here is that the reciprocal velocity field bv needs only to satisfy the linear momentum equations and incompressibility condition. The director field b n is chosen to be instantaneously the same as that in the real fluid. In Subsection 4.1 an incompressible isothermal viscous fluid in the isotropic viscosity phase of a nematic liquid crystal is considered. Similar solutions to those of Newtonian viscous fluids are obtained. The high viscosity limit is studied in Subsection 4.2. The equations necessary to model these boundary and initial value problems are deduced and a reciprocal material associated with a nematic liquid crystal with no elastic effects is defined. Here we find that the viscous stress tensor is symmetric and an equation for the indeterminacy c (see Section 2) is also determined. We also consider the case where the viscous stress tensor is only composed of the isotropic viscosity l4. The high elasticity limit is investigated in Subsection 4.3. The governing equations are obtained and an integral relation between the velocity and stress fields of both flows is found. Finally, a generalization of our main result (1.3) to biaxial nematic liquid crystals is outlined in Section 5. The equations in Cartesian coordinates governing the hydrodynamic behaviour of biaxial nematic liquid crystals are given in Appendix A. A generalization of the result of biaxial nematics to the high elasticity limit is given in Appendix B.
2. Governing partial differential equations The equations describing the hydrodynamic behaviour of a nematic liquid crystal are well known [8–10]. For completeness, these equations are presented here in Cartesian coordinates. The flow of a nematic liquid crystal is described by the velocity field v together with a director field n, which is considered in this work as a non-polar unit vector giving the orientation of the anisotropic axis in these transversely isotropic liquids. With the assumption of incompressibility, the equations are the constraints div v ¼ 0;
jnj ¼ 1
ð2:1Þ
together with the balance laws: linear momentum equations q_vi ¼ F i þ tij;j
ð2:2Þ
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angular momentum equations r€ ni ¼ Gi þ gi þ S ij;j
ð2:3Þ
and the constitutive relations: the equations for the stress tensor t tij ¼ pdij
oW nk;i þ et ij onk;j
ð2:4Þ
the equations for the tensor S S ij ¼ ni bj þ
oW oni;j
ð2:5Þ
the equations for the vector g gi ¼ cni ðni bj Þ;j
oW þe gi oni
ð2:6Þ
where the energy function is defined by 2W ¼ K 1 ðni;i Þ2 þ K 2 ðijk ni nk;j Þ2 þ K 3 ni nj nk;i nk;j þ ðK 2 þ K 4 Þ½ni;j nj;i ðni;i Þ2
ð2:7Þ
the viscous stress tensor et is given by et ij ¼ l1 nk np Akp ni nj þ l2 N i nj þ l3 N j ni þ l4 Aij þ l5 Aik nk nj þ l6 Ajk nk ni
ð2:8Þ
and the vector e g is given by e g i ¼ k1 N i k2 Aik nk
ð2:9Þ
the rate of strain tensor A is defined by 2Aij ¼ vi;j þ vj;i
ð2:10Þ
the vector N is given by 2N i ¼ 2n_ i 2Aij nj
ð2:11Þ
where the vorticity tensor A* is defined by 2Aij ¼ vi;j vj;i
ð2:12Þ
the material time derivatives of the director and velocity fields are given by n_ i ¼
oni þ vj ni;j ; ot
v_ i ¼
ovi þ vj vi;j ot
ð2:13Þ
and k1 ¼ l3 l2 ;
k2 ¼ l6 l5
ð2:14Þ
The usual summation convention (summation over repeated indices) is used. A comma denotes partial differentiation and the superposed dot a material time derivative. In the conservation of linear momentum Eq. (2.2), q is the density, F any external body force and t the stress tensor
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(asymmetric). In the conservation of angular momentum Eq. (2.3), r is a constant inertial coefficient, G a generalized body force arising from a body couple present due to magnetic or electric fields, g is a generalized intrinsic body force that depends upon the director vector field n through Eq. (2.6), and S is a generalized stress tensor. The inertial term in these equations, rn€i , is usually assumed negligible and hence omitted in most treatments. In the constitutive relations (2.4), the scalar p is the arbitrary hydrostatic pressure following from the assumption of incompressibility, while c in (2.6) and the vector b in (2.5) represent similar indeterminacies due to the constraint on the additional kinematic variable n. The vector N in (2.11) is the co-rotational time flux of n, which describes the internal motion of the director with respect to the fluid. In (2.10) A is the velocity gradient tensor (symmetric), i.e., the rate of strain tensor, whereas A* in (2.12) is the vorticity tensor (antisymmetric). In (2.8) et is the viscous stress tensor, which is an even function of n. The vector e g in (2.9) is the hydrodynamic part of g and ðn e g Þ define the viscous torque on the director with coefficients of friction given by k1 and k2, which are defined by (2.14) and usually called rotational and torsional viscosities, respectively. In (2.4) dij is the well-known Kronecker symbol and in (2.7) ijk is the Levi–Civita symbol. The elastic torque on the director field is given by n · h, where h is the molecular vector field defined by oW oW hi ¼ þ ð2:15Þ oni oni;j ;j Here W is the energy function and Kp (p = 1, 2, 3, 4) are the usual elastic coefficients related by Eq. (2.7), whereas in (2.8) lm (m = 1, 2, . . ., 6) are the viscosity coefficients. The elastic and viscosity coefficients are assumed constant if thermal gradient effects are ignored. Various authors have discussed the possibility of restrictions on these coefficients for physically acceptable behaviour. In particular, the coefficients in (2.8) are to be consistent with the thermodynamic inequality et ij vi;j e g i n_ i P 0
ð2:16Þ
Other restrictions have been suggested, e.g., the Parodi relationship [11] among the viscosity coefficients l2 þ l3 ¼ l6 l5
ð2:17Þ
If in (2.2) and (2.3) we neglect both fluid and director inertial terms we obtain the equations F i þ tij;j ¼ 0
ð2:18Þ
Gi þ gi þ S ij;j ¼ 0
ð2:19Þ
and which together with Eqs. (2.4)–(2.14) as well as (2.1) and appropriate boundary and initial conditions enable (in principle) the velocity and director fields to be determined. A slightly different form of these equations can be deduced. We consider an incompressible isothermal nematic liquid crystal with external body forces given by Fi ¼
o1ðxi ; ni Þ oxi
and Gi ¼
o1ðxi ; ni Þ oni
ð2:20Þ
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where 1(xi, ni) is some scalar energy density function, e.g., 1(xi, ni) can be the magnetic or electric energy. Note that the case of no external body forces is described by 1ðxi ; ni Þ ¼ C 1 ðC 1 2 RÞ. Differentiating (2.4) we obtain, tij;j
o oW ¼ p;j dij nk;i þ et ij;j oxj onk;j
Inserting (2.5) and (2.6) into the angular momentum Eq. (2.19) we obtain oW o oW þe gi þ þ Gi ¼ 0 cni oni oxj oni;j
ð2:21Þ
ð2:22Þ
Assuming that the energy function only depends on W = W(nk, nk,j), we have oW oW oW ¼ nk;i þ nk;ji oxi onk onk;j Using (2.22), (2.21) becomes o oW oW nk;ij þ et ij;j tij;j ¼ p;j dij nk;i oxj onk;j onk;j oW oW ¼ p;j dij nk;i e g k cnk Gk nk;ij þ et ij;j onk onk;j
ð2:23Þ
ð2:24Þ
Considering a non-polar and unit director field we can write nk nk ¼ 1 () nk;i nk ¼ 0
ð2:25Þ
Thus, from (2.18), (2.20), (2.23), (2.24) as well as (2.25) we obtain, et ij;j ¼
o gk ðp þ W 1Þ nk;i e oxi
ð2:26Þ
Thus, the angular and linear momentum equations can be written in the form of (2.22) and (2.26), respectively.
3. Reciprocal materials for anisotropic nematic liquid crystals: The general case For the general case the full viscous stress tensor is asymmetric and the linear and angular momentum equations are also coupled through the energy function terms. Here we consider flow regimes with very small Reynolds number, i.e., the fluid inertial terms can be neglected. We can rewrite (2.8) in the form et ij ¼ C ijkp vk;p þ l2 n_ i nj þ l3 n_ j ni
ð3:1Þ
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where the fourth order rank tensor Cijkp is defined as 1 2l1 nk np ni nj þ l4 ðdki dpj þ dkj dpi Þ þ l5 ðnk nj dip þ np nj dik Þ C ijkp ¼ 2
1191
þl6 ðnk ni dpj þ np ni dkj Þ þ l3 ðni nk dpj ni np dkj Þ þ l2 ðnj nk dpi nj np dki Þ Similar expressions follow for a reciprocal material where (2.26) is replaced by o ob nk b b b b b b b et ij;j ¼ ðb p þ W b1 Þ b n k;i k 1 n k;p þ k 1 A kp b n p k 2 A kp b np þ bv p b oxi ot
ð3:2Þ
ð3:3Þ
where ob n ob n i j b b ijkp bv k;p þ b et ij ¼ C nj n i;p þ b ni n j;p þ bv p b þ bv p b l2 b l3b ot ot
ð3:4Þ
b ijkp as in (3.2) with all viscosity coefficients and director field components with the overbar with C circumflex. We now show that a reciprocal material can be defined such that the following result holds: Result 1. If ½v; n; et; p; W ; 1; lm ; K l (m = 1, 2, . . ., 6 and l = 1, 2, 3, 4) is an incompressible isothermal viscous flow of an anisotropic nematic liquid crystal associated with the full asymmetric viscous stress tensor, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then at a fixed time ZZ ZZ b b1 Þvj dS j et ij ðb e ½bv i t ij ðp þ W 1Þbv j dS j ¼ ½vib pþW S
S
oni onj bv i;j l2 þ nj þ l3 ni ot ot QðSÞ ob ni ob nj onk ob nk b b b l3 vi;j b l2 n k;i k 1 nj þ b n i þ bv i nk;i k1 vi b dV ot ot ot ot ZZZ
ð3:5Þ where the bv field is only required to satisfy the incompressibility condition bv i;i ¼ 0 and the linear momentum equations given by (3.3) with the auxiliary viscous stress tensor defined by (3.4). b ijkp are satisfied b ijkp ¼ C kpij . It can be shown that the conditions on C Result (3.5) then follows if C provided b n ¼ n, b l 1 ¼ l1 , b l 4 ¼ l4 , b k 1 ¼ k1 ¼ l3 l2 , b k 2 ¼ l2 þ l3 , 2b l 2 ¼ l6 l3 l5 þ l2 , l 5 ¼ l5 l2 þ l6 l3 , and 2b l 6 ¼ l3 þ l2 þ l5 þ l6 . Note that the re2b l 3 ¼ l3 l2 l5 þ l6 , 2b ciprocal director field is identical with the real director field at any fixed time for which (3.5) is applied and the terms involving ootbn i can be set to zero in (3.3), (3.4) as well as in (3.5) if required. Proof. Using the divergence theorem, the incompressibility conditions for both flows vj;j ¼ bv j;j ¼ 0, and (2.26), (3.1), (3.3) as well as (3.4), we can write
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ZZ S
b b1 Þvj dS j et ij þ ðb ½bv iet ij ðp þ W 1Þbv j vib pþW
ZZZ
oni onj bv i;j l2 nj þ l3 ni ot ot QðSÞ
ob ni ob nj onk ob nk b2 b b vi;j l l3 k1 n k;i b nj þ b n i þ bv i nk;i k1 vi b dV ot ot ot ot ZZZ ob nk b b b b b þ bv p b g k þ vi b C ijkp vk;p bv i;j bv i nk;i e n k;i k 1 n k;p þ k 1 A kp b n p k 2 A kp b np ¼ ot QðSÞ ob n ob n i j b ijkp bv k;p vi;j þ bv i;j l2 n_ i nj vi;j b þ bv p b þ bv p b C l2 n i;p b l3 n j;p b n j þ bv i;j l3 n_ j ni vi;j b ni ot ot oni onj ob ni ob nj onk ob nk b b bv i;j l2 l3 k1 l2 n k;i b nj þ l3 ni þ vi;j b nj þ b n i bv i nk;i k1 þ vi b dV ot ot ot ot ot ot
ð3:6Þ Q(S) is an arbitrary region bounded by a closed surface S, so the right hand side of (3.6) is zero provided b ijkp bv k;p vi;j ¼ 0 C ijkp vk;p bv i;j C
ð3:7Þ
and bv i;j ½l2 vp ni;p nj þ l3 vp nj;p ni bv i nk;i ½k1 vp nk;p þ k1 Akp np k2 Akp np b b np ¼ 0 b b vi;j ½b l 2 bv p b l 3 bv p b k 1 bv p b k1 A n i;p b nj þ b n j;p b n i þ vi b n k;i ½b n k;p þ b kp n p k 2 A kp b
ð3:8Þ
where (2.9), (2.11) and (2.13) have been used to obtain (3.8). Eq. (3.7) is satisfied provided b ijkp ¼ C kpij C
ð3:9Þ
To satisfy (3.8) we need to balance terms involving partial derivatives of the real and reciprocal velocity field components, i.e., ( " #) b k1 b kq b k2 A n k;p b nq n q ðbv q;k bv k;q Þ b vp bv i;j l2 ni;p nj þ l3 nj;p ni þ b 2 k1 bv p vi;j b l3 b ¼0 ð3:10Þ l2 b n i;p b nj þ b n j;p b n i þ nk;p nq ðvq;k vk;q Þ k2 Akq nq 2 as well as the remaining terms involving products of the real and reciprocal velocity field components bv i nk;i k1 vp nk;p vi b k 1 bv p b n k;i b n k;p ¼ 0
ð3:11Þ
In (3.10) we have used (2.12) and a similar equation for the reciprocal flow. From (3.11) we obtain a relation among the partial derivatives of the director field components k 1 ) of both flows, i.e., and coefficients of friction (k1 and b b n k;i b k1b n k;p ¼ nk;i k1 nk;p
ð3:12Þ
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We choose the director field of the reciprocal material b n to be identical to that of the real material, i.e., b n¼n
ð3:13Þ
Then (3.12) is satisfied provided b k 1 ¼ k1 ¼ l3 l2
ð3:14Þ
where we have used (2.14). Using (2.10) and a similar equation for the reciprocal flow as well as (3.10) we can write, ( " #) b b b b k1 k1 k2 k2 ¼0 n k;p b n q bv q;k þ b n q bv k;q b n q bv k;q b n q bv q;k vp bv i;j ½l2 ni;p nj þ l3 nj;p ni þ b 2 2 2 2 k1 k1 k2 k2 bv p vi;j ½b l2 b l3 b ¼0 n i;p b nj þ b n j;p b n i þ nk;p nq vq;k þ nq vk;q nq vk;q nq vq;k 2 2 2 2 ð3:15Þ From (3.13) and (3.15) we obtain ! ! b b k1 b k2 k1 b k2 ni;p nj þ l3 nj;p ni ¼ 0 l2 þ 2 2 2 2 k1 k2 k1 k2 b l2 þ l3 ni;p nj þ b nj;p ni ¼ 0 2 2 2 2
ð3:16Þ
which requires b l3 b l 2 ¼ l3 l2 k 1 ¼ k1 ¼ b b k 2 ¼ l2 þ l3
ð3:17Þ
k2 ¼ b l2 þ b l 3 ¼ l6 l5 Using the result that b n is identical with the actual director field n at the time of the calculations in (3.5), it can be shown that (3.9) is satisfied provided the reciprocal material coefficients are chosen as follows: b l 1 ¼ l1 b l 4 ¼ l4 2b l 2 ¼ l6 l3 l5 þ l2 2b l 3 ¼ l3 þ l6 l5 l2 2b l 5 ¼ l5 l2 þ l6 l3 2b l 6 ¼ l3 þ l2 þ l5 þ l6 where we have used (3.2) and a similar equation for the reciprocal flow.
ð3:18Þ
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To sum up, we have deduced the following conditions among the components of both flows: b n¼n b k 1 ¼ k1 ¼ l3 l2 b k2 ¼ l þ l 2
ð3:19Þ
3
together with the conditions shown in (3.18).
h
Note that if we use the Parodi relationship (2.17) [11] as well as the conditions among the viscosity coefficients of both flows given by (3.18) and (3.19), we find that b n¼n b k 1 ¼ k1 ¼ l3 l2 b k 2 ¼ k2 ¼ l2 þ l3 ¼ l6 l5 b l i ¼ li ði ¼ 1; 2; . . . ; 6Þ
ð3:20Þ
However, the Parodi relationship [11] comes from Onsagers reciprocal relations in irreversible processes. Some authors suggest that Onsagers relations do not apply to phenomena like heat conduction, viscosity and diffusion since there is no unambiguous way of selecting the fluxes and forces (see [12,13]). Even though available data indicate that (2.17) is satisfied within experimental limits this raises doubts about the universal validity of the Parodi relationship [11]. 4. Reciprocal materials for nematic liquid crystals: Limit cases 4.1. Isotropic viscosity phase In the isotropic viscosity phase only the term involving l4 remains in the viscous stress tensor equation. The Ericksen–Leslie partial differential equations reduce to those of Newtonian incompressible isothermal viscous fluids. Thus, (2.8) turns out to be equal to et ij ¼ l4 Aij
ð4:1Þ
Using (2.2), (2.4) as well as (4.1) and assuming no external body forces and that the inertial terms q_vi are neglected, we can write tij;j ¼ ðpdij Þ;j þ et ij;j ¼ ðpdij Þ;j þ l4 Aij;j ¼ 0 ð4:2Þ
where the energy terms oxoj onoWk;j nk;i have been dropped. Using the incompressibility condition (2.1) one can see that the pressure is as usual a harmonic function, i.e., satisfies the Laplace equation ,2p = 0. Thus, similar solutions to those of Newtonian viscous fluids are obtained. A similar result to that outlined in Section 1 for Newtonian viscous fluids can be stated as follows: Result 2. If ½v; n; et; p; l4 is an incompressible isothermal viscous flow in the isotropic viscosity phase of a nematic liquid crystal, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then at a fixed time
C. Atkinson, P.J.S. Pereira / International Journal of Engineering Science 43 (2005) 1185–1205
ZZ S
vibt ij dS j ¼
ZZ S
bv i tij dS j
1195
ð4:3Þ
which is satisfied if b l 4 ¼ l4 . Since it is available in standard textbooks (see, e.g., [1,2]) the proof is omitted here. 4.2. High viscosity limit In this subsection we investigate reciprocal materials associated with a symmetric viscous stress tensor where the interaction of the director field and the viscous flow is maintained through the viscous stress tensor equation. To satisfy this purpose, we assume a model flow regime where the elastic and fluid inertial terms are neglected, which is plausible in the high viscosity limit, i.e., very large Ericksen number and very small Reynolds number. The Reynolds number is described by the ratio of the inertial forces to the viscous forces, Re = qUL/l, whereas the Ericksen number is defined by the ratio of the viscous forces to the elastic forces, Er = lUL/K. In this last equation, K is a representative elastic constant and lU/L the stress. 4.2.1. Governing partial differential equations Using the general equations of Section 2 the equations governing the hydrodynamic behaviour of nematic liquid crystals in the high viscosity limit can be deduced. Using (2.2) and (2.4) and assuming no external body forces as well as no inertial terms q_vi (see Section 1), the linear momentum equations in the high viscosity limit are given by tij;j ¼ ðpdij Þ;j þ et ij;j ¼ 0 where the energy terms have been dropped. From (2.22) (see Section 2) we can write, oW o oW cni þe gi þ ¼0 oni oxj oni;j
ð4:4Þ
ð4:5Þ
since the director inertial terms r€ ni have been neglected. As Kj ! 0 ( j = 1, 2, 3, 4) in the high viscosity limit the energy terms vanish, so we obtain gi ¼ 0 cni þ e
ð4:6Þ
Using (2.9), Eq. (4.6) reduces to the following equation: cni k1 N i k2 Aik nk ¼ 0
ð4:7Þ
giving
k2 ni Aik nk c Ni ¼ k1 k2
ð4:8Þ
Using (4.7) a further equation can be deduced to define the indeterminacy c. As a non-polar and unit director field is considered, we have nini = 1 and, consequently, nini,k = 0. Using this result and (2.11) as well as the scalar product of (4.7) with ni, we obtain c ¼ k1 Aij nj ni þ k2 Aik nk ni
ð4:9Þ
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where the terms k1 n_ i ni ¼ ðk1 =2ÞðD=DtÞðni ni Þ ¼ 0
ðni ni ¼ 1Þ
Furthermore, the terms Aij nj ni ¼
3 X 3 X
Amp nm np
m¼1 p¼1
¼ n21 A11 þ n22 A22 þ n23 A33 þ n1 n2 ðA12 þ A21 Þ þ n2 n3 ðA23 þ A32 Þ þ n3 n1 ðA13 þ A31 Þ ¼ 0 since A11 ¼ A22 ¼ A33 ¼ 0
and Aij þ Aji ¼ Aij Aij ¼ 0
Thus, the indeterminacy c is given by c ¼ k2 Akp nk np
ð4:10Þ
Using (4.8) and (4.10), the internal motion of the director with respect to the fluid is given by k2 ð4:11Þ ½Aik nk Akp nk np ni Ni ¼ k1 Inserting (4.11) into (2.8) and using (2.14) as well as l5 (k2/k1)l2 = l6 (k2/k1)l3, we obtain the viscous stress tensor equations, which are valid in the high viscosity limit, i.e., k2 k et ij ¼ l1 þ ðl2 þ l3 Þ nk np Akp ni nj þ l5 2 l2 ½Ajk nk ni þ Aik nk nj þ l4 Aij ð4:12Þ k1 k1 Here the viscous stress tensor is symmetric but the interaction between the directors and the viscous fluid is also maintained through (4.12). Using (2.3), (2.5), (2.6), (2.9), (2.11), (2.13) as well as (4.10) and assuming no body forces, the angular momentum equations in the high viscosity limit where Kj ! 0 (j = 1, 2, 3, 4) associated with an incompressible isothermal nematic liquid crystal are given by oni k2 ½Aij nj Akp nk np ni ð4:13Þ þ vj ni;j ¼ Aij nj n_ i ¼ ot k1 where the director inertial terms r€ ni are neglected and the energy terms vanish in this limiting formulation. To close this set of equations a further equation is defined, div v ¼ 0
ð4:14Þ
since we have considered an
incompressible isothermal viscous flow. Moreover, if we assume that l1 þ kk21 ðl2 þ l3 Þ ¼ l5
k2 k1
l2 ¼ 0 in (4.12), the high viscosity limit equations reduce to those of
Newtonian incompressible isothermal viscous flows with an additional equation giving the orientation of the director field. After some simplifications using the Parodi relationship (2.17) [11] the coefficients of friction in the approximation mentioned above are given by k1 ¼ ðl1 l22 =l25 Þ and k2 ¼ ðl1 l2 =l5 Þ, so (k2/k1) = (l5/l2).
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4.2.2. Reciprocal materials for nematics in the high viscosity limit We can rewrite the result (4.12) in the form et ij ¼ C ijkp vk;p
ð4:15Þ
where the fourth order rank tensor Cijkp is defined as 1 k2 k2 C ijkp ¼ l l4 ðdki dpj þ dkj dpi Þ þ 2 l1 þ ðl2 þ l3 Þ ni nj nk np þ l5 2 k1 k1 2 ½ni nk dpj þ ni np dkj þ nj nk dpi þ nj np dki
ð4:16Þ
Here, tij ¼ pdij þ et ij
ð4:17Þ
We define similar expressions for a reciprocal material b b ijkp bv k;p et ij ¼ C et ij bt ij ¼ b p dij þ b
ð4:18Þ
bt ij;j ¼ ðb et ij;j ¼ 0 p dij Þ;j þ b b ijkp is as in (4.16) with all viscosity coefficients, director field and coefficients of friction where C with the overbar circumflex. We now show that a reciprocal material can be defined such that the following result holds: Result 3. If ½v; n; et; p; lm (m = 1, 2, . . ., 6) is an incompressible isothermal viscous flow of a nematic liquid crystal in the high viscosity limit, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then at a fixed time ZZ ZZ bv i tij dS j vibt ij dS j ¼ ð4:19Þ S
S
b ijkp ¼ C kpij ¼ C ijkp . The reciprocal director field b which is satisfied provided C n is identical with n at any fixed time for which (4.19) is applied and the reciprocal velocity field bv satisfies the incompressibility condition bv i;i ¼ 0 and (4.18). b are satisfied provided b l 4 ¼ l4 , b l 1 þ bk 2 ðb l þb l3Þ ¼ It can be shown that the conditions on C
ijkp
bk 1 2 l1 þ kk21 ðl2 þ l3 Þ, and b l 5 bk 2 b l ¼ l5 kk21 l2 . bk 1 2 Proof. By an application of the divergence theorem and using the linear momentum equations of both flows, i.e., tij,j = 0 and bt ij;j ¼ 0, we obtain ZZ ZZZ ½vibt ij bv i tij dS j ¼ ½vi;jbt ij bv i;j tij dV S
QðSÞ
¼
ZZZ
QðSÞ
b ijkp bv k;p b ½ð C p dij Þvi;j ðC ijkp vk;p pdij Þbv i;j dV
ð4:20Þ
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From the incompressibility conditions given by vi;i ¼ bv i;i ¼ 0, we obtain ZZ ZZZ b ijkp bv k;p vi;j C ijkp vk;p bv i;j dV ½vibt ij bv i tij dS j ¼ ½C S
ð4:21Þ
QðSÞ
Q(S) is an arbitrary region bounded by a closed surface S, so the right hand side of (4.21) is zero provided b ijkp bv k;p vi;j C ijkp vk;p bv i;j ¼ 0 C
ð4:22Þ
This requires, b ijkp ¼ C kpij ¼ C ijkp C
ð4:23Þ
where in the last equality of (4.23) we have used the symmetry properties of these tensors. From (4.16) and a similar equation for the reciprocal flow as well as (4.23), we can deduce the following conditions for the viscosity coefficients and director fields of both flows: b n¼n b 4 ¼ l4 l b k2 k2 b l 1 þ ðb l2 þ b l 3 Þ ¼ l1 þ ðl2 þ l3 Þ b k1 k1 ! b k2 k2 b b l l5 l 2 ¼ l5 b k1 2 k1
ð4:24Þ
For nematic liquid crystals with no elastic effects and l1 þ kk21 ðl2 þ l3 Þ ¼ l5 kk21 l2 ¼ 0, a reciprocal material can be defined using (4.19) or the result (4.3) obtained in Subsection 4.1 since et ij ¼ l4 Aij . 4.3. High elasticity limit Here we try to establish a model flow regime where the viscous forces do not distort the director field. In the high elasticity limit, i.e., very small Reynolds and Ericksen numbers (Re 1 and Er 1), we can neglect the fluid inertial terms as well as the response of the director to the flow field and just use the static director field for v = 0. This means that the flow field does not create any distortion to the director but whenever a director field gradient occurs in the nematic sample a flow perturbation can be induced. For zero velocity field the static director defines a static pressure ps via the linear momentum balance. This pressure is only a function of the director field and xi and, therefore, does not depend on the velocity field. For v 5 0 we subdivide the pressure as p = ph ps, where ph is the hydrodynamic contribution which is a function of the director and velocity fields as well as xi. 4.3.1. Governing partial differential equations Using (2.22) (see Section 2), the angular momentum equations in the high elasticity limit for a ¼ 0 and v = 0 can be written as follows: stationary director field on ot
C. Atkinson, P.J.S. Pereira / International Journal of Engineering Science 43 (2005) 1185–1205
oW o oW cni þ þ Gi ¼ 0 oni oxj oni;j
1199
ð4:25Þ
where the director inertial terms have been dropped (see Section 2) and e g ¼ 0. Note that (4.25) is the Euler–Lagrange equilibrium equation of the static theory (see, e.g., [10]). Using (2.4), (2.8), (2.10)–(2.12), (2.13), (2.18), (2.20), (2.23) and (2.25) and the scalar product of ¼ 0 and v = 0 is given by (4.25) with ni,k, we find that the static pressure for on ot ps ¼ C þ 1ðxi ; ni Þ W
ð4:26Þ
where C 2 R. From (2.18), (2.20), (2.21), (2.23) and (2.25) as well as the scalar product of (4.25) with ni,k, we obtain the linear momentum equations in the high elasticity limit for a static director field associated with an incompressible isothermal nematic liquid crystal F i þ tij;j ¼ ðph þ W 1Þ;i þ et ij;j ¼ 0
ð4:27Þ
where (ph + W 1),i = (ph ps + C),i. As before the velocity field satisfies the incompressibility condition vi,i = 0 and the director field is constrained by nini = 1. Note that in the stationary case, the convective derivative v Æ $n is still present in the viscous ¼ 0. Thus, et ij ¼ stress tensor given by (2.8) whenever $n 5 0 and li 5 0 (i = 2, 3) but on ot C ijkp vk;p þ l2 vp ni;p nj þ l3 vp nj;p ni , where the fourth order rank tensor Cijkp is given by (3.2) (see Section 3). 4.3.2. Reciprocal materials for nematics in the high elasticity limit Similar expressions to those given above follow for the reciprocal material with all components with the overbar circumflex. For example, the angular and linear momentum equations are given by ! b b oW o oW bi ¼ 0 bc b þG þ ð4:28Þ ni ob n i oxj ob n i;j and b b1 Þ þ b et ij;j ¼ 0 Fb i þ bt ij;j ¼ ðb ph þ W ;i
ð4:29Þ
Using a similar procedure to that outlined in Section 3, a reciprocal material can be established such that the following result holds: Result 4. If ½v; n; et; p; W ; 1; lm ; K l (m = 1, 2, . . ., 6 and l = 1, 2, 3, 4) is an incompressible isothermal viscous flow of a nematic liquid crystal in the high elasticity limit, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then
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ZZ S
½bv iet ij ðph þ W 1Þbv j dS j ¼
ZZ
b b1 Þvj dS j et ij ðb ½vib ph þ W S ZZZ þ ½bv i;j ðl2 vp ni;p nj þ l3 vp nj;p ni Þ QðSÞ
vi;j ðb l 2 bv p b l 3 bv p b n i;p b nj þ b n j;p b n i Þ dV
ð4:30Þ
where the reciprocal and real velocity fields satisfy the incompressibility conditions and the linear b ijkp ¼ C kpij . It can be shown that the conmomentum equations. The result (4.30) then follows if C b ijkp are satisfied provided: ditions on C b n¼n b 1 ¼ l1 l b l 4 ¼ l4 2b l 2 ¼ l6 l3 l5 þ l2
ð4:31Þ
2b l 3 ¼ l3 þ l6 l5 l2 2b l 5 ¼ l5 l2 þ l6 l3 2b l 6 ¼ l3 þ l2 þ l5 þ l6 b i ¼ li ði ¼ 1; 2; . . . ; 6Þ and If in addition we use the Parodi relationship (2.17) [11], we obtain l b k i ¼ ki ði ¼ 1; 2Þ. In the high elasticity limit a physically attainable director field to the reciprocal material that also satisfies the angular momentum balance given by (4.28) can be defined. Note that the real and auxiliary velocity fields are such that they do not create any distortion to the director field but it can induce some perturbation on both flows. As the reciprocal director field is chosen to be the same to that of the real material it also has to satisfy the angular momentum balance (4.25). The velocity fields v and bv of the real and auxiliary incompressible flows satisfy the linear momentum equations given by (4.27) and (4.29), respectively.
5. Applications and concluding remarks A reciprocal like theorem described by ZZ ZZ b bv i tij dS j vi t ij dS j ¼ S
ð5:1Þ
S
can be used as an auxiliary tool to find the coefficients of leading order terms in velocity fields near sharp edges or corners. Applications of this kind have been made by Atkinson and Bastero [4] for anisotropic elastic fields and by Barone and Robinson [14] for isotropic elastic fields. For slow viscous flows analogous results hold, i.e., if the actual velocity field at a sharp corner has the form vj = rkfj(h) with (r, h) a plane polar coordinate system with origin at the corner then the auxiliary field bv j can be chosen so that bv j ¼ rk gj (h) with fj(h) and gj(h) given. Furthermore, the auxiliary field can be chosen to satisfy appropriate boundary conditions on wedge sides. For example, if one writes down the eigenvalue equation for a viscous flow problem such as that considered by
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Moffatt [15] this equation would be given by sin(2ka) = ± k sin(2a), where a is the included half angle of the wedge and the minus sign corresponds to the antisymmetric flow, whereas the plus sign corresponds to the symmetric flow. In this case bv j ¼ rk gj (h) and satisfies the same boundary conditions as vj. Note that these eigenvalue problems apply also to traction boundary conditions. Furthermore, the eigenvalue problems will in principle apply to more general boundary conditions where there is a third component of velocity (v1, v2, v3) even though these velocity fields depend only on (r, h) and are independent of z. Such flows have been considered by Atkinson and Pereira [16] for the general non-linear equations. However, in these cases the eigensolutions are only local solutions valid close to the corner. An approximate version of the above applications could be devised for these local solutions. However, our main result described by (3.5) (see Section 3) is valid for the full asymmetric stress tensor and non-linear field. This theorem can also be used as an auxiliary tool when generating numerical solutions. The result (3.5) can be generalized to biaxial nematic liquid crystals. If one uses the theory outlined in Leslie et al. [17] and Leslie [18] one simply needs to identify the two sets of orthonormal b of the reciprocal material so b b ¼ m. director vector fields n, m of the real field and b n, m n ¼ n and m Using the biaxial nematic liquid crystal equations presented in Appendix A, we find the following result: ZZ ½bv iet ij ðp þ W 1 Þbv j dS j S
¼
ZZ S
c 1b Þvj dS j et ij ðb ½vib pþW
oni onj omi bv i;j l2 þ nj þ l3 ni þ b2 mj ot ot ot QðSÞ omj onp onp þb3 mi þ a1 mi mp nj þ a2 mj mp ni ot ot ot bi bj ob ni ob nj om om b b bj þ b bi l3 b2 b3 vi;j b l2 nj þ b ni þ b m m ot ot ot ot ob np ob np onk onp b pb b pb b im b jm þ k3 mp mk a2 nj þ b n i þ bv i nk;i k1 þb a1 m m ot ot ot ot bk ob n k b ob np omk om b k þ bv i mk;i c1 b k;i bc 1 bpm vi b k1 n k;i b þ k3 m vi m dV ot ot ot ot ZZZ
ð5:2Þ where p is the hydrostatic pressure, 1*(xi, ni, mi) is some scalar energy density function, which represents some external body forces, and W * the elastic energy function of the biaxial material. In (5.2) the several coefficients denote viscosities of the real and reciprocal materials. Again the terms mi involving ootbn i and ootb could be set to zero if required. Using a similar procedure to that outlined in Section 3, the following relations between the sets of viscosity coefficients can be shown to be necessary for (5.2) to hold in this case:
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b b ¼m n ¼ n; m b b bc 1 ¼ c1 ¼ b3 b2 ; k 3 ¼ k3 ¼ a2 a1 k 1 ¼ k1 ¼ l3 l2 ; b b bc 2 ¼ b2 þ b3 ; k 4 ¼ a1 þ a2 k 2 ¼ l2 þ l3 ; b b b ¼b l ¼l ; 1
1
b l 4 ¼ l4 ;
1
1
b a 5 ¼ a5 ¼ 0
ð5:3Þ 2b a 1 ¼ a4 a2 a3 þ a1
2b l 5 ¼ l5 l2 þ l6 l3 ;
2b b 2 ¼ b6 b3 b5 þ b2 ; 2b b 3 ¼ b3 þ b6 b5 b2 ; b ¼b b þb b ; 2b
2b l 6 ¼ l3 þ l2 þ l5 þ l6 ;
2b b 6 ¼ b3 þ b2 þ b5 þ b6 ;
2b a 4 ¼ a1 þ a2 þ a3 þ a4
2b l 2 ¼ l6 l3 l5 þ l2 ; 2b l 3 ¼ l3 þ l6 l5 l2 ;
5
5
2
6
3
2b a 2 ¼ a2 þ a4 a1 a3 2b a 3 ¼ a4 a2 þ a3 a1
For these incompressible isothermal biaxial nematics the Onsager–Parodi relationships are defined by l2 þ l3 ¼ l6 l5 ; b2 þ b3 ¼ b6 b5 ; a1 þ a2 ¼ a4 a3 ; a5 ¼ 0 ð5:4Þ From (5.3) one can see that the condition, a5 = 0, necessary to establish the result (5.2) is in accordance with the last Onsager–Parodi relationship of (5.4). Using (5.3) as well as (5.4) we obtain the following conditions for the viscosity coefficients of both flows: b b ¼m n ¼ n; m b b bc 1 ¼ c1 ¼ b3 b2 ; k 3 ¼ k3 ¼ a2 a1 k 1 ¼ k1 ¼ l3 l2 ; b bc 2 ¼ c2 ¼ b2 þ b3 ¼ b6 b5 k 2 ¼ k2 ¼ l2 þ l3 ¼ l6 l5 ; b k 4 ¼ k4 ¼ a1 þ a2 ¼ a4 a3 b l i ¼ li ði ¼ 1; 2; . . . ; 6Þ;
b b j ¼ bj
ðj ¼ 1; 2; 3; 5; 6Þ;
b a p ¼ ap
ð5:5Þ ðp ¼ 1; 2; . . . ; 4Þ
b a 5 ¼ a5 ¼ 0 Note that the result for biaxial nematic liquid crystals is true for any set of viscosity coefficients except that a5 must be zero, so the result (5.2) generalized does not require the Parodi relationships (5.4) to be true except for the last one, i.e., a5 = 0. The results established in Section 4 can be generalized to biaxial nematic liquid crystals. In Appendix B, the result (4.30) (see Subsection 4.3) is generalized to biaxial nematics in the high elasticity limit. In the general case as well as in the high viscosity and elasticity limits, where the auxiliary director fields are chosen to be the same to those of the real material, integral relations between velocity and stress fields of the real and reciprocal materials are obtained.
Acknowledgements The author P.J.S. Pereira thanks to the Instituto Superior de Engenharia de Lisboa and Ministry of Education of Portugal for a scholarship.
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Appendix A Using the theory outlined in Leslie et al. [17] and Leslie [18] the equations in Cartesian coordinates describing the hydrodynamic behaviour of biaxial nematic liquid crystals can be written in a similar form to that deduced in Section 2. We consider an incompressible isothermal biaxial nematic liquid crystal with no director inertial terms and external body forces given by Fi ¼
o1 ðxi ; ni ; mi Þ ; oxi
Gni ¼
o1 ðxi ; ni ; mi Þ oni
and Gmi ¼
o1 ðxi ; ni ; mi Þ omi
ðA:1Þ
where F is any external body force and Gn as well as Gm are generalized body forces arising from a body couple present due to magnetic or electric fields and associated with the directors n and m, respectively. As before the angular momentum equations can be written as follows: oW o oW n e þ gi þ þ Gni ¼ 0 c ni þ j mi oxj oni;j oni ðA:2Þ oW o oW m m þe gi þ þ Gi ¼ 0 s mi þ j ni oxj omi;j omi where c*, j*, and s* are arbitrary scalars. The linear momentum equations can be written as follows: g nk þ mk;i e g mk þ et ij;j q_vi ¼ ðp þ W 1 Þ;i þ nk;i e
ðA:3Þ
Using (A.3) we have et ij;j ¼
o ðp þ W 1 Þ nk;i e g nk mk;i e g mk oxi
ðA:4Þ
where the fluid inertial terms q_vi have been neglected (see Section 1). Here the asymmetric viscous stress tensor is given by et ij ¼ l1 nk np Akp ni nj þ l2 N i nj þ l3 N j ni þ l4 Aij þ l5 Aik nk nj þ l6 Ajk nk ni þ b1 mk mp Akp mi mj þ b2 M i mj þ b3 M j mi þ b5 Aik mk mj þ b6 Ajk mk mi þ N p mp ða1 mi nj þ a2 mj ni Þ ðA:5Þ þ nk Akp mp ða3 mi nj þ a4 mj ni Þ þ a5 mk mp Akp ni nj with 2N i ¼ 2n_ i 2Aij nj 2M i ¼ 2m_ i 2Aij mj
ðA:6Þ
and li (i = 1, 2, . . ., 6), bj (j = 1, 2, 3, 5, 6), as well as ap (p = 1, 2, . . ., 5) are viscosity coefficients. The n m g are given by vectors e g and e e g ni ¼ ðk1 N i þ k2 Aip np þ k3 N p mp mi þ k4 mk Akp np mi Þ e g mi ¼ ðc1 M i þ c2 Aip mp Þ
ðA:7Þ
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where k1 ¼ l3 l2 ;
k2 ¼ l6 l5 ;
c1 ¼ b3 b2 ;
c2 ¼ b6 b5
k3 ¼ a2 a1 ;
k4 ¼ a4 a3
ðA:8Þ
and Aij and Aij are given by (2.10) and (2.12), respectively (see Section 2). The elastic free energy density W * for hard biaxial nematic liquid crystals, i.e., considering materials in the hydrodynamic limit where both order parameters associated with the director vector fields n and m are nearly constant and only these orthonormal director fields depend on position, is given by (see Longa et al. [19]) 2
2
2
2
2W ¼ K l1 ðdiv lÞ þ K l2 ðl curl lÞ þ K l3 jl curl lj2 þ K m1 ðdiv mÞ þ K m2 ðm curl mÞ þ K m3 jm curl mj2 þ K n1 ðdiv nÞ2 þ K n2 ðn curl nÞ2 þ K n3 jn curl nj2
þ K mn ðm curl nÞ2 þ K nl ðn curl lÞ2 þ K lm ðl curl mÞ2 þ K l4 div ½ðl 5Þl ldiv l þ K m4 div ½ðm 5Þm mdiv m þ K n4 div ½ðn 5Þn ndiv n
ðA:9Þ
where Kli (i = 1, 2, 3, 4, m), Kmj (j = 1, 2, 3, 4, n) and Knp (p = 1, 2, 3, 4, l) denote the elastic constants and l = m · n. Note that (A.9) reduces to (2.7) (see Section 2) with Kni = Ki and Kn4 = (K2 + K4) (i = 1, 2, 3) by disregarding the terms that involve the secondary director fields m and l. For these biaxial nematics the Onsager–Parodi relationships are defined by (5.4) (see Section 5). Finally, the velocity field satisfies the incompressibility condition (vi,i = 0) and the director fields n and m as well as l are mutually orthonormal vectors, i.e., nini = 1, mimi = 1, lili = 1, n Æ m = 0, n Æ l = 0 and m Æ l = 0.
Appendix B The result (4.30) (see Subsection 4.3) can be generalized to biaxial nematic liquid crystals in the high elasticity limit as follows: ZZ ½bv iet ij ðph þ W 1 Þbv j dS j S ZZ ZZZ b c ¼ ½viet ij ðc ph þ W 1b Þvj dS j þ fbv i;j ½l2 vp ni;p nj þ l3 vp nj;p ni S
QðSÞ
n i;p b nj þ b n j;p b ni þ b2 vp mi;p mj þ b3 vp mj;p mi þ ða1 mi nj þ a2 mj ni Þmp vk np;k vi;j ½b l 2 bv p b l 3 bv p b b bv p m b p bv k b b i;p m bj þ b b j;p m b i þ ðb b ib b jb þb b 3 bv p m a1 m a2 m ðB:1Þ nj þ b niÞm n p;k g dV 2 It can be shown that the result (B.1) holds if the conditions (5.3) are satisfied with exception to those involving the coefficients of friction of both flows, i.e., ki, b k j , c1 and bc p (i = 1, 3; j = 1, 2, 3, 4; p = 1, 2). In addition, if one uses the Onsager–Parodi relationships given by (5.4), we obtain the conditions (5.5) (see Section 5). As before the angular momentum equations are obtained from n m g ¼ 0. The linear momentum equa(A.2) for static director fields and v = 0, so the vectors e g ¼e n m g and replactions are obtained from (A.4) disregarding the terms involving the vectors e g and e
C. Atkinson, P.J.S. Pereira / International Journal of Engineering Science 43 (2005) 1185–1205
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ing p by ph. Moreover, the viscous stress tensor in the linear momentum balance is given by (A.5) ¼ om ¼ 0 but the convective derivatives v Æ $n and v Æ $m are still present. (see Appendix A) with on ot ot Similarly, we subdivide the pressure for biaxial nematics as p = ph ps where ps = C2 + 1*(xi, ni, mi) W * (C 2 2 R; see Subsection 4.3).
References [1] S. Kim, S. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth, London, 1991. [2] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague, The Netherlands, 1983. [3] F.R.N. Nabarro, Theory of Crystal Dislocations, Clarendon Press, Oxford, 1967. [4] C. Atkinson, C. Bastero, On the use of Bettis reciprocal theorem for computing the coefficients of stress singularities in anisotropic media, Int. J. Eng. Sci. 29 (1991) 727–741. [5] H.B. Glenn, Advances in Liquid Crystals, Academic Press, New York, 1978. [6] W. Helfrich, Conduction-induced alignment of nematic liquid crystals: basic model and stability considerations, J. Chem. Phys. 51 (1969) 4092–4105. [7] E. Dubois Violette, P.G. de Gennes, O. Parodi, Hydrodynamic instabilities of nematic liquid crystals under a.c. electric fields, J. Phys. (Paris) 32 (1971) 305–317. [8] J.L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol. 5 (1961) 23–34. [9] F.M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal. 28 (1968) 265–283. [10] I.W. Stewart, The static and dynamic continuum theory of liquid crystals, The Liquid Crystals Book Series, Taylor & Francis, London, 2004. [11] O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. (Paris) 31 (1970) 581–584. [12] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, 1992. [13] C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969. [14] M.R. Barone, A.R. Robinson, Determination of elastic stresses at notches and corners by integral equations, Int. J. Solids Struct. 8 (1972) 1319–1338. [15] H.K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18 (1964) 1–18. [16] C. Atkinson, P.J.S. Pereira, Flow of a nematic liquid crystal near the leading edge of an infinite prism, Q. J. Mech. Appl. Math., in press. [17] F.M. Leslie, J.S. Laverty, Continuum theory for biaxial nematic liquid crystals, Q. J. Mech. Appl. Math. 45 (1992) 595–606. [18] F.M. Leslie, Flow alignment in biaxial nematic liquid crystals, J. Non-Newtonian Fluid Mech. 54 (1994) 241–250. [19] L. Longa, J. Stelzer, Density functional approach to study the elastic constants of biaxial nematic liquid crystals, J. Chem. Phys. 109 (1998) 1555–1566.
Reciprocal and related theorems for nematic liquid crystals C. Atkinson
a,*
, P.J.S. Pereira
a,b
a
b
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK ´ dio A´rea Cientı´fica de Matema´tica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emı Navarro, 1949-014 Lisboa, Portugal Received 4 March 2005; accepted 4 March 2005 Available online 21 September 2005
Abstract Reciprocal and related theorems are considered for uniaxial and biaxial nematic liquid crystals. In the general case, where the directors of real and reciprocal materials are assumed to coincide, integral relations between velocity and stress fields of the real and reciprocal materials are obtained. Simplified expressions are also obtained for the high viscosity and elasticity limits. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Reciprocal theorems for the classical equations of elasticity and fluid mechanics are well known. For example, for a Newtonian viscous fluid contained in the region Q bounded by a closed surface S with orientation given by an outward unit vector u normal to S, one has the result ZZ ZZ b bv i tij dS j vi t ij dS j ¼ ð1:1Þ S
*
S
Corresponding author. E-mail address: [email protected] (C. Atkinson).
0020-7225/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.03.008
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where both sets of velocity fields and stress tensors of the real and reciprocal materials (vi, tij) and ðbv i ; bt ij Þ (i, j = 1, 2, 3) satisfy the same constitutive equations and equations of motion (dSj = uj dS). The result is easily confirmed by means of the divergence theorem (see, e.g., [1,2]). A similar result for elasticity is ZZ ZZ b d ibt ij dS j ¼ ð1:2Þ d i tij dS j S
S
where tij is the stress tensor associated with the displacement field di (i, j = 1, 2, 3) in anisotropic bodies with the equilibrium equations tij,j = 0 and bt ij;j ¼ 0 also being satisfied. In this case, the elastic constants of both materials are the same. In all of the above equations the summation convention over repeated indices is used and a comma denotes partial differentiation with respect to an appropriate Cartesian coordinate xi (i = 1, 2, 3). Reciprocal theorems of this kind are the basis of the boundary element method and can be used to obtain information concerning solutions to boundary value problems and can occasionally be used to find the full solution, e.g., it provides a way to compute fields for arbitrarily shaped dislocation loops in an infinite solid (see [3]) or to find the singular stress fields in the neighbourhood of notches and corners in anisotropic media (see d i field of the reciprocal material can be selected for some [4]). In the above formulations the bv i or b auxiliary purpose and it does not have to be a physically attainable velocity or displacement field. It is the purpose of this paper to try to extend the above results to uniaxial and biaxial nematic liquid crystals. The hydrodynamic behaviour of a uniaxial nematic liquid crystal depends on both a velocity vector field v and a unit director vector field n with the spatial orientation defined by the twist and tilt angles. The director field is also considered to be non-polar such that the states n and n are indistinguishable. The velocity field is constrained by the incompressibility condition vi,i = 0 and the director field by nini = 1. For biaxial nematic liquid crystals there is one pair of orthonormal director vector fields n and m. The most general result derived (see Section 3) is of the form ZZ h ZZ i b b1 Þvj dS j et ij ðb e bv i t ij ðp þ W 1Þbv j dS j ¼ vi b pþW S ZSZ Z oni onj bv i;j l2 nj þ l3 ni þ ot ot QðSÞ ob ni ob nj onk ob nk b b b l3 vi;j b l2 n k;i k 1 nj þ b n i þ bv i nk;i k1 vi b dV ot ot ot ot ð1:3Þ where the director fields of both real and reciprocal materials are assumed to coincide at the time at which the relation above holds. In (1.3) et is the second order viscous stress tensor, p(xi, t) the hydrostatic pressure, W(ni, ni,j) the elastic energy of the director, and the scalar energy density function 1(xi, ni) represents some external body forces (i, j = 1, 2, 3). The constants lm (m = 1, 2, . . ., 6) are viscosity coefficients, k1 is the rotational viscosity coefficient, Kp (p = 1, 2, 3, 4) are elastic constants, xi (i = 1, 2, 3) Cartesian coordinates and t the time. The components of the reciprocal fluid are identified by a circumflex accent. Throughout the paper, unless otherwise stated, indices take the values 1, 2, and 3. For this theorem to hold various relations must hold between the viscosities of the real and reciprocal materials (see Section 3).
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In all cases we consider that the fluid inertial terms of both flows in the linear momentum equations are neglected when compared with the pressure, viscous stress tensor and elastic terms. In fact, the order of magnitude of a typical term of the force per unit volume due to the viscous stress is lU/L2, where U is the characteristic velocity, L the length, which is of the order of the thickness of the sample (10–100 lm) down to 1 lm, and l the viscosity, which is of the order of 101 poise. Therefore, in most cases the fluid inertial term qU_ ¼ wqU can be neglected up to angular frequencies, w, of the order of 106 Hz and similarly for the reciprocal flow (see [5–7]). In this last expression q is the mass density and U_ is the material time derivative of U. In addition to the main result (1.3) simplified results are obtained in the high viscosity and elasticity limits. The result (1.3) is also generalized to biaxial nematic liquid crystals (see Section 5). The plan of this paper is as follows. We begin with the Ericksen–Leslie partial differential equations in Cartesian coordinates (see Section 2). In Section 3, we investigate a reciprocal material associated with an anisotropic nematic liquid crystal described by the full asymmetric viscous stress tensor. A key feature here is that the reciprocal velocity field bv needs only to satisfy the linear momentum equations and incompressibility condition. The director field b n is chosen to be instantaneously the same as that in the real fluid. In Subsection 4.1 an incompressible isothermal viscous fluid in the isotropic viscosity phase of a nematic liquid crystal is considered. Similar solutions to those of Newtonian viscous fluids are obtained. The high viscosity limit is studied in Subsection 4.2. The equations necessary to model these boundary and initial value problems are deduced and a reciprocal material associated with a nematic liquid crystal with no elastic effects is defined. Here we find that the viscous stress tensor is symmetric and an equation for the indeterminacy c (see Section 2) is also determined. We also consider the case where the viscous stress tensor is only composed of the isotropic viscosity l4. The high elasticity limit is investigated in Subsection 4.3. The governing equations are obtained and an integral relation between the velocity and stress fields of both flows is found. Finally, a generalization of our main result (1.3) to biaxial nematic liquid crystals is outlined in Section 5. The equations in Cartesian coordinates governing the hydrodynamic behaviour of biaxial nematic liquid crystals are given in Appendix A. A generalization of the result of biaxial nematics to the high elasticity limit is given in Appendix B.
2. Governing partial differential equations The equations describing the hydrodynamic behaviour of a nematic liquid crystal are well known [8–10]. For completeness, these equations are presented here in Cartesian coordinates. The flow of a nematic liquid crystal is described by the velocity field v together with a director field n, which is considered in this work as a non-polar unit vector giving the orientation of the anisotropic axis in these transversely isotropic liquids. With the assumption of incompressibility, the equations are the constraints div v ¼ 0;
jnj ¼ 1
ð2:1Þ
together with the balance laws: linear momentum equations q_vi ¼ F i þ tij;j
ð2:2Þ
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angular momentum equations r€ ni ¼ Gi þ gi þ S ij;j
ð2:3Þ
and the constitutive relations: the equations for the stress tensor t tij ¼ pdij
oW nk;i þ et ij onk;j
ð2:4Þ
the equations for the tensor S S ij ¼ ni bj þ
oW oni;j
ð2:5Þ
the equations for the vector g gi ¼ cni ðni bj Þ;j
oW þe gi oni
ð2:6Þ
where the energy function is defined by 2W ¼ K 1 ðni;i Þ2 þ K 2 ðijk ni nk;j Þ2 þ K 3 ni nj nk;i nk;j þ ðK 2 þ K 4 Þ½ni;j nj;i ðni;i Þ2
ð2:7Þ
the viscous stress tensor et is given by et ij ¼ l1 nk np Akp ni nj þ l2 N i nj þ l3 N j ni þ l4 Aij þ l5 Aik nk nj þ l6 Ajk nk ni
ð2:8Þ
and the vector e g is given by e g i ¼ k1 N i k2 Aik nk
ð2:9Þ
the rate of strain tensor A is defined by 2Aij ¼ vi;j þ vj;i
ð2:10Þ
the vector N is given by 2N i ¼ 2n_ i 2Aij nj
ð2:11Þ
where the vorticity tensor A* is defined by 2Aij ¼ vi;j vj;i
ð2:12Þ
the material time derivatives of the director and velocity fields are given by n_ i ¼
oni þ vj ni;j ; ot
v_ i ¼
ovi þ vj vi;j ot
ð2:13Þ
and k1 ¼ l3 l2 ;
k2 ¼ l6 l5
ð2:14Þ
The usual summation convention (summation over repeated indices) is used. A comma denotes partial differentiation and the superposed dot a material time derivative. In the conservation of linear momentum Eq. (2.2), q is the density, F any external body force and t the stress tensor
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(asymmetric). In the conservation of angular momentum Eq. (2.3), r is a constant inertial coefficient, G a generalized body force arising from a body couple present due to magnetic or electric fields, g is a generalized intrinsic body force that depends upon the director vector field n through Eq. (2.6), and S is a generalized stress tensor. The inertial term in these equations, rn€i , is usually assumed negligible and hence omitted in most treatments. In the constitutive relations (2.4), the scalar p is the arbitrary hydrostatic pressure following from the assumption of incompressibility, while c in (2.6) and the vector b in (2.5) represent similar indeterminacies due to the constraint on the additional kinematic variable n. The vector N in (2.11) is the co-rotational time flux of n, which describes the internal motion of the director with respect to the fluid. In (2.10) A is the velocity gradient tensor (symmetric), i.e., the rate of strain tensor, whereas A* in (2.12) is the vorticity tensor (antisymmetric). In (2.8) et is the viscous stress tensor, which is an even function of n. The vector e g in (2.9) is the hydrodynamic part of g and ðn e g Þ define the viscous torque on the director with coefficients of friction given by k1 and k2, which are defined by (2.14) and usually called rotational and torsional viscosities, respectively. In (2.4) dij is the well-known Kronecker symbol and in (2.7) ijk is the Levi–Civita symbol. The elastic torque on the director field is given by n · h, where h is the molecular vector field defined by oW oW hi ¼ þ ð2:15Þ oni oni;j ;j Here W is the energy function and Kp (p = 1, 2, 3, 4) are the usual elastic coefficients related by Eq. (2.7), whereas in (2.8) lm (m = 1, 2, . . ., 6) are the viscosity coefficients. The elastic and viscosity coefficients are assumed constant if thermal gradient effects are ignored. Various authors have discussed the possibility of restrictions on these coefficients for physically acceptable behaviour. In particular, the coefficients in (2.8) are to be consistent with the thermodynamic inequality et ij vi;j e g i n_ i P 0
ð2:16Þ
Other restrictions have been suggested, e.g., the Parodi relationship [11] among the viscosity coefficients l2 þ l3 ¼ l6 l5
ð2:17Þ
If in (2.2) and (2.3) we neglect both fluid and director inertial terms we obtain the equations F i þ tij;j ¼ 0
ð2:18Þ
Gi þ gi þ S ij;j ¼ 0
ð2:19Þ
and which together with Eqs. (2.4)–(2.14) as well as (2.1) and appropriate boundary and initial conditions enable (in principle) the velocity and director fields to be determined. A slightly different form of these equations can be deduced. We consider an incompressible isothermal nematic liquid crystal with external body forces given by Fi ¼
o1ðxi ; ni Þ oxi
and Gi ¼
o1ðxi ; ni Þ oni
ð2:20Þ
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where 1(xi, ni) is some scalar energy density function, e.g., 1(xi, ni) can be the magnetic or electric energy. Note that the case of no external body forces is described by 1ðxi ; ni Þ ¼ C 1 ðC 1 2 RÞ. Differentiating (2.4) we obtain, tij;j
o oW ¼ p;j dij nk;i þ et ij;j oxj onk;j
Inserting (2.5) and (2.6) into the angular momentum Eq. (2.19) we obtain oW o oW þe gi þ þ Gi ¼ 0 cni oni oxj oni;j
ð2:21Þ
ð2:22Þ
Assuming that the energy function only depends on W = W(nk, nk,j), we have oW oW oW ¼ nk;i þ nk;ji oxi onk onk;j Using (2.22), (2.21) becomes o oW oW nk;ij þ et ij;j tij;j ¼ p;j dij nk;i oxj onk;j onk;j oW oW ¼ p;j dij nk;i e g k cnk Gk nk;ij þ et ij;j onk onk;j
ð2:23Þ
ð2:24Þ
Considering a non-polar and unit director field we can write nk nk ¼ 1 () nk;i nk ¼ 0
ð2:25Þ
Thus, from (2.18), (2.20), (2.23), (2.24) as well as (2.25) we obtain, et ij;j ¼
o gk ðp þ W 1Þ nk;i e oxi
ð2:26Þ
Thus, the angular and linear momentum equations can be written in the form of (2.22) and (2.26), respectively.
3. Reciprocal materials for anisotropic nematic liquid crystals: The general case For the general case the full viscous stress tensor is asymmetric and the linear and angular momentum equations are also coupled through the energy function terms. Here we consider flow regimes with very small Reynolds number, i.e., the fluid inertial terms can be neglected. We can rewrite (2.8) in the form et ij ¼ C ijkp vk;p þ l2 n_ i nj þ l3 n_ j ni
ð3:1Þ
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where the fourth order rank tensor Cijkp is defined as 1 2l1 nk np ni nj þ l4 ðdki dpj þ dkj dpi Þ þ l5 ðnk nj dip þ np nj dik Þ C ijkp ¼ 2
1191
þl6 ðnk ni dpj þ np ni dkj Þ þ l3 ðni nk dpj ni np dkj Þ þ l2 ðnj nk dpi nj np dki Þ Similar expressions follow for a reciprocal material where (2.26) is replaced by o ob nk b b b b b b b et ij;j ¼ ðb p þ W b1 Þ b n k;i k 1 n k;p þ k 1 A kp b n p k 2 A kp b np þ bv p b oxi ot
ð3:2Þ
ð3:3Þ
where ob n ob n i j b b ijkp bv k;p þ b et ij ¼ C nj n i;p þ b ni n j;p þ bv p b þ bv p b l2 b l3b ot ot
ð3:4Þ
b ijkp as in (3.2) with all viscosity coefficients and director field components with the overbar with C circumflex. We now show that a reciprocal material can be defined such that the following result holds: Result 1. If ½v; n; et; p; W ; 1; lm ; K l (m = 1, 2, . . ., 6 and l = 1, 2, 3, 4) is an incompressible isothermal viscous flow of an anisotropic nematic liquid crystal associated with the full asymmetric viscous stress tensor, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then at a fixed time ZZ ZZ b b1 Þvj dS j et ij ðb e ½bv i t ij ðp þ W 1Þbv j dS j ¼ ½vib pþW S
S
oni onj bv i;j l2 þ nj þ l3 ni ot ot QðSÞ ob ni ob nj onk ob nk b b b l3 vi;j b l2 n k;i k 1 nj þ b n i þ bv i nk;i k1 vi b dV ot ot ot ot ZZZ
ð3:5Þ where the bv field is only required to satisfy the incompressibility condition bv i;i ¼ 0 and the linear momentum equations given by (3.3) with the auxiliary viscous stress tensor defined by (3.4). b ijkp are satisfied b ijkp ¼ C kpij . It can be shown that the conditions on C Result (3.5) then follows if C provided b n ¼ n, b l 1 ¼ l1 , b l 4 ¼ l4 , b k 1 ¼ k1 ¼ l3 l2 , b k 2 ¼ l2 þ l3 , 2b l 2 ¼ l6 l3 l5 þ l2 , l 5 ¼ l5 l2 þ l6 l3 , and 2b l 6 ¼ l3 þ l2 þ l5 þ l6 . Note that the re2b l 3 ¼ l3 l2 l5 þ l6 , 2b ciprocal director field is identical with the real director field at any fixed time for which (3.5) is applied and the terms involving ootbn i can be set to zero in (3.3), (3.4) as well as in (3.5) if required. Proof. Using the divergence theorem, the incompressibility conditions for both flows vj;j ¼ bv j;j ¼ 0, and (2.26), (3.1), (3.3) as well as (3.4), we can write
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ZZ S
b b1 Þvj dS j et ij þ ðb ½bv iet ij ðp þ W 1Þbv j vib pþW
ZZZ
oni onj bv i;j l2 nj þ l3 ni ot ot QðSÞ
ob ni ob nj onk ob nk b2 b b vi;j l l3 k1 n k;i b nj þ b n i þ bv i nk;i k1 vi b dV ot ot ot ot ZZZ ob nk b b b b b þ bv p b g k þ vi b C ijkp vk;p bv i;j bv i nk;i e n k;i k 1 n k;p þ k 1 A kp b n p k 2 A kp b np ¼ ot QðSÞ ob n ob n i j b ijkp bv k;p vi;j þ bv i;j l2 n_ i nj vi;j b þ bv p b þ bv p b C l2 n i;p b l3 n j;p b n j þ bv i;j l3 n_ j ni vi;j b ni ot ot oni onj ob ni ob nj onk ob nk b b bv i;j l2 l3 k1 l2 n k;i b nj þ l3 ni þ vi;j b nj þ b n i bv i nk;i k1 þ vi b dV ot ot ot ot ot ot
ð3:6Þ Q(S) is an arbitrary region bounded by a closed surface S, so the right hand side of (3.6) is zero provided b ijkp bv k;p vi;j ¼ 0 C ijkp vk;p bv i;j C
ð3:7Þ
and bv i;j ½l2 vp ni;p nj þ l3 vp nj;p ni bv i nk;i ½k1 vp nk;p þ k1 Akp np k2 Akp np b b np ¼ 0 b b vi;j ½b l 2 bv p b l 3 bv p b k 1 bv p b k1 A n i;p b nj þ b n j;p b n i þ vi b n k;i ½b n k;p þ b kp n p k 2 A kp b
ð3:8Þ
where (2.9), (2.11) and (2.13) have been used to obtain (3.8). Eq. (3.7) is satisfied provided b ijkp ¼ C kpij C
ð3:9Þ
To satisfy (3.8) we need to balance terms involving partial derivatives of the real and reciprocal velocity field components, i.e., ( " #) b k1 b kq b k2 A n k;p b nq n q ðbv q;k bv k;q Þ b vp bv i;j l2 ni;p nj þ l3 nj;p ni þ b 2 k1 bv p vi;j b l3 b ¼0 ð3:10Þ l2 b n i;p b nj þ b n j;p b n i þ nk;p nq ðvq;k vk;q Þ k2 Akq nq 2 as well as the remaining terms involving products of the real and reciprocal velocity field components bv i nk;i k1 vp nk;p vi b k 1 bv p b n k;i b n k;p ¼ 0
ð3:11Þ
In (3.10) we have used (2.12) and a similar equation for the reciprocal flow. From (3.11) we obtain a relation among the partial derivatives of the director field components k 1 ) of both flows, i.e., and coefficients of friction (k1 and b b n k;i b k1b n k;p ¼ nk;i k1 nk;p
ð3:12Þ
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We choose the director field of the reciprocal material b n to be identical to that of the real material, i.e., b n¼n
ð3:13Þ
Then (3.12) is satisfied provided b k 1 ¼ k1 ¼ l3 l2
ð3:14Þ
where we have used (2.14). Using (2.10) and a similar equation for the reciprocal flow as well as (3.10) we can write, ( " #) b b b b k1 k1 k2 k2 ¼0 n k;p b n q bv q;k þ b n q bv k;q b n q bv k;q b n q bv q;k vp bv i;j ½l2 ni;p nj þ l3 nj;p ni þ b 2 2 2 2 k1 k1 k2 k2 bv p vi;j ½b l2 b l3 b ¼0 n i;p b nj þ b n j;p b n i þ nk;p nq vq;k þ nq vk;q nq vk;q nq vq;k 2 2 2 2 ð3:15Þ From (3.13) and (3.15) we obtain ! ! b b k1 b k2 k1 b k2 ni;p nj þ l3 nj;p ni ¼ 0 l2 þ 2 2 2 2 k1 k2 k1 k2 b l2 þ l3 ni;p nj þ b nj;p ni ¼ 0 2 2 2 2
ð3:16Þ
which requires b l3 b l 2 ¼ l3 l2 k 1 ¼ k1 ¼ b b k 2 ¼ l2 þ l3
ð3:17Þ
k2 ¼ b l2 þ b l 3 ¼ l6 l5 Using the result that b n is identical with the actual director field n at the time of the calculations in (3.5), it can be shown that (3.9) is satisfied provided the reciprocal material coefficients are chosen as follows: b l 1 ¼ l1 b l 4 ¼ l4 2b l 2 ¼ l6 l3 l5 þ l2 2b l 3 ¼ l3 þ l6 l5 l2 2b l 5 ¼ l5 l2 þ l6 l3 2b l 6 ¼ l3 þ l2 þ l5 þ l6 where we have used (3.2) and a similar equation for the reciprocal flow.
ð3:18Þ
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To sum up, we have deduced the following conditions among the components of both flows: b n¼n b k 1 ¼ k1 ¼ l3 l2 b k2 ¼ l þ l 2
ð3:19Þ
3
together with the conditions shown in (3.18).
h
Note that if we use the Parodi relationship (2.17) [11] as well as the conditions among the viscosity coefficients of both flows given by (3.18) and (3.19), we find that b n¼n b k 1 ¼ k1 ¼ l3 l2 b k 2 ¼ k2 ¼ l2 þ l3 ¼ l6 l5 b l i ¼ li ði ¼ 1; 2; . . . ; 6Þ
ð3:20Þ
However, the Parodi relationship [11] comes from Onsagers reciprocal relations in irreversible processes. Some authors suggest that Onsagers relations do not apply to phenomena like heat conduction, viscosity and diffusion since there is no unambiguous way of selecting the fluxes and forces (see [12,13]). Even though available data indicate that (2.17) is satisfied within experimental limits this raises doubts about the universal validity of the Parodi relationship [11]. 4. Reciprocal materials for nematic liquid crystals: Limit cases 4.1. Isotropic viscosity phase In the isotropic viscosity phase only the term involving l4 remains in the viscous stress tensor equation. The Ericksen–Leslie partial differential equations reduce to those of Newtonian incompressible isothermal viscous fluids. Thus, (2.8) turns out to be equal to et ij ¼ l4 Aij
ð4:1Þ
Using (2.2), (2.4) as well as (4.1) and assuming no external body forces and that the inertial terms q_vi are neglected, we can write tij;j ¼ ðpdij Þ;j þ et ij;j ¼ ðpdij Þ;j þ l4 Aij;j ¼ 0 ð4:2Þ
where the energy terms oxoj onoWk;j nk;i have been dropped. Using the incompressibility condition (2.1) one can see that the pressure is as usual a harmonic function, i.e., satisfies the Laplace equation ,2p = 0. Thus, similar solutions to those of Newtonian viscous fluids are obtained. A similar result to that outlined in Section 1 for Newtonian viscous fluids can be stated as follows: Result 2. If ½v; n; et; p; l4 is an incompressible isothermal viscous flow in the isotropic viscosity phase of a nematic liquid crystal, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then at a fixed time
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ZZ S
vibt ij dS j ¼
ZZ S
bv i tij dS j
1195
ð4:3Þ
which is satisfied if b l 4 ¼ l4 . Since it is available in standard textbooks (see, e.g., [1,2]) the proof is omitted here. 4.2. High viscosity limit In this subsection we investigate reciprocal materials associated with a symmetric viscous stress tensor where the interaction of the director field and the viscous flow is maintained through the viscous stress tensor equation. To satisfy this purpose, we assume a model flow regime where the elastic and fluid inertial terms are neglected, which is plausible in the high viscosity limit, i.e., very large Ericksen number and very small Reynolds number. The Reynolds number is described by the ratio of the inertial forces to the viscous forces, Re = qUL/l, whereas the Ericksen number is defined by the ratio of the viscous forces to the elastic forces, Er = lUL/K. In this last equation, K is a representative elastic constant and lU/L the stress. 4.2.1. Governing partial differential equations Using the general equations of Section 2 the equations governing the hydrodynamic behaviour of nematic liquid crystals in the high viscosity limit can be deduced. Using (2.2) and (2.4) and assuming no external body forces as well as no inertial terms q_vi (see Section 1), the linear momentum equations in the high viscosity limit are given by tij;j ¼ ðpdij Þ;j þ et ij;j ¼ 0 where the energy terms have been dropped. From (2.22) (see Section 2) we can write, oW o oW cni þe gi þ ¼0 oni oxj oni;j
ð4:4Þ
ð4:5Þ
since the director inertial terms r€ ni have been neglected. As Kj ! 0 ( j = 1, 2, 3, 4) in the high viscosity limit the energy terms vanish, so we obtain gi ¼ 0 cni þ e
ð4:6Þ
Using (2.9), Eq. (4.6) reduces to the following equation: cni k1 N i k2 Aik nk ¼ 0
ð4:7Þ
giving
k2 ni Aik nk c Ni ¼ k1 k2
ð4:8Þ
Using (4.7) a further equation can be deduced to define the indeterminacy c. As a non-polar and unit director field is considered, we have nini = 1 and, consequently, nini,k = 0. Using this result and (2.11) as well as the scalar product of (4.7) with ni, we obtain c ¼ k1 Aij nj ni þ k2 Aik nk ni
ð4:9Þ
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where the terms k1 n_ i ni ¼ ðk1 =2ÞðD=DtÞðni ni Þ ¼ 0
ðni ni ¼ 1Þ
Furthermore, the terms Aij nj ni ¼
3 X 3 X
Amp nm np
m¼1 p¼1
¼ n21 A11 þ n22 A22 þ n23 A33 þ n1 n2 ðA12 þ A21 Þ þ n2 n3 ðA23 þ A32 Þ þ n3 n1 ðA13 þ A31 Þ ¼ 0 since A11 ¼ A22 ¼ A33 ¼ 0
and Aij þ Aji ¼ Aij Aij ¼ 0
Thus, the indeterminacy c is given by c ¼ k2 Akp nk np
ð4:10Þ
Using (4.8) and (4.10), the internal motion of the director with respect to the fluid is given by k2 ð4:11Þ ½Aik nk Akp nk np ni Ni ¼ k1 Inserting (4.11) into (2.8) and using (2.14) as well as l5 (k2/k1)l2 = l6 (k2/k1)l3, we obtain the viscous stress tensor equations, which are valid in the high viscosity limit, i.e., k2 k et ij ¼ l1 þ ðl2 þ l3 Þ nk np Akp ni nj þ l5 2 l2 ½Ajk nk ni þ Aik nk nj þ l4 Aij ð4:12Þ k1 k1 Here the viscous stress tensor is symmetric but the interaction between the directors and the viscous fluid is also maintained through (4.12). Using (2.3), (2.5), (2.6), (2.9), (2.11), (2.13) as well as (4.10) and assuming no body forces, the angular momentum equations in the high viscosity limit where Kj ! 0 (j = 1, 2, 3, 4) associated with an incompressible isothermal nematic liquid crystal are given by oni k2 ½Aij nj Akp nk np ni ð4:13Þ þ vj ni;j ¼ Aij nj n_ i ¼ ot k1 where the director inertial terms r€ ni are neglected and the energy terms vanish in this limiting formulation. To close this set of equations a further equation is defined, div v ¼ 0
ð4:14Þ
since we have considered an
incompressible isothermal viscous flow. Moreover, if we assume that l1 þ kk21 ðl2 þ l3 Þ ¼ l5
k2 k1
l2 ¼ 0 in (4.12), the high viscosity limit equations reduce to those of
Newtonian incompressible isothermal viscous flows with an additional equation giving the orientation of the director field. After some simplifications using the Parodi relationship (2.17) [11] the coefficients of friction in the approximation mentioned above are given by k1 ¼ ðl1 l22 =l25 Þ and k2 ¼ ðl1 l2 =l5 Þ, so (k2/k1) = (l5/l2).
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4.2.2. Reciprocal materials for nematics in the high viscosity limit We can rewrite the result (4.12) in the form et ij ¼ C ijkp vk;p
ð4:15Þ
where the fourth order rank tensor Cijkp is defined as 1 k2 k2 C ijkp ¼ l l4 ðdki dpj þ dkj dpi Þ þ 2 l1 þ ðl2 þ l3 Þ ni nj nk np þ l5 2 k1 k1 2 ½ni nk dpj þ ni np dkj þ nj nk dpi þ nj np dki
ð4:16Þ
Here, tij ¼ pdij þ et ij
ð4:17Þ
We define similar expressions for a reciprocal material b b ijkp bv k;p et ij ¼ C et ij bt ij ¼ b p dij þ b
ð4:18Þ
bt ij;j ¼ ðb et ij;j ¼ 0 p dij Þ;j þ b b ijkp is as in (4.16) with all viscosity coefficients, director field and coefficients of friction where C with the overbar circumflex. We now show that a reciprocal material can be defined such that the following result holds: Result 3. If ½v; n; et; p; lm (m = 1, 2, . . ., 6) is an incompressible isothermal viscous flow of a nematic liquid crystal in the high viscosity limit, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then at a fixed time ZZ ZZ bv i tij dS j vibt ij dS j ¼ ð4:19Þ S
S
b ijkp ¼ C kpij ¼ C ijkp . The reciprocal director field b which is satisfied provided C n is identical with n at any fixed time for which (4.19) is applied and the reciprocal velocity field bv satisfies the incompressibility condition bv i;i ¼ 0 and (4.18). b are satisfied provided b l 4 ¼ l4 , b l 1 þ bk 2 ðb l þb l3Þ ¼ It can be shown that the conditions on C
ijkp
bk 1 2 l1 þ kk21 ðl2 þ l3 Þ, and b l 5 bk 2 b l ¼ l5 kk21 l2 . bk 1 2 Proof. By an application of the divergence theorem and using the linear momentum equations of both flows, i.e., tij,j = 0 and bt ij;j ¼ 0, we obtain ZZ ZZZ ½vibt ij bv i tij dS j ¼ ½vi;jbt ij bv i;j tij dV S
QðSÞ
¼
ZZZ
QðSÞ
b ijkp bv k;p b ½ð C p dij Þvi;j ðC ijkp vk;p pdij Þbv i;j dV
ð4:20Þ
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From the incompressibility conditions given by vi;i ¼ bv i;i ¼ 0, we obtain ZZ ZZZ b ijkp bv k;p vi;j C ijkp vk;p bv i;j dV ½vibt ij bv i tij dS j ¼ ½C S
ð4:21Þ
QðSÞ
Q(S) is an arbitrary region bounded by a closed surface S, so the right hand side of (4.21) is zero provided b ijkp bv k;p vi;j C ijkp vk;p bv i;j ¼ 0 C
ð4:22Þ
This requires, b ijkp ¼ C kpij ¼ C ijkp C
ð4:23Þ
where in the last equality of (4.23) we have used the symmetry properties of these tensors. From (4.16) and a similar equation for the reciprocal flow as well as (4.23), we can deduce the following conditions for the viscosity coefficients and director fields of both flows: b n¼n b 4 ¼ l4 l b k2 k2 b l 1 þ ðb l2 þ b l 3 Þ ¼ l1 þ ðl2 þ l3 Þ b k1 k1 ! b k2 k2 b b l l5 l 2 ¼ l5 b k1 2 k1
ð4:24Þ
For nematic liquid crystals with no elastic effects and l1 þ kk21 ðl2 þ l3 Þ ¼ l5 kk21 l2 ¼ 0, a reciprocal material can be defined using (4.19) or the result (4.3) obtained in Subsection 4.1 since et ij ¼ l4 Aij . 4.3. High elasticity limit Here we try to establish a model flow regime where the viscous forces do not distort the director field. In the high elasticity limit, i.e., very small Reynolds and Ericksen numbers (Re 1 and Er 1), we can neglect the fluid inertial terms as well as the response of the director to the flow field and just use the static director field for v = 0. This means that the flow field does not create any distortion to the director but whenever a director field gradient occurs in the nematic sample a flow perturbation can be induced. For zero velocity field the static director defines a static pressure ps via the linear momentum balance. This pressure is only a function of the director field and xi and, therefore, does not depend on the velocity field. For v 5 0 we subdivide the pressure as p = ph ps, where ph is the hydrodynamic contribution which is a function of the director and velocity fields as well as xi. 4.3.1. Governing partial differential equations Using (2.22) (see Section 2), the angular momentum equations in the high elasticity limit for a ¼ 0 and v = 0 can be written as follows: stationary director field on ot
C. Atkinson, P.J.S. Pereira / International Journal of Engineering Science 43 (2005) 1185–1205
oW o oW cni þ þ Gi ¼ 0 oni oxj oni;j
1199
ð4:25Þ
where the director inertial terms have been dropped (see Section 2) and e g ¼ 0. Note that (4.25) is the Euler–Lagrange equilibrium equation of the static theory (see, e.g., [10]). Using (2.4), (2.8), (2.10)–(2.12), (2.13), (2.18), (2.20), (2.23) and (2.25) and the scalar product of ¼ 0 and v = 0 is given by (4.25) with ni,k, we find that the static pressure for on ot ps ¼ C þ 1ðxi ; ni Þ W
ð4:26Þ
where C 2 R. From (2.18), (2.20), (2.21), (2.23) and (2.25) as well as the scalar product of (4.25) with ni,k, we obtain the linear momentum equations in the high elasticity limit for a static director field associated with an incompressible isothermal nematic liquid crystal F i þ tij;j ¼ ðph þ W 1Þ;i þ et ij;j ¼ 0
ð4:27Þ
where (ph + W 1),i = (ph ps + C),i. As before the velocity field satisfies the incompressibility condition vi,i = 0 and the director field is constrained by nini = 1. Note that in the stationary case, the convective derivative v Æ $n is still present in the viscous ¼ 0. Thus, et ij ¼ stress tensor given by (2.8) whenever $n 5 0 and li 5 0 (i = 2, 3) but on ot C ijkp vk;p þ l2 vp ni;p nj þ l3 vp nj;p ni , where the fourth order rank tensor Cijkp is given by (3.2) (see Section 3). 4.3.2. Reciprocal materials for nematics in the high elasticity limit Similar expressions to those given above follow for the reciprocal material with all components with the overbar circumflex. For example, the angular and linear momentum equations are given by ! b b oW o oW bi ¼ 0 bc b þG þ ð4:28Þ ni ob n i oxj ob n i;j and b b1 Þ þ b et ij;j ¼ 0 Fb i þ bt ij;j ¼ ðb ph þ W ;i
ð4:29Þ
Using a similar procedure to that outlined in Section 3, a reciprocal material can be established such that the following result holds: Result 4. If ½v; n; et; p; W ; 1; lm ; K l (m = 1, 2, . . ., 6 and l = 1, 2, 3, 4) is an incompressible isothermal viscous flow of a nematic liquid crystal in the high elasticity limit, contained in the region Q bounded by a closed surface S regular or piecewise regular, with orientation given by an outward unit vector u normal to S, then
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ZZ S
½bv iet ij ðph þ W 1Þbv j dS j ¼
ZZ
b b1 Þvj dS j et ij ðb ½vib ph þ W S ZZZ þ ½bv i;j ðl2 vp ni;p nj þ l3 vp nj;p ni Þ QðSÞ
vi;j ðb l 2 bv p b l 3 bv p b n i;p b nj þ b n j;p b n i Þ dV
ð4:30Þ
where the reciprocal and real velocity fields satisfy the incompressibility conditions and the linear b ijkp ¼ C kpij . It can be shown that the conmomentum equations. The result (4.30) then follows if C b ijkp are satisfied provided: ditions on C b n¼n b 1 ¼ l1 l b l 4 ¼ l4 2b l 2 ¼ l6 l3 l5 þ l2
ð4:31Þ
2b l 3 ¼ l3 þ l6 l5 l2 2b l 5 ¼ l5 l2 þ l6 l3 2b l 6 ¼ l3 þ l2 þ l5 þ l6 b i ¼ li ði ¼ 1; 2; . . . ; 6Þ and If in addition we use the Parodi relationship (2.17) [11], we obtain l b k i ¼ ki ði ¼ 1; 2Þ. In the high elasticity limit a physically attainable director field to the reciprocal material that also satisfies the angular momentum balance given by (4.28) can be defined. Note that the real and auxiliary velocity fields are such that they do not create any distortion to the director field but it can induce some perturbation on both flows. As the reciprocal director field is chosen to be the same to that of the real material it also has to satisfy the angular momentum balance (4.25). The velocity fields v and bv of the real and auxiliary incompressible flows satisfy the linear momentum equations given by (4.27) and (4.29), respectively.
5. Applications and concluding remarks A reciprocal like theorem described by ZZ ZZ b bv i tij dS j vi t ij dS j ¼ S
ð5:1Þ
S
can be used as an auxiliary tool to find the coefficients of leading order terms in velocity fields near sharp edges or corners. Applications of this kind have been made by Atkinson and Bastero [4] for anisotropic elastic fields and by Barone and Robinson [14] for isotropic elastic fields. For slow viscous flows analogous results hold, i.e., if the actual velocity field at a sharp corner has the form vj = rkfj(h) with (r, h) a plane polar coordinate system with origin at the corner then the auxiliary field bv j can be chosen so that bv j ¼ rk gj (h) with fj(h) and gj(h) given. Furthermore, the auxiliary field can be chosen to satisfy appropriate boundary conditions on wedge sides. For example, if one writes down the eigenvalue equation for a viscous flow problem such as that considered by
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Moffatt [15] this equation would be given by sin(2ka) = ± k sin(2a), where a is the included half angle of the wedge and the minus sign corresponds to the antisymmetric flow, whereas the plus sign corresponds to the symmetric flow. In this case bv j ¼ rk gj (h) and satisfies the same boundary conditions as vj. Note that these eigenvalue problems apply also to traction boundary conditions. Furthermore, the eigenvalue problems will in principle apply to more general boundary conditions where there is a third component of velocity (v1, v2, v3) even though these velocity fields depend only on (r, h) and are independent of z. Such flows have been considered by Atkinson and Pereira [16] for the general non-linear equations. However, in these cases the eigensolutions are only local solutions valid close to the corner. An approximate version of the above applications could be devised for these local solutions. However, our main result described by (3.5) (see Section 3) is valid for the full asymmetric stress tensor and non-linear field. This theorem can also be used as an auxiliary tool when generating numerical solutions. The result (3.5) can be generalized to biaxial nematic liquid crystals. If one uses the theory outlined in Leslie et al. [17] and Leslie [18] one simply needs to identify the two sets of orthonormal b of the reciprocal material so b b ¼ m. director vector fields n, m of the real field and b n, m n ¼ n and m Using the biaxial nematic liquid crystal equations presented in Appendix A, we find the following result: ZZ ½bv iet ij ðp þ W 1 Þbv j dS j S
¼
ZZ S
c 1b Þvj dS j et ij ðb ½vib pþW
oni onj omi bv i;j l2 þ nj þ l3 ni þ b2 mj ot ot ot QðSÞ omj onp onp þb3 mi þ a1 mi mp nj þ a2 mj mp ni ot ot ot bi bj ob ni ob nj om om b b bj þ b bi l3 b2 b3 vi;j b l2 nj þ b ni þ b m m ot ot ot ot ob np ob np onk onp b pb b pb b im b jm þ k3 mp mk a2 nj þ b n i þ bv i nk;i k1 þb a1 m m ot ot ot ot bk ob n k b ob np omk om b k þ bv i mk;i c1 b k;i bc 1 bpm vi b k1 n k;i b þ k3 m vi m dV ot ot ot ot ZZZ
ð5:2Þ where p is the hydrostatic pressure, 1*(xi, ni, mi) is some scalar energy density function, which represents some external body forces, and W * the elastic energy function of the biaxial material. In (5.2) the several coefficients denote viscosities of the real and reciprocal materials. Again the terms mi involving ootbn i and ootb could be set to zero if required. Using a similar procedure to that outlined in Section 3, the following relations between the sets of viscosity coefficients can be shown to be necessary for (5.2) to hold in this case:
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b b ¼m n ¼ n; m b b bc 1 ¼ c1 ¼ b3 b2 ; k 3 ¼ k3 ¼ a2 a1 k 1 ¼ k1 ¼ l3 l2 ; b b bc 2 ¼ b2 þ b3 ; k 4 ¼ a1 þ a2 k 2 ¼ l2 þ l3 ; b b b ¼b l ¼l ; 1
1
b l 4 ¼ l4 ;
1
1
b a 5 ¼ a5 ¼ 0
ð5:3Þ 2b a 1 ¼ a4 a2 a3 þ a1
2b l 5 ¼ l5 l2 þ l6 l3 ;
2b b 2 ¼ b6 b3 b5 þ b2 ; 2b b 3 ¼ b3 þ b6 b5 b2 ; b ¼b b þb b ; 2b
2b l 6 ¼ l3 þ l2 þ l5 þ l6 ;
2b b 6 ¼ b3 þ b2 þ b5 þ b6 ;
2b a 4 ¼ a1 þ a2 þ a3 þ a4
2b l 2 ¼ l6 l3 l5 þ l2 ; 2b l 3 ¼ l3 þ l6 l5 l2 ;
5
5
2
6
3
2b a 2 ¼ a2 þ a4 a1 a3 2b a 3 ¼ a4 a2 þ a3 a1
For these incompressible isothermal biaxial nematics the Onsager–Parodi relationships are defined by l2 þ l3 ¼ l6 l5 ; b2 þ b3 ¼ b6 b5 ; a1 þ a2 ¼ a4 a3 ; a5 ¼ 0 ð5:4Þ From (5.3) one can see that the condition, a5 = 0, necessary to establish the result (5.2) is in accordance with the last Onsager–Parodi relationship of (5.4). Using (5.3) as well as (5.4) we obtain the following conditions for the viscosity coefficients of both flows: b b ¼m n ¼ n; m b b bc 1 ¼ c1 ¼ b3 b2 ; k 3 ¼ k3 ¼ a2 a1 k 1 ¼ k1 ¼ l3 l2 ; b bc 2 ¼ c2 ¼ b2 þ b3 ¼ b6 b5 k 2 ¼ k2 ¼ l2 þ l3 ¼ l6 l5 ; b k 4 ¼ k4 ¼ a1 þ a2 ¼ a4 a3 b l i ¼ li ði ¼ 1; 2; . . . ; 6Þ;
b b j ¼ bj
ðj ¼ 1; 2; 3; 5; 6Þ;
b a p ¼ ap
ð5:5Þ ðp ¼ 1; 2; . . . ; 4Þ
b a 5 ¼ a5 ¼ 0 Note that the result for biaxial nematic liquid crystals is true for any set of viscosity coefficients except that a5 must be zero, so the result (5.2) generalized does not require the Parodi relationships (5.4) to be true except for the last one, i.e., a5 = 0. The results established in Section 4 can be generalized to biaxial nematic liquid crystals. In Appendix B, the result (4.30) (see Subsection 4.3) is generalized to biaxial nematics in the high elasticity limit. In the general case as well as in the high viscosity and elasticity limits, where the auxiliary director fields are chosen to be the same to those of the real material, integral relations between velocity and stress fields of the real and reciprocal materials are obtained.
Acknowledgements The author P.J.S. Pereira thanks to the Instituto Superior de Engenharia de Lisboa and Ministry of Education of Portugal for a scholarship.
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Appendix A Using the theory outlined in Leslie et al. [17] and Leslie [18] the equations in Cartesian coordinates describing the hydrodynamic behaviour of biaxial nematic liquid crystals can be written in a similar form to that deduced in Section 2. We consider an incompressible isothermal biaxial nematic liquid crystal with no director inertial terms and external body forces given by Fi ¼
o1 ðxi ; ni ; mi Þ ; oxi
Gni ¼
o1 ðxi ; ni ; mi Þ oni
and Gmi ¼
o1 ðxi ; ni ; mi Þ omi
ðA:1Þ
where F is any external body force and Gn as well as Gm are generalized body forces arising from a body couple present due to magnetic or electric fields and associated with the directors n and m, respectively. As before the angular momentum equations can be written as follows: oW o oW n e þ gi þ þ Gni ¼ 0 c ni þ j mi oxj oni;j oni ðA:2Þ oW o oW m m þe gi þ þ Gi ¼ 0 s mi þ j ni oxj omi;j omi where c*, j*, and s* are arbitrary scalars. The linear momentum equations can be written as follows: g nk þ mk;i e g mk þ et ij;j q_vi ¼ ðp þ W 1 Þ;i þ nk;i e
ðA:3Þ
Using (A.3) we have et ij;j ¼
o ðp þ W 1 Þ nk;i e g nk mk;i e g mk oxi
ðA:4Þ
where the fluid inertial terms q_vi have been neglected (see Section 1). Here the asymmetric viscous stress tensor is given by et ij ¼ l1 nk np Akp ni nj þ l2 N i nj þ l3 N j ni þ l4 Aij þ l5 Aik nk nj þ l6 Ajk nk ni þ b1 mk mp Akp mi mj þ b2 M i mj þ b3 M j mi þ b5 Aik mk mj þ b6 Ajk mk mi þ N p mp ða1 mi nj þ a2 mj ni Þ ðA:5Þ þ nk Akp mp ða3 mi nj þ a4 mj ni Þ þ a5 mk mp Akp ni nj with 2N i ¼ 2n_ i 2Aij nj 2M i ¼ 2m_ i 2Aij mj
ðA:6Þ
and li (i = 1, 2, . . ., 6), bj (j = 1, 2, 3, 5, 6), as well as ap (p = 1, 2, . . ., 5) are viscosity coefficients. The n m g are given by vectors e g and e e g ni ¼ ðk1 N i þ k2 Aip np þ k3 N p mp mi þ k4 mk Akp np mi Þ e g mi ¼ ðc1 M i þ c2 Aip mp Þ
ðA:7Þ
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where k1 ¼ l3 l2 ;
k2 ¼ l6 l5 ;
c1 ¼ b3 b2 ;
c2 ¼ b6 b5
k3 ¼ a2 a1 ;
k4 ¼ a4 a3
ðA:8Þ
and Aij and Aij are given by (2.10) and (2.12), respectively (see Section 2). The elastic free energy density W * for hard biaxial nematic liquid crystals, i.e., considering materials in the hydrodynamic limit where both order parameters associated with the director vector fields n and m are nearly constant and only these orthonormal director fields depend on position, is given by (see Longa et al. [19]) 2
2
2
2
2W ¼ K l1 ðdiv lÞ þ K l2 ðl curl lÞ þ K l3 jl curl lj2 þ K m1 ðdiv mÞ þ K m2 ðm curl mÞ þ K m3 jm curl mj2 þ K n1 ðdiv nÞ2 þ K n2 ðn curl nÞ2 þ K n3 jn curl nj2
þ K mn ðm curl nÞ2 þ K nl ðn curl lÞ2 þ K lm ðl curl mÞ2 þ K l4 div ½ðl 5Þl ldiv l þ K m4 div ½ðm 5Þm mdiv m þ K n4 div ½ðn 5Þn ndiv n
ðA:9Þ
where Kli (i = 1, 2, 3, 4, m), Kmj (j = 1, 2, 3, 4, n) and Knp (p = 1, 2, 3, 4, l) denote the elastic constants and l = m · n. Note that (A.9) reduces to (2.7) (see Section 2) with Kni = Ki and Kn4 = (K2 + K4) (i = 1, 2, 3) by disregarding the terms that involve the secondary director fields m and l. For these biaxial nematics the Onsager–Parodi relationships are defined by (5.4) (see Section 5). Finally, the velocity field satisfies the incompressibility condition (vi,i = 0) and the director fields n and m as well as l are mutually orthonormal vectors, i.e., nini = 1, mimi = 1, lili = 1, n Æ m = 0, n Æ l = 0 and m Æ l = 0.
Appendix B The result (4.30) (see Subsection 4.3) can be generalized to biaxial nematic liquid crystals in the high elasticity limit as follows: ZZ ½bv iet ij ðph þ W 1 Þbv j dS j S ZZ ZZZ b c ¼ ½viet ij ðc ph þ W 1b Þvj dS j þ fbv i;j ½l2 vp ni;p nj þ l3 vp nj;p ni S
QðSÞ
n i;p b nj þ b n j;p b ni þ b2 vp mi;p mj þ b3 vp mj;p mi þ ða1 mi nj þ a2 mj ni Þmp vk np;k vi;j ½b l 2 bv p b l 3 bv p b b bv p m b p bv k b b i;p m bj þ b b j;p m b i þ ðb b ib b jb þb b 3 bv p m a1 m a2 m ðB:1Þ nj þ b niÞm n p;k g dV 2 It can be shown that the result (B.1) holds if the conditions (5.3) are satisfied with exception to those involving the coefficients of friction of both flows, i.e., ki, b k j , c1 and bc p (i = 1, 3; j = 1, 2, 3, 4; p = 1, 2). In addition, if one uses the Onsager–Parodi relationships given by (5.4), we obtain the conditions (5.5) (see Section 5). As before the angular momentum equations are obtained from n m g ¼ 0. The linear momentum equa(A.2) for static director fields and v = 0, so the vectors e g ¼e n m g and replactions are obtained from (A.4) disregarding the terms involving the vectors e g and e
C. Atkinson, P.J.S. Pereira / International Journal of Engineering Science 43 (2005) 1185–1205
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ing p by ph. Moreover, the viscous stress tensor in the linear momentum balance is given by (A.5) ¼ om ¼ 0 but the convective derivatives v Æ $n and v Æ $m are still present. (see Appendix A) with on ot ot Similarly, we subdivide the pressure for biaxial nematics as p = ph ps where ps = C2 + 1*(xi, ni, mi) W * (C 2 2 R; see Subsection 4.3).
References [1] S. Kim, S. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth, London, 1991. [2] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague, The Netherlands, 1983. [3] F.R.N. Nabarro, Theory of Crystal Dislocations, Clarendon Press, Oxford, 1967. [4] C. Atkinson, C. Bastero, On the use of Bettis reciprocal theorem for computing the coefficients of stress singularities in anisotropic media, Int. J. Eng. Sci. 29 (1991) 727–741. [5] H.B. Glenn, Advances in Liquid Crystals, Academic Press, New York, 1978. [6] W. Helfrich, Conduction-induced alignment of nematic liquid crystals: basic model and stability considerations, J. Chem. Phys. 51 (1969) 4092–4105. [7] E. Dubois Violette, P.G. de Gennes, O. Parodi, Hydrodynamic instabilities of nematic liquid crystals under a.c. electric fields, J. Phys. (Paris) 32 (1971) 305–317. [8] J.L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol. 5 (1961) 23–34. [9] F.M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal. 28 (1968) 265–283. [10] I.W. Stewart, The static and dynamic continuum theory of liquid crystals, The Liquid Crystals Book Series, Taylor & Francis, London, 2004. [11] O. Parodi, Stress tensor for a nematic liquid crystal, J. Phys. (Paris) 31 (1970) 581–584. [12] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, 1992. [13] C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969. [14] M.R. Barone, A.R. Robinson, Determination of elastic stresses at notches and corners by integral equations, Int. J. Solids Struct. 8 (1972) 1319–1338. [15] H.K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18 (1964) 1–18. [16] C. Atkinson, P.J.S. Pereira, Flow of a nematic liquid crystal near the leading edge of an infinite prism, Q. J. Mech. Appl. Math., in press. [17] F.M. Leslie, J.S. Laverty, Continuum theory for biaxial nematic liquid crystals, Q. J. Mech. Appl. Math. 45 (1992) 595–606. [18] F.M. Leslie, Flow alignment in biaxial nematic liquid crystals, J. Non-Newtonian Fluid Mech. 54 (1994) 241–250. [19] L. Longa, J. Stelzer, Density functional approach to study the elastic constants of biaxial nematic liquid crystals, J. Chem. Phys. 109 (1998) 1555–1566.