Dynamic Equations for SmC Liquid Crystals

The dynamic theory for SmC liquid crystals is based upon the same three conservation laws for mass, linear momentum and angular momentum that were employed in the derivation of the dynamic theory of nematics developed in Section 4.2.2. They are given by equations (4.29), (4.30) and (4.31) and, as before, the incompressibihty assumption applied to the mass conservation law leads to the familiar constraint that the velocity vector v satisfies 

Motivated by the static theory for SmC, it seems reasonable, from equations (6.70) and (6.90), to set 

Material frame-indifference requires that the dependence in equation (6.192) be replaced by (cf. the nematic case in Section 4.2.3 at equation (4.67)) 

The above constitutive assumptions (6.190) and (6.191) enable the identity (6.188) to be reduced to the inequality 

Under the assumption that iij is a linear function of A and the rate of strain Dij and that the symmetry requirement (6.11) must also hold, it can be shown that the dependence in (6.193) forces Uj to be an isotropic function of the variables listed there, analogous to the case for nematic liquid crystals (cf. Section 4.2.3). It then follows that iij consists of forty-one terms [266] they can be obtained by arguments analogous to those used to obtain the nematic viscous stress at equations (4.71) to (4.74). Four of these terms equate to zero by a simple apphcation of the inequality (6.199), analogous to showing /i2 = 0 in nematic theory at equation (4.84). A further five can be shown to be linear combinations of the other terms by means of rather involved vector identities, which leaves thirty-two terms. However, this number can be reduced further to twenty terms if we apply Onsager relations [266]. Employing the notation 

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