5.2.1 Shear and director transverse modes

Accordingly, the shear and director transverse modes are the roots of pTð Þ¼ <sup>λ</sup> 0, and are given by Eq. (63) in Ref. [26] (or by Eq. (157) in [25]). Following again the approximate method of small quantities used previously, the quantities σ4, KIIk<sup>2</sup> <sup>=</sup>γ<sup>1</sup> and <sup>λ</sup>þKIIk<sup>2</sup> k2 <sup>∥</sup>=ρ0, may be identified in this equation. In terms of them, we have another set of anisotropic coefficients given by the viscosity σ4, the elasticity KII, and symmetry λ<sup>þ</sup> (see, respectively, Eqs. (27), (29), and (31) in [26]). We also define the small or reduced dimensionless quantities, analogous to those defined in Eq. (5), namely, <sup>a</sup><sup>4</sup> � <sup>σ</sup>4k<sup>2</sup> <sup>ω</sup> , a<sup>0</sup> <sup>5</sup> � KIIk<sup>2</sup> <sup>γ</sup>1<sup>ω</sup> , a<sup>0</sup> <sup>6</sup> � <sup>λ</sup><sup>2</sup> <sup>þ</sup>KII <sup>ρ</sup>0ω<sup>2</sup> <sup>k</sup><sup>2</sup> k2 <sup>∥</sup>, where again ω � csk. It should be noted that the viscous coefficient σ<sup>4</sup> only depends on the viscous coefficients ν2, ν3, while the elastic coefficient KII depends on the two Frank elastic constants K<sup>2</sup> and <sup>K</sup>3: Since for typical nematics <sup>λ</sup><sup>þ</sup> � 1, <sup>γ</sup><sup>1</sup> � <sup>10</sup>�<sup>1</sup> , <sup>σ</sup><sup>4</sup> � <sup>10</sup>�<sup>2</sup> , KII � <sup>10</sup>�<sup>6</sup> [32], and also by taking into account that cs � <sup>10</sup><sup>5</sup> , k � <sup>10</sup><sup>5</sup> , <sup>g</sup> � <sup>10</sup><sup>3</sup> , the quantities a4, a<sup>0</sup> 5 and a<sup>0</sup> <sup>6</sup> have the orders of magnitude <sup>a</sup><sup>4</sup> � <sup>10</sup>�<sup>2</sup> , a<sup>0</sup> <sup>5</sup> � <sup>10</sup>�<sup>5</sup> , and a<sup>0</sup> <sup>6</sup> � <sup>10</sup>�6. According to Eqs. (64) and (65) in Ref. [26] (or Eqs. (167) and (168) in [25]), up to first order in such small amounts, these two roots can be written as:

$$
\lambda\_6 = \sigma\_4 \boldsymbol{k}^2 - \frac{\lambda\_+^2 \boldsymbol{K}\_{\boldsymbol{I}\boldsymbol{k}} \boldsymbol{k}^2 \boldsymbol{k}\_{\parallel}^2}{\rho\_0 \sigma\_4 \boldsymbol{k}^2}, \quad \lambda\_7 = \frac{\boldsymbol{K}\_{\boldsymbol{I}\boldsymbol{k}} \boldsymbol{k}^2}{\gamma\_1} + \frac{\lambda\_+^2 \boldsymbol{K}\_{\boldsymbol{I}\boldsymbol{k}} \boldsymbol{k}^2 \boldsymbol{k}\_{\parallel}^2}{\rho\_0 \sigma\_4 \boldsymbol{k}^2} \tag{14}$$

It should be noted that these shear and director diffusive transverse modes also match completely with those already reported for nematics [22, 31, 32].
