5.1.2 Visco-heat and director longitudinal modes

These modes are the roots of the characteristic equation pL YYð Þ¼ λ 0. In Ref. [26] (or in [25]), it is shown that, up to first order in the small quantities (Eq. (5)), these roots can be written approximately as:

$$\lambda\_{3,4} = \frac{1}{2} \left( D\_T k^2 + \sigma\_3 k^2 - \frac{\Omega^2 K\_l k^4}{\rho\_0 \sigma\_3 k^2} \right)$$

$$\mp \frac{1}{2} \sqrt{\left( D\_T k^2 + \sigma\_3 k^2 - \frac{\Omega^2 K\_l k^4}{\rho\_0 \sigma\_3 k^2} \right)^2 - 4 D\_T k^2 \sigma\_3 k^2 \left( 1 - \frac{R}{R\_c} \right)},\tag{7}$$

and

$$
\lambda\_5 \simeq \frac{K\_I k^2}{\varkappa\_1} + \frac{\Omega^2 K\_I k^4}{\rho\_0 \sigma\_3 k^2},
\tag{8}
$$

with

$$\frac{R\left(\overrightarrow{k}\right)}{R\_{\varepsilon}} \equiv -\frac{\text{g}\beta\hat{k}\_{\perp}^{2}}{D\_{T}\sigma\_{3}k^{4}}\left[\text{X} + \frac{\alpha\Omega\chi\_{a}}{D\_{T}\sigma\_{3}}(\sigma\_{3} + D\_{T})\right],\tag{9}$$

where ^ k2 <sup>⊥</sup> � <sup>k</sup><sup>2</sup> ⊥=k<sup>2</sup> . In Eq. (7), <sup>R</sup> � <sup>β</sup>gΔTd<sup>3</sup> <sup>σ</sup>3<sup>χ</sup> is the Rayleigh number and Rc denotes its critical value above which convection sets in. It should be emphasized that our results are expressed in terms of the ratio R k � �! =Rc and are, therefore, independent of the value of the separation d between the plates. However, the appropriate value of d in an experiment should be chosen with an experimental criterion [42].

The Rayleigh-number ratio R k � �! =Rc contains two contributions: the first term is due to the presence of the effective temperature gradient X, which depends on both, the temperature gradient α and the gravity field g. The second term is entirely a contribution due to α and the nematic anisotropy χa. For typical nematics and conventional light scattering experiments, both contributions are approximately of order 10�16.

The decay rates λ<sup>3</sup> and λ<sup>4</sup> for an inhomogeneous nematic given by Eq. (7) are called visco-heat modes, because they are composed of the coupling between the thermal DTk<sup>2</sup> and shear <sup>σ</sup>3k<sup>2</sup> � <sup>Ω</sup><sup>2</sup>KIk<sup>4</sup> <sup>ρ</sup>0σ3k<sup>2</sup> diffusive modes through the ratio R k � �! =Rc. The nature of these modes may be propagative or diffuse, as will be shown below.

### 5.1.3 Values of R k � �! =Rc

The three nematic modes (7) and (8) could be two propagative and one diffusive, or all of them completely diffusive; its nature depends on the values assumed by the ratio R k � �! =Rc. For simple fluids, these features have been predicted theoretically and corroborated experimentally, but to our knowledge, not for an NLC. In this sense, the following results suggest that it might be feasible to be also verified experimentally for nematics.
