6.1 Nematic in equilibrium

while the third, λ5, is identical to Eq. (8). This behavior for the decay rates λ<sup>3</sup>

It should be noted that our expressions for these three decay rates are not in agreement with those reported for an NLC in the literature [44, 45]. In these works, the director mode tends to zero, the shear mode does not change and there is an additional mode which is the sum of the thermal and director modes. In contrast, we have found that the thermal mode λ<sup>3</sup> vanishes, the director mode λ<sup>5</sup> is virtually unchanged, while λ<sup>4</sup> has contributions from the thermal and shear diffusive modes. We know that this phenomenon also occurs in the simple fluid, where there are two diffusive modes, the thermal mode also vanishes and the other one has contributions from the shear and thermal modes. In other words, our results reduce to the corresponding one for a simple fluid as R reaches its critical value Rc. Because for a simple fluid, these features have been predicted theoretically, our results suggest that it might be feasible to verify them experimentally also for nematics [8, 37, 38].

As mentioned earlier, pTð Þ<sup>λ</sup> is the characteristic polynomial of second order in <sup>λ</sup> of the matrix N<sup>T</sup>. The corresponding transverse hydrodynamic modes are the roots

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,

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

> <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

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>

According to Eqs. (64) and (65) in Ref. [26] (or Eqs. (167) and (168) in [25]), up to

It should be noted that these shear and director diffusive transverse modes also

From the hydrodynamic modes calculated for an NLC in a NESS determined by a Rayleigh-Bénard system, it is possible to obtain, as limit cases, the corresponding

<sup>∥</sup>=ρ0, may be identified in this equation. In terms of them, we

, k � <sup>10</sup><sup>5</sup>

, a<sup>0</sup>

γ1 þ λ2 þKIIk<sup>2</sup> k2 ∥ <sup>ρ</sup>0σ4k<sup>2</sup> (14)

, <sup>σ</sup><sup>4</sup> � <sup>10</sup>�<sup>2</sup>

, <sup>g</sup> � <sup>10</sup><sup>3</sup>

, and a<sup>0</sup>

<sup>5</sup> � <sup>10</sup>�<sup>5</sup>

<sup>∥</sup>, where again ω � csk. It should

, KII � <sup>10</sup>�<sup>6</sup> [32],

<sup>6</sup> � <sup>10</sup>�6.

, the quantities a4, a<sup>0</sup>

5

and λ<sup>4</sup> is also shown in Figure 3.

Non-Equilibrium Particle Dynamics

5.2 Transverse modes

of this equation pTð Þ¼ <sup>λ</sup> 0.

<sup>=</sup>γ<sup>1</sup> and <sup>λ</sup>þKIIk<sup>2</sup>

Eq. (5), namely, <sup>a</sup><sup>4</sup> � <sup>σ</sup>4k<sup>2</sup>

KIIk<sup>2</sup>

and a<sup>0</sup>

156

5.2.1 Shear and director transverse modes

k2

<sup>ω</sup> , a<sup>0</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>6</sup> have the orders of magnitude <sup>a</sup><sup>4</sup> � <sup>10</sup>�<sup>2</sup>

<sup>λ</sup><sup>6</sup> <sup>¼</sup> <sup>σ</sup>4k<sup>2</sup> � <sup>λ</sup><sup>2</sup>

6. The equilibrium and simple fluid limits

and also by taking into account that cs � <sup>10</sup><sup>5</sup>

<sup>5</sup> � KIIk<sup>2</sup> <sup>γ</sup>1<sup>ω</sup> , a<sup>0</sup>

first order in such small amounts, these two roots can be written as:

þKIIk<sup>2</sup> k2 ∥ <sup>ρ</sup>0σ4k<sup>2</sup> , <sup>λ</sup><sup>7</sup> <sup>¼</sup> KIIk<sup>2</sup>

match completely with those already reported for nematics [22, 31, 32].

It has been found that for an NLC in a NESS, the effects of the external gradients α and g are only manifested in the coupling of the thermal diffusive and shear longitudinal modes, which gives rise to the visco-heat modes λ3,<sup>4</sup> indicated, respectively, by means of Eq. (7). If the nematic layer is in a state of homogeneous thermodynamic equilibrium, g ¼ 0 and α ¼ 0, and therefore X ¼ 0 and R k ! =Rc ¼ 0: Thus, the hydrodynamic modes of a nematic, in the state of equilibrium (denoted by the superscript e), are composed of five longitudinal and two transverse modes. The longitudinal modes are integrated by the two acoustic propagatives λ<sup>e</sup> <sup>1</sup> and λ<sup>e</sup> <sup>2</sup> given by Eq. (6); as well as by the three diffusives, which consist of one thermal:

$$
\lambda\_3^{\epsilon} = D\_T k^2,\tag{15}
$$

another of shear:

$$
\lambda\_4^\varepsilon = \sigma\_3 k^2 - \frac{\Omega^2 K\_I k^2}{\rho\_0 \sigma\_3} \tag{16}
$$

and one more of the director, λ<sup>e</sup> <sup>5</sup>, which is the same as Eq. (8). The longitudinal diffusive modes (15) and (16) are obtained precisely from Eq. (7), since in this, the Rayleigh ratio, given by Eq. (9), is zero if α and g vanish. Moreover, the pair of transverse modes consist of the shear and director modes λ<sup>e</sup> <sup>6</sup> and λ<sup>e</sup> <sup>7</sup> which are equal to the Eq. (14). It is necessary to mention that the decay rates λ<sup>e</sup> <sup>i</sup> ð Þ i ¼ 1…7 are well known in the literature [22, 31, 46]. Note that λ<sup>e</sup> <sup>3</sup> and λ<sup>e</sup> <sup>4</sup> are shown in the middle part of Figure 3.
