6.2 Simple fluid in a Rayleigh-Bénard system

Given that in the isotropic limit (simple fluid limit), the degree of nematic order goes to zero, ni is no longer a hydrodynamic variable, and the elastic constants Ki (for i ¼ 1, 2, 3) and the kinetic parameters γ1, λ vanish. Also, χ<sup>⊥</sup> and χ<sup>∥</sup> are reduced to the coefficient of thermal diffusivity χ and χ<sup>a</sup> ¼ 0. On the other hand, the nematic viscosities are reduced in the following way: ν<sup>1</sup> ! η, ν<sup>2</sup> ! η, ν<sup>3</sup> ! η, <sup>ν</sup><sup>4</sup> ! <sup>ζ</sup> <sup>þ</sup> <sup>1</sup> <sup>3</sup> <sup>η</sup>, <sup>ν</sup><sup>5</sup> ! � <sup>2</sup> <sup>3</sup> η þ ζ, where η and ζ denote, respectively, the shear and volumetric viscosities of the simple fluid. As a result, from Eqs. (23)–(31) in Ref. [26] (or Eqs. (74)–(82) in [25]), it follows that in the isotropic limit DT ! χ, <sup>σ</sup><sup>1</sup> ! <sup>1</sup> ρ0 4 <sup>3</sup> <sup>η</sup> <sup>þ</sup> <sup>ζ</sup> , <sup>σ</sup><sup>2</sup> ! 0, <sup>σ</sup><sup>3</sup> ! <sup>ν</sup>, <sup>σ</sup><sup>4</sup> ! <sup>ν</sup>, where <sup>ν</sup> � <sup>η</sup>=ρ<sup>0</sup> is the kinematic viscosity, whereas KI ! 0, KII ! 0, and Ω ! 0. Consequently, by making the identifications indicated above, the corresponding hydrodynamic modes of a simple fluid can be obtained when it is in a Rayleigh-Bénard system. Thus, according to Eq. (6), a simple fluid has the two acoustic propagative modes:

$$
\lambda\_1 \simeq \Gamma' k^2 + i c\_\nu k, \quad \lambda\_2 \simeq \Gamma' k^2 - i c\_\nu k,\tag{17}
$$

where cs corresponds to the adiabatic velocity of the sound in this medium and <sup>Γ</sup><sup>0</sup> � <sup>1</sup> <sup>2</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>χ</sup> <sup>þ</sup> <sup>1</sup> ρ0 4 <sup>3</sup> <sup>η</sup> <sup>þ</sup> <sup>ζ</sup> � � h i is the corresponding coefficient of sound attenuation. On the other hand, according to the Eq. (7), the longitudinal visco-heat modes are:

$$
\lambda\_{3,4} \simeq \frac{1}{2} (\chi + \nu) k^2 \mp \frac{1}{2} \sqrt{\left(\chi + \nu\right)^2 k^4 - 4\chi\nu k^4 \left(1 - \frac{R}{R\_c}\right)}.\tag{18}
$$

In the isotropic limit of the simple fluid, λ<sup>5</sup> ¼ λ<sup>7</sup> ¼ 0, so that, according to the Eq. (14), the only transverse mode of this substance in a Rayleigh-Bénard system is:

$$
\lambda\_{\mathsf{G}} = \mathsf{\nu}k^{\mathsf{Z}}.\tag{19}
$$

special interest. If R k !

DOI: http://dx.doi.org/10.5772/intechopen.82609

equal to:

diffusive mode:

and the shear mode:

reaches its critical value Rc), R k !

acquire the values:

and

reference [37].

7. Conclusions

159

ing comments may be useful.

=Rc ¼ R0, then both visco-heat modes (18) are identical and

: (21)

<sup>3</sup> <sup>¼</sup> <sup>χ</sup>k<sup>2</sup> (22)

=Rc ¼ 1, and the two visco-heat modes (18)

λ<sup>3</sup> ¼ 0 (24)

: (25)

: (23)

=Rc ¼ 0; conse-

=Rc

<sup>λ</sup>3,<sup>4</sup> <sup>¼</sup> <sup>1</sup> 2

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

namic equilibrium, <sup>g</sup> <sup>¼</sup> 0 and <sup>α</sup> <sup>¼</sup> 0, so that <sup>X</sup> <sup>¼</sup> 0 and R k !

ð Þ <sup>χ</sup> <sup>þ</sup> <sup>ν</sup> <sup>k</sup><sup>2</sup>

On the other hand, if the simple fluid is in a state of homogeneous thermody-

quently, in this equilibrium state (identified by the superscript e), there is a thermal

λe

λe <sup>4</sup> <sup>¼</sup> <sup>ν</sup>k<sup>2</sup>

a simple fluid, commonly <sup>ν</sup> is greater than <sup>χ</sup>, according to Eq. (18), and as R k !

grows and approaches to 1, the magnitude of the thermal diffusive mode decreases, while the shear mode grows. At the threshold of the convective regime (when R k !

<sup>λ</sup><sup>4</sup> <sup>¼</sup> ð Þ <sup>χ</sup> <sup>þ</sup> <sup>ν</sup> <sup>k</sup><sup>2</sup>

In this work, we have used the standard formulation of FH to describe the dynamics of the fluctuations of a NLC layer in a NESS characterized by the simultaneous action of a uniform temperature gradient α and a constant gravitational field g, which corresponds to a Rayleigh-Bénard system. The analysis carried out takes into account only the nonconvective regime. The most important results are the analytic expressions for the seven nematic hydrodynamic modes. The explicit details of several of the calculations can be found in Refs. [25, 26]. To summarize the results obtained in this work and to put them into a proper context, the follow-

First, in our analysis, the symmetry properties of the nematic are taken into consideration, and this allowed us to separate its hydrodynamic variables into two completely independent sets: one longitudinal, composed of five variables, and the

These three cases are consistent with those obtained in analytical studies already reported for simple fluids in this regime [8, 37, 38]. Schematically, its behavior is very similar to that illustrated in Figure 3, and this can be seen in Figure 1 of the

These decay rates are well known in the literature [8, 37, 38]. Finally, because in

In Eq. (18), the ratio R k � �! =Rc is defined as:

$$\frac{R\left(\stackrel{\rightarrow}{k}\right)}{R\_{\epsilon}} \equiv -\frac{\text{g}\rho \text{X}\hat{k}\_{\perp}^{2}}{\chi\nu k^{4}},\tag{20}$$

which, in this limit case, can be derived from Eq. (9). It should be pointed out that Eq. (20) coincides with the Eq. (2.21) of reference [37]. The modes (17)–(19) are in complete concordance with those analytically calculated in [8, 37, 38].

Moreover, if in the coefficient matrix M of the stochastic system given by Eq. (20) in Ref. [26], the simple fluid limit is taken, it reduces to a matrix that is a generalization of the one given by the Eq. (6) in [38]. Additionally, if in the corresponding matrix M found for the simple fluid, the equilibrium limit is now considered, i. e., when α and g vanish, the resulting matrix is also reduced to that given by Eq. (4) of [38].

#### 6.2.1 Values of R k � �! =Rc

The two visco-heat mode, as in the nematic, could be propagative or diffusive. These characteristics depend on the values assumed by the ratio R k � �! =Rc. For simple fluids, these have been predicted theoretically and corroborated experimentally.
