6.2.1.1 Propagative modes

If R k � �! =Rc , R0, where R<sup>0</sup> � �ð Þ ν � χ 2 =ð Þ 4χν , 0, the visco-heat modes (18) will be propagative. According to Eq. (20), this occurs again if the α contained in X changes its sign and increases by several orders of magnitude, a situation that is achieved by inverting the temperature gradient (when heated from below and its intensity is increased). There are analytical [8, 37, 38] and experimental [43] studies that report, for simple fluids in these conditions, the presence of visco-heat propagative modes.

### 6.2.1.2 Pure diffusive modes

When <sup>R</sup><sup>0</sup> <sup>≤</sup> R k � �! =Rc ≤1, the visco-heat modes preserve the form (Eq. (18)), they are real and completely diffusive. In this regime, there are again three cases of Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal DOI: http://dx.doi.org/10.5772/intechopen.82609

special interest. If R k ! =Rc ¼ R0, then both visco-heat modes (18) are identical and equal to:

$$
\lambda\_{3,4} = \frac{1}{2}(\chi + \nu)k^2. \tag{21}
$$

On the other hand, if the simple fluid is in a state of homogeneous thermodynamic 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 ! =Rc ¼ 0; consequently, in this equilibrium state (identified by the superscript e), there is a thermal diffusive mode:

$$
\lambda\_3^{\epsilon} = \chi k^2 \tag{22}
$$

and the shear mode:

where cs corresponds to the adiabatic velocity of the sound in this medium and

On the other hand, according to the Eq. (7), the longitudinal visco-heat modes are:

ð Þ χ þ ν 2

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:

<sup>λ</sup><sup>6</sup> <sup>¼</sup> <sup>ν</sup>k<sup>2</sup>

� � <sup>g</sup>βX^ k 2 ⊥

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

The two visco-heat mode, as in the nematic, could be propagative or diffusive.

simple fluids, these have been predicted theoretically and corroborated experimen-

2

=Rc ≤1, the visco-heat modes preserve the form (Eq. (18)),

will be propagative. According to Eq. (20), this occurs again if the α contained in X changes its sign and increases by several orders of magnitude, a situation that is achieved by inverting the temperature gradient (when heated from below and its intensity is increased). There are analytical [8, 37, 38] and experimental [43] studies that report, for simple fluids in these conditions, the presence of visco-heat propa-

they are real and completely diffusive. In this regime, there are again three cases of

These characteristics depend on the values assumed by the ratio R k

=Rc , R0, where R<sup>0</sup> � �ð Þ ν � χ

=Rc is defined as:

R k � �!

Rc

is the corresponding coefficient of sound attenuation.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s � �

<sup>k</sup><sup>4</sup> � <sup>4</sup>χνk<sup>4</sup> <sup>1</sup> � <sup>R</sup>

Rc

: (19)

χνk<sup>4</sup> , (20)

� �!

=ð Þ 4χν , 0, the visco-heat modes (18)

=Rc. For

: (18)

<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

Non-Equilibrium Particle Dynamics

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

In Eq. (18), the ratio R k

given by Eq. (4) of [38].

6.2.1.1 Propagative modes

6.2.1.2 Pure diffusive modes

� �!

When R<sup>0</sup> ≤ R k

� �! =Rc

6.2.1 Values of R k

tally.

If R k � �!

gative modes.

158

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

� �!

∓ 1 2

$$
\lambda\_4^\epsilon = \nu k^2. \tag{23}
$$

These decay rates are well known in the literature [8, 37, 38]. Finally, because in a simple fluid, commonly <sup>ν</sup> is greater than <sup>χ</sup>, according to Eq. (18), and as R k ! =Rc 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 ! reaches its critical value Rc), R k ! =Rc ¼ 1, and the two visco-heat modes (18) acquire the values:

$$
\lambda\_3 = \mathbf{0} \tag{24}
$$

and

$$
\lambda\_4 = (\chi + \nu)k^2. \tag{25}
$$

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 reference [37].
