3.1 Stationary state

The external gradients drive the nematic layer into a nonequilibrium steady state. We shall assume that the temperature difference T<sup>1</sup> � T<sup>2</sup> amounts only to a few degrees, so that there are no nematic layer flows vst <sup>i</sup> <sup>¼</sup> <sup>0</sup> , nor instabilities of the Rayleigh-Bénard type. In this NESS, we choose as the nematodynamic variables the set <sup>Ψ</sup> <sup>¼</sup> <sup>ρ</sup>; <sup>s</sup>; vi f g ; ni , where s r!; <sup>t</sup> is the specific entropy density (entropy per unit mass), the hydrodynamic velocity is vi r !; t and ni <sup>r</sup> !; t is the director field. It is to be expected that in this steady state, the changes in Ψst will only occur in the <sup>z</sup> direction, so that <sup>Ψ</sup>st <sup>¼</sup> <sup>Ψ</sup>½ � p zð Þ; T zð Þ , where <sup>p</sup> is the local pressure. We assume that Ψst admits an expansion of the Taylor series around an equilibrium state T0; p<sup>0</sup> at z<sup>0</sup> ¼ 0, and we consider only first-order terms in the gradients. Thus, by setting the values of the temperature at the plates, T<sup>1</sup> ¼ T zð Þ ¼ �d=2 and T<sup>2</sup> ¼ T zð Þ ¼ d=2 , the steady temperature profile is completely determined by:

$$T^{\text{et}} = T(\mathbf{z}) = T\mathbf{o} + \frac{dT}{dz}\mathbf{z} = T\mathbf{o}\left(\mathbf{1} - \frac{a}{T\_0}\mathbf{z}\right),\tag{1}$$

where <sup>T</sup><sup>0</sup> � <sup>T</sup>stð Þ¼ <sup>z</sup> <sup>¼</sup> <sup>0</sup> ð Þ <sup>T</sup><sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>=</sup><sup>d</sup> and <sup>α</sup> � <sup>Δ</sup>T=d, with <sup>Δ</sup><sup>T</sup> � <sup>T</sup><sup>1</sup> � <sup>T</sup>2. In what follows, we shall only consider <sup>T</sup>0≈<sup>3</sup> � <sup>10</sup><sup>2</sup> K, and it will be convenient to introduce the effective temperature gradient <sup>∇</sup>zTst � <sup>X</sup>^<sup>z</sup> as [37],

$$X \equiv -a + \frac{\mathbf{g}\beta T\_0}{c\_p},\tag{2}$$

new state variables. In the case of the present model, owing to the initial orientation

around the z axis, symmetry under inversions with respect to both, the xy plane and with respect to reflections on planes containing the z axis. A proper set of variables

> ! k ! ; t � �<sup>þ</sup> <sup>Θ</sup> ! k ! ; t

!<sup>L</sup> k ! ; t

. The superscript t denotes the transpose, while L and T

! k ! ; t

with δX

!<sup>L</sup>; Θ !T � �<sup>t</sup>

two sets completely independent: the five longitudinal <sup>δ</sup> <sup>e</sup>p; δφe; <sup>δ</sup>es; <sup>δ</sup>eξ; <sup>δ</sup>e<sup>f</sup> <sup>1</sup>

define a new set of variables having the same dimensionality, δzj k

s � ��1=<sup>2</sup>

> with Z<sup>L</sup> ! k ! ; t

<sup>s</sup> <sup>k</sup>�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

indicate, respectively, the longitudinal and transverse sets of variables. In Eq. (3), M stands for a 7 � 7 hydrodynamic matrix which is diagonal in the 5 � <sup>5</sup> <sup>N</sup><sup>L</sup> and the <sup>2</sup> � <sup>2</sup> <sup>N</sup><sup>T</sup> blocks. The explicit form of these matrices is not necessary in our discussion; however, they are given explicitly by Eqs. (21) and (22) in Ref. [26] (see also

can be found in Eqs. (32) and (33) in Ref. [26] (or Eqs. (84) and (85) of Ref. [25]). It is important to emphasize that as a consequence of this change of representation, in this last system, it can be clearly seen how the nematic variables are separated in

However, in order to facilitate the calculation of the hydrodynamic modes, we

terms of these new variables, the system of equations (3) is rewritten in the more

diagonal in the 5 � <sup>5</sup> <sup>N</sup><sup>L</sup> and the 2 � <sup>2</sup> <sup>N</sup><sup>T</sup> blocks. Again, the explicit form of these matrices is not necessary in our discussion, but they are given explicitly by Eqs. (39)–(41) in Ref. [26] (see also Eqs. (94)–(96) in [25]). In Eq. (4),

<sup>δ</sup>ep, <sup>z</sup><sup>2</sup> � <sup>ρ</sup>0k�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

<sup>δ</sup>e<sup>f</sup> <sup>1</sup> , <sup>z</sup><sup>6</sup> � <sup>ρ</sup>0k�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

k ! ; t � �<sup>þ</sup> <sup>Ξ</sup> ! k ! ; t

� �, (3)

� �, in Eq. (3) are given

n o and the

! ; t h i � � <sup>¼</sup>

<sup>s</sup> <sup>k</sup>�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

p � �<sup>1</sup>=<sup>2</sup>

δes,

δ ef <sup>2</sup> . In

δφe, <sup>z</sup><sup>3</sup> � <sup>ρ</sup>0T0c�<sup>1</sup>

� �, (4)

� � <sup>¼</sup> <sup>ζ</sup>6; <sup>ζ</sup><sup>7</sup> ð Þ<sup>t</sup> noise vectors. The

δψe, <sup>z</sup><sup>7</sup> � <sup>ρ</sup>0c<sup>2</sup>

� � <sup>¼</sup> ð Þ <sup>z</sup>1; <sup>z</sup>2; <sup>z</sup>3; <sup>z</sup>4; <sup>z</sup><sup>5</sup> <sup>t</sup> and

. In Eq. (4), N stands for a 7 � 7 hydrodynamic matrix which is

is the stochastic term, composed by the longitudinal

!<sup>T</sup> k ! ; t and

� �<sup>t</sup>

which explicit form of its components

� � <sup>¼</sup> <sup>δ</sup> <sup>e</sup>p; δφe; <sup>δ</sup>es; <sup>δ</sup>eξ; <sup>δ</sup>e<sup>f</sup> <sup>1</sup>

for this purpose was proposed long ago [38, 39], in terms of the variables

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

the Fourier transformation of this set of equations is given by:

!<sup>L</sup>; δX !T � �<sup>t</sup>

Eqs. (72) and (73) in Ref. [25]). The stochastic terms, Θ

! k ! ; t � � <sup>¼</sup> <sup>Θ</sup>

n o.

<sup>t</sup> (<sup>j</sup> <sup>¼</sup> 1,…, 7): <sup>z</sup><sup>1</sup> � <sup>ρ</sup>0c<sup>2</sup>

<sup>δ</sup>eξ, <sup>z</sup><sup>5</sup> � <sup>ρ</sup>0c<sup>2</sup>

∂ ∂t Z ! k ! ; t � � ¼ �N Z!

!<sup>L</sup>; Z !T � �<sup>t</sup>

� � <sup>¼</sup> <sup>ζ</sup>1; <sup>ζ</sup>2; <sup>ζ</sup>3; <sup>ζ</sup>4; <sup>ζ</sup><sup>5</sup> ð Þ<sup>t</sup> and transverse <sup>Ξ</sup>

∂ ∂t δ X ! k ! ; t � � ¼ �M<sup>δ</sup> <sup>X</sup>

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

� �, defined in detail in Eqs. (6)–(10) in Ref. [26] (or Eqs. (53)–(57) in [25]). In this new representation, the complete set of stochastic hydrodynamic equations for the fluctuations takes an alternative form given by Eqs. (11)–(17) in Ref. [26] (or Eqs. (58)–(64) in [25]). The matrix representation of

<sup>i</sup> , the NLC exhibits several symmetries: rotational invariances

of the director n^st

where δ X ! k ! ; t � � <sup>¼</sup> <sup>δ</sup><sup>X</sup>

� � <sup>¼</sup> δψe; <sup>δ</sup> <sup>e</sup><sup>f</sup> <sup>2</sup>

by the column vector Θ

two transverse δψe; <sup>δ</sup> <sup>e</sup><sup>f</sup> <sup>2</sup>

� �<sup>t</sup>

δX !<sup>T</sup> k ! ; t

M<sup>1</sup>=<sup>2</sup>

Z !<sup>T</sup> k ! ; t � � <sup>¼</sup> ð Þ <sup>z</sup>6; <sup>z</sup><sup>7</sup> <sup>t</sup>

Ξ ! k ! ; t � � <sup>¼</sup> <sup>Ξ</sup>

Ξ !<sup>L</sup> k ! ; t

151

L�1=<sup>2</sup>

<sup>z</sup><sup>4</sup> <sup>¼</sup> <sup>ρ</sup>0k�<sup>4</sup> � �<sup>1</sup>=<sup>2</sup>

compact form as:

where Z ! k ! ; t � � <sup>¼</sup> <sup>Z</sup>

> !<sup>L</sup>; Ξ !T � �<sup>t</sup>

δp; δφ; δs; δξ; δf <sup>1</sup>; δψ; δf <sup>2</sup>

which contains explicitly the contributions of both external forces. In Eq. (2), cp is the specific heat at constant pressure, β is the thermal expansion coefficient, which satisfies the relationship <sup>β</sup><sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup> cp=T0c<sup>2</sup> <sup>s</sup> , where cs is the adiabatic sound velocity in the nematic, <sup>γ</sup> � cp=cv <sup>¼</sup> <sup>c</sup><sup>2</sup> <sup>s</sup> =c<sup>2</sup> <sup>T</sup>, being cv the specific heat at constant volume and cT the isothermic sound velocity in the nematic.
