Nomenclature



Ξ<sup>L</sup> ! k ! ; t

Ξ<sup>T</sup> ! k ! ; t

Z !<sup>L</sup> <sup>X</sup> k ! ; t and <sup>Z</sup>

pL

a0, a<sup>0</sup>

a1, a<sup>0</sup>

^

a4, a<sup>0</sup>

λe

163

5, and a<sup>0</sup>

0, and a<sup>00</sup>

XXð Þ<sup>λ</sup> and pL

longitudinal component of <sup>Ξ</sup>

transverse component of <sup>Ξ</sup>

DT anisotropic thermal coefficient σ1, σ2, σ3, and σ<sup>4</sup> anisotropic viscous coefficients

χ<sup>a</sup> anisotropic thermal diffusivity ν<sup>i</sup> nematic viscosities (i ¼ 1, …, 5)

K1, K<sup>2</sup> and K<sup>3</sup> elastic coefficients of Frank KI and KII anisotropic elastic coefficients

λ<sup>3</sup> and λ<sup>4</sup> visco-heat longitudinal modes λ<sup>5</sup> director diffusive longitudinal mode

χ thermal diffusivity of a simple fluid

ν kinetic viscosity of a simple fluid

γ<sup>1</sup> torsion viscosity ω auxiliary parameter

<sup>k</sup>⊥and <sup>k</sup><sup>∥</sup> components of <sup>k</sup>

<sup>k</sup><sup>⊥</sup> � <sup>k</sup>⊥=<sup>k</sup> unit vector of ^

pð Þλ characteristic polynomial of the matrix N pLð Þ<sup>λ</sup> characteristic polynomial of the submatrix <sup>N</sup><sup>L</sup> pTð Þ<sup>λ</sup> characteristic polynomial of the submatrix <sup>N</sup><sup>T</sup>

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

components of the vector <sup>Z</sup><sup>L</sup>

Ω and λ<sup>þ</sup> anisotropic adimensional nematic coefficients

1, and a<sup>2</sup> small dimensionless longitudinal quantities a3, a5, and a<sup>6</sup> small dimensionless longitudinal quantities λ<sup>1</sup> and λ<sup>2</sup> acoustic propagative longitudinal modes Γ anisotropic sound attenuation coefficient

YYð Þ<sup>λ</sup> polynomials in which pLð Þ<sup>λ</sup> is broken down

χ<sup>∥</sup> and χ<sup>⊥</sup> thermal diffusivities parallel and perpendicular to n

<sup>0</sup> small dimensionless longitudinal quantities

!

k⊥ R<sup>0</sup> reference value of the Rayleigh ratio below which viscocaloric modes are propagative

<sup>6</sup> small dimensionless transverse quantities λ<sup>6</sup> and λ<sup>7</sup> shear and director diffusive transverse modes

<sup>i</sup> nematic modes in the state of equilibrium (i ¼ 1, …, 7)

η and ζ shear and volumetric viscosities of a simple fluid

Γ<sup>0</sup> attenuation coefficient of sound in a simple fluid

<sup>ζ</sup><sup>i</sup> noise components of <sup>Ξ</sup><sup>L</sup>

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

λ eigenvalues of pð Þλ

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

(<sup>i</sup> <sup>¼</sup> 1,…, 5) and <sup>Ξ</sup><sup>T</sup>

perpendicular and parallel to n^<sup>0</sup>

!

(i ¼ 6, 7)

!

! k ! ; t 

> ! k ! ; t

!

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal DOI: http://dx.doi.org/10.5772/intechopen.82609

Ξ<sup>L</sup> ! k ! ; t Ξ<sup>T</sup> ! k ! ; t transverse component of <sup>Ξ</sup> <sup>ζ</sup><sup>i</sup> noise components of <sup>Ξ</sup><sup>L</sup> λ eigenvalues of pð Þλ Z !<sup>L</sup> <sup>X</sup> k ! ; t and <sup>Z</sup> !<sup>L</sup> <sup>Y</sup> k ! ; t pL XXð Þ<sup>λ</sup> and pL χ<sup>a</sup> anisotropic thermal diffusivity K1, K<sup>2</sup> and K<sup>3</sup> elastic coefficients of Frank KI and KII anisotropic elastic coefficients γ<sup>1</sup> torsion viscosity ω auxiliary parameter a0, a<sup>0</sup> 0, and a<sup>00</sup> a1, a<sup>0</sup> λ<sup>3</sup> and λ<sup>4</sup> visco-heat longitudinal modes <sup>k</sup>⊥and <sup>k</sup><sup>∥</sup> components of <sup>k</sup> ^ <sup>k</sup><sup>⊥</sup> � <sup>k</sup>⊥=<sup>k</sup> unit vector of ^ a4, a<sup>0</sup> 5, and a<sup>0</sup> λe

g

v

r

δ X ! k ! ; t

δX<sup>L</sup> ! k ! ; t

δX<sup>T</sup> ! k ! ; t

Θ ! k ! ; t 

Θ<sup>L</sup> ! k ! ; t

Θ<sup>T</sup> ! k ! ; t

zi k ! ; t

Z ! k ! ; t 

ZL ! k ! ; t

ZT ! k ! ; t

Ξ ! k ! ; t 

162

! constant gravitational force of magnitude g

α temperature gradient of magnitude ∇zT

s specific density of entropy (entropy per unit mass)

ΔT temperature difference between the plates of the cell

!

!

vector whose components are the spatial Fourier trans-

form of the variables δp, δφ, δs, δψ, δξ, δf <sup>1</sup> and δf <sup>2</sup>

stochastic vector of the linear system for δ X

! k ! ; t 

! k ! ; t 

! k ! ; t 

! k ! ; t 

! ; t 

> ! k ! ; t

! k ! ; t  !

!

!

! k ! ; t 

! k ! ; t 

> ! k ! ; t

! k ! ; t 

x, ^ ^y, ^z Cartesian unitary vectors x, y, z Cartesian coordinates

ρ volumetric density of mass

X effective temperature gradient β coefficient of thermal expansivity cp specific heat at constant pressure cv specific heat at constant volume

δψ component z of the rotational of δ v

δf <sup>2</sup> component z of the rotational of δ n

longitudinal component of δ X

transverse component of δ X

longitudinal component of Θ

transverse component of Θ

longitudinal component of Z

transverse component of Z

δξ component z of the double rotational of δ v

t as superscript, indicates transpose matrix

<sup>M</sup> coefficient matrix of the linear system for <sup>δ</sup> <sup>X</sup>

M<sup>L</sup> and M<sup>T</sup> longitudinal and transverse submatrices of M

variables of same dimensionality (i ¼ 1,…, 7)

<sup>N</sup> coefficient matrix of the linear system for <sup>δ</sup> <sup>Z</sup>

N<sup>L</sup> and N<sup>T</sup> longitudinal and transverse submatrices of N

vector of the variables zi k

noise vector of the linear system for Z

p hydrostatic pressure ∇zp pressure gradient

T temperature

Non-Equilibrium Particle Dynamics

! flow velocity

! position vector

γ ratio of specific heats cs adiabatic sound velocity cT isothermic sound velocity Ψ set of nematodynamic variables

δφ divergence of δ v

δf <sup>1</sup> divergence of δ n

 longitudinal component of <sup>Ξ</sup> ! k ! ; t ! k ! ; t ! (<sup>i</sup> <sup>¼</sup> 1,…, 5) and <sup>Ξ</sup><sup>T</sup> ! (i ¼ 6, 7) pð Þλ characteristic polynomial of the matrix N pLð Þ<sup>λ</sup> characteristic polynomial of the submatrix <sup>N</sup><sup>L</sup> pTð Þ<sup>λ</sup> characteristic polynomial of the submatrix <sup>N</sup><sup>T</sup> components of the vector <sup>Z</sup><sup>L</sup> ! k ! ; t YYð Þ<sup>λ</sup> polynomials in which pLð Þ<sup>λ</sup> is broken down DT anisotropic thermal coefficient σ1, σ2, σ3, and σ<sup>4</sup> anisotropic viscous coefficients χ<sup>∥</sup> and χ<sup>⊥</sup> thermal diffusivities parallel and perpendicular to n ! ν<sup>i</sup> nematic viscosities (i ¼ 1, …, 5) Ω and λ<sup>þ</sup> anisotropic adimensional nematic coefficients <sup>0</sup> small dimensionless longitudinal quantities 1, and a<sup>2</sup> small dimensionless longitudinal quantities a3, a5, and a<sup>6</sup> small dimensionless longitudinal quantities λ<sup>1</sup> and λ<sup>2</sup> acoustic propagative longitudinal modes Γ anisotropic sound attenuation coefficient λ<sup>5</sup> director diffusive longitudinal mode ! perpendicular and parallel to n^<sup>0</sup> k⊥ R<sup>0</sup> reference value of the Rayleigh ratio below which viscocaloric modes are propagative <sup>6</sup> small dimensionless transverse quantities λ<sup>6</sup> and λ<sup>7</sup> shear and director diffusive transverse modes <sup>i</sup> nematic modes in the state of equilibrium (i ¼ 1, …, 7) χ thermal diffusivity of a simple fluid η and ζ shear and volumetric viscosities of a simple fluid ν kinetic viscosity of a simple fluid Γ<sup>0</sup> attenuation coefficient of sound in a simple fluid

Non-Equilibrium Particle Dynamics
