NonEquilibrium Thermodynamics of Flowing Systems: 3

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1 *Following the developent in Beris and Edwards, 1994, Chapter 11 NonEquilibriu Therodynaics of Flowing Systes: 3 Antony N. Beris Schedule: Multiscale Modeling and Siulation of Coplex Fluids Center for Scientific Coputing and Matheatical Modeling University of Maryland, College Park 1. 4/13/07, 9:30 a Introduction. One ode viscoelasticity. 2. 4/13/07, 10:15 a Coupled transport: Two-fluid odel. 3. 4/14/07, 2:00 p Modeling under constraints: Liquid crystals*. 4. 4/14/07, 3:00 a Non-hoogeneous systes: Surface effects.

2 The ynaical Theory of Liquid Crystals Several levels of description: epending on the structural variable(s) used: irector description, n (unit vector) irector-scalar order paraeter description, n, s Tensor order paraeter description, (unit trace) epending on whether or not inertial coponents are kept in the structural evolution equations: Inertial forulations (where both c and ċ are considered as variables, c representing any structural paraeter) Inertialess forulations Interconnectivity between various forulations: Coplexity increases as the nuber of structural variables increase (but also the capability of representing ore states!) Inertial forulations useful to deduce the for of dissipation in inertialess odels

3 Hailtonian/dissipation structure in the presence of constraints It is very iportant to take into account the constraints of the structural variables irector n: unit vector, n n = 1 Variations constrained to be perpendicular to the director: n n = 0 This affects the definition of the Volterra derivatives of a functional F with respect to n, (δf/δn) c, since those also need to be in the sae subspace as n : This is defined fro the unconstrained functional (δf/δn) u by taking its projection to the noral to n space : (δf/δn) c = (δf/δn) u ((δf/δn) u n)n It also affects (potentially) the structure of the brackets. Forally, those can be constructed fro the equivalent brackets obtained in the absence of constraints through the following substitution: δ F 1 δf δf δf δ nn = p δ δ n n δn ( p p) This is obtained by exploring the relationship between dn/dt and dp/dt when n is forally obtained fro the unconstrained (still considered unit) variable p as: p dn 1 dp dp n= ; = dt n n dt dt ( p p) ( p p)

4 Hailtonian/dissipation structure in the presence of constraints (2) Order paraeter tensor : Syetric and of unit trace atrix, αβ = βα ; tr() = 1 Variations constrained to be syetric and traceless: tr( ) = 0 This affects the definition of the Volterra derivatives of a functional F with respect to, (δf/δ) c, since those also need to be in the sae subspace as : This is defined fro the unconstrained functional (δf/δ) u by taking its projection to a syetric and traceless space: (δf/δ) c = ½( (δf/δ) u + (δf/δ) ut ) - 1/3(tr ( (δf/δ) u ))δ. It also affects (potentially) the structure of the brackets. Forally, those can be constructed fro the equivalent brackets obtained in the absence of constraints through the following substitution: δf 1 δf δf δf δf : : δ tr( ) δ = δ δ δ c c c c δc δc This is obtained by exploring the relationship between d/dt and dc/dt when is forally obtained fro the unconstrained variable c (still considered of unit trace) as: c d 1 dc dc = ; = : tr( ) dt tr( ) δ c c dt dt

5 Inertial irector Theory: Variables For an incopressible, syste we have v, the velocity s, the entropy density (alternatively, T, teperature) n, the director (constrained to be a unit vector field) w, the oentu of the director (w = σ dn/dt)

6 Inertial irector Theory: Hailtonian The Hailtonian (extended Helholtz free energy of the syste) is assued to have the for: V 2 2 ( ) 1 2ρ 1 2σ ψ A= v + w + W + where W is the elastic (Oseen/Frank) distortion free energy density : W = ( k (div n) + k ( n curl n) + k (( n ) n) ) dv and ψ the effects of an external field. For exaple, for agnetically susceptible aterial it is given as: ( 2 ( ) ) n H H H ψ = 1 2 ( χ χ ) + χ where χ and χ are the agnetic susceptibilities perpendicular and parallel to n

7 Inertial irector Theory : Reversible equations For an isotheral syste, we get the standard reversible dynaics for a Hailtonian syste endowed with a vector structural paraeter and its (aterial) tie derivative: where F ρ t W vα = Fα p, α n β, α n βγ, =Φ H and Φ = ( χ χ ) n Hn + χ H α β α β 1 t n = w σ w n n, γ α α α ( ) α α β β α t δ H w = δ n

8 Inertial irector Theory : issipation Bracket Upper convected Lower convected where Q δf δg [ FG, ] Qαβγε α γ dω δv β δvε δf δf δg δg + α2 nβ β nγ γ dω δ wα δ vα δ wα δ vα δf δf δg δg α3 + nβ α + nγ α dω δ wα δ v β δ wα δ v γ ( ) = α n n n n + α δ δ + δ δ 1 αβγε 1 α β γ ε 2 4 αγ βε βγ αε ( α α )( δ nn nn) βε α γ δβγ α ε ( α α )( δ nn nn) αε β γ δαγ β ε

9 Inertial irector Theory : Final equations ρ t W v = F p n + t α α, α β, α αγ, γ nβγ,, γ where t αγ is exactly the Leslie/Ericksen stress 1 t n = w σ w n n ( ) α α β β α δ H w α 1 w n v 1 w n v t = + σ σ + ( ) α ( ) α 2 α γ γ α 3 α γ α γ δ nα

10 Inertialess irector Theory: Variables For an incopressible, syste we have v, the velocity s, the entropy density (alternatively, T, teperature) n, the director (constrained to be a unit vector field)

11 Inertialess irector Theory: Hailtonian The Hailtonian (extended Helholtz free energy of the syste) is assued to have the for: V 1 2 ( 2 ρ ψ) A= v + W + where W is the elastic (Oseen/Frank) distortion free energy density : W = ( k (div n) + k ( n curl n) + k (( n ) n) ) dv and ψ the effects of an external field. For exaple, for agnetically susceptible aterial it is given as: ( n H H H) ψ = 1 2 ( χ χ ) + χ where χ and χ are the agnetic susceptibilities perpendicular and parallel to n

12 Inertialess irector Theory : Reversible equations For an isotheral syste, we get the standard reversible dynaics for a Hailtonian syste endowed with a vector structural paraeter: where F ρ t W v = F p n + t α α, α β, α αγ, γ nβγ,, γ =Φ H and Φ = ( χ χ ) n Hn + χ H α β α β t αγ = δ H δ n 0 t n n v n n n v α β β α + α γ β β γ = α α α α n γ

13 Inertialess irector Theory : issipation Bracket where Q P L δf δg [ FG, ] Qαβγε α γ dω δv β δvε δf δg Pαβγε nβ nε dω δn α δn γ δf δg δg δf Lαβγε α nγ α nγ dω δv β δnε δv β δn ε n n n n 1 2 ( ) 1 2 β3 ( δβεnn α γ δβγnn α ε δαεnn β γ δαγnn β ε ) ( ) ( ) = β + β δ δ + δ δ αβγε 1 α β γ ε 2 αγ βε βγ αε = β δ δ + δ δ αβγε 4 αγ βε βγ αε = β δ δ + δ δ αβγε 5 αγ βε βγ αε

14 Inertialess irector Theory : Final equations ρ t W v = F p n + t α α, α β, α αγ, γ nβγ,, γ where t αγ is exactly the Leslie/Ericksen stress for suitably selected paraeters t δ H nα (1 + β5) ( nβ βvα nαnγnβ βvγ ) β5( nβ αvβ nαnγnβ γvβ) = β4 δ n α

15 Inertial Tensor Theory: Variables For an incopressible, syste we have v, the velocity s, the entropy density (alternatively, T, teperature), the tensor order paraeter (constrained to be of unit trace), = <nn> w, the oentu of the tensor order paraeter (w = σ d/dt)

16 Inertial Tensor Theory: Hailtonian The Hailtonian (extended Helholtz free energy of the syste) is assued to have the for: V ( ) 2ρ 2σ ψ A= v + w + W + + a b where W, the elastic (Oseen/Frank) distortion free energy density, is written in ters of gradients of, for exaple: ψ represents an external field. For exaple, for agnetically susceptible aterial it is given as: dv W = ( b( ) + b ( ) ) ( HH H H) ψ = 1 2 ( χ χ ) : + χ where χ and χ are the agnetic susceptibilities perpendicular and parallel to n Finally, a b represents the bulk free energy that can be represented through a phenoenological Landau/de Gennes expansion of S = 1/3 (tr)δ

17 Inertial Tensor Theory : Reversible equations For an isotheral syste, we get the standard reversible dynaics for a Hailtonian syste endowed with a tensor structural paraeter and its (aterial) tie derivative: where F ρ t W vα = Fα p, α βε, α βε, γ =Φ H and Φ = ( χ χ ) H + χ H α β α β t 1 t = w σ w α βα β α ( ) αβ αβ γβ αβ w δh δh = + : δ δ δ, γ

18 Inertial Tensor Theory : issipation Bracket δf δg [ FG, ] Rαβγε α γ dω δv β δvε δf δf δf δg δg δg Upper convected + α2 αγ γ βγ γ αγ γ βγ γ dω δwαβ δvβ δv α δwαβ δvβ δv α δf δf δf δg δg δg Lower convected α3 + αγ β + βγ α + αγ β + βγ α dω δwαβ δvγ δv γ δwαβ δvγ δv γ where ( ) ( ) ( α )( αγδβε αεδβγ βεδαγ βγδαε ) ( α )( αζ ζγδβε αζ ζεδβγ βζ ζεδαγ βζ ζγδαε ) ( α )( αζ ζγβε αζζεβγ βζζεαγ βζζγαε) ( α )( αζ ζγ βηηε αζ ζε βηηγ ) R = α + n + α δ δ + δ δ 1 1 αβγε 2 1 αγ βε αε βγ 2 4 αγ βε βγ αε

19 Inertial Tensor Theory : Final equations where ρ t W v = F p + T α α, α βε, α βα, β βε, γ, γ 1 1 Tαβ = Rαβγε ( vγ, ε + vε, γ ) + 2a2 βγ wαγ αεvγ, ε εγvα, ε + 2a3 αγ wβγ+ γεvε, β + βεvε, γ σ σ 1 t = w σ w ( ) αβ αβ γβ αβ δh δh 1 w = + : δ + a w v v t αβ αβ 2 αβ αγ β, γ βγ α, γ δαβ δ σ 1 + a w + v + v σ 3 αβ αγ γ, β βγ γ, α

20 Inertialess Tensor Theory: Variables For an incopressible, syste we have v, the velocity s, the entropy density (alternatively, T, teperature), the tensor order paraeter (constrained to be of unit trace), = <nn>

21 Inertialess Tensor Theory: Hailtonian The Hailtonian (extended Helholtz free energy of the syste) is assued to have the for: V ( 2 ) 1 2 ρ ψ A= v + W + + a b where W, the elastic (Oseen/Frank) distortion free energy density, is written in ters of gradients of, for exaple: ψ represents an external field. For exaple, for agnetically susceptible aterial it is given as: dv W = ( b( ) + b ( ) ) ( HH H H) ψ = 1 2 ( χ χ ) : + χ where χ and χ are the agnetic susceptibilities perpendicular and parallel to n Finally, a b represents the bulk free energy that can be represented through a phenoenological Landau/de Gennes expansion of S = 1/3 (tr)δ

22 Inertialess Tensor Theory : Reversible equations For an isotheral syste, we get the standard reversible dynaics for a Hailtonian syste endowed with a tensor, constrained, structural paraeter: ρ t W v = F p + T α α, α βε, α βα, β βε, γ, γ where δ H T = 2 2 δ αβ βγ αβ γε γα δ H δ γε ( ),, 2, 0 t v v v αβ αγ β γ + βγ α γ + αβ γε γ ε =

23 Inertialess Tensor Theory : issipation Bracket δf δg δf δg [ F, G] Rαβγε α γ dω Pαβγε dω δv β δv ε δ αβ δ γε δf δg δg δf Lαβγε α α dω δv β δγε δv β δ γε δf δg δg δf Lηζγγ αβ η η dω δvζ δαβ δvζ δ αβ where = β1 ( + ) + 2β4 ( δ δ + δ δ ) + 2( β2 )( δ + δ + δ + δ ) ( β3 )( αζζγδβε αζζεδβγ βζζεδαγ βζζγδαε) ( β6 )( αζζγβηηε αζζεβηηγ ) ( β )( αζ ζγ βε αζ ζεβγ βζ ζεαγ βζ ζγ αε ) Rαβγε αγβε αεnβγ αγ βε βγ αε αγ βε αε βγ βε αγ βγ αε P = + + ( δ δ δ δ 6 ) 1 αβγε 2 αγ βε βγ αε αβ γε β7 ( ) L = β δ + δ + δ + δ 1 αβγε 2 8 αγ βε αε βγ βε αγ βγ αε

24 Inertialess Tensor Theory : Final equations where ρ t W v = F p + T α α, α βε, α βα, β βε, γ, γ δ H δh δh (,, ) ( 2 β8 ) ( β8 ) ( 2 β8 ) Tαβ = Rαβγε vγ ε + vε γ + + βγ + αγ + αβγε δγα δγβ δγε ( 2 + β 8 ) ( ) β8,, (,, ) 21 ( β8 ) αβ αγ vβ γ + βγ vα γ αγ vβ γ + βγ vα γ + + αβ γε vγε = t δh δh 3 ( 1 3 ) + γε αβ δαβ β 7 δαβ δ γε

25 Conclusions The ost iportant benefit: To be able to draw coparisons between different levels of descriptions and in this way fill up the blanks. Most iportant exaple: The deonstration of the possibility of a generalized convected derivative for irect coparison between the inertialess and the inertial foraliss gives: β 8 = ( 2α ) 3 ( α ) 2 α3 Even in the dissipationless liit, this paraeter (being undeterined) can still be non-zero! This is a crucial paraeter as it regulates tubling