Tensors for Physics

332 16 Constitutive Relations and (eμxeλx−eyμeyλ)ezλezν =− 1 3 (exμexν−eμyeνy), lead to two coupled equations forΠ−andΠ 0 : τM ∂ ...

16.4 Viscosity and Alignment in Nematics 333 pμν =− 2 η∇μvν− 2 η ̃ 1 nμnλ ∇λvν − 2 η ̃ 3 nμnνnκnλ∇λvκ − 2 η ̃ 2 ( −εμλκωλnκnν ) ...

334 16 Constitutive Relations The three Miesowicz coefficients do not involve the viscosity coefficientη ̃ 3 .This is different ...

16.4 Viscosity and Alignment in Nematics 335 viscous behavior of liquid crystals can also be analyzed by computer simulations. N ...

336 16 Constitutive Relations The friction pressurepνμA, in the affine space, obeys the constitutive law −pAνμ= 2 ηAΓνμA+ηAV∇Aλv ...

16.4 Viscosity and Alignment in Nematics 337 Miesowicz coefficients obey the inequalitiesη 2 >η 3 >η 1 andγ 2 <0. For o ...

338 16 Constitutive Relations and its gradient. In particular, for the geometry used above, e.g. in Sect.12.4.6, one hasnx=cosχ, ...

16.4 Viscosity and Alignment in Nematics 339 Ω ̇λ=−νrΩλ+θ−^1 Tλsyst+θ−^1 Tλflct. Hereνr>0 is a rotational damping coefficient ...

340 16 Constitutive Relations for uniaxial ellipsoids with the axis ratioQ. ValuesQ>1 andQ<1 pertain to prolate, i.e. rod- ...

16.4 Viscosity and Alignment in Nematics 341 cf. Sect.12.2.2. Thus the moment equation for the second rank alignment tensor beco ...

342 16 Constitutive Relations temperatureTni, from above. Notice thatTniis slightly larger than the pseudo-critical temperatureT ...

16.4 Viscosity and Alignment in Nematics 343 velocity gradient does not alter the uniaxial character of the alignment nor affect ...

344 16 Constitutive Relations is made. Then the entropy production is given by − ρ m T ( δs δt )( 2 ) irrev =pνμ∇νvμ+ ρ m kBTΦμν ...

16.4 Viscosity and Alignment in Nematics 345 vector associated with the antisymmetric part of the pressure tensor, cf. Sect.16.3 ...

346 16 Constitutive Relations The non-Newtonian viscosity coefficientη+, cf., Sect.16.3.9, is found to be η+=ηNewH+(Γ ), H+(Γ )= ...

16.4 Viscosity and Alignment in Nematics 347 pμ=εμνλnν ( γ 1 Nλ+γ 2 ∇λvκnκ ) , (16.145) γ 1 = 3 ρ m kBTa^2 eqτa,γ 2 = ρ m kBT ( ...

348 16 Constitutive Relations Shear rates, in units ofτref−^1 are now denoted byΓ, thus one hasΓ=γτref. Further- more, the tumbl ...

16.4 Viscosity and Alignment in Nematics 349 The model parameters occurring in the scaled equations are the reduced temper- atur ...

350 16 Constitutive Relations isotropic phase, whereΦμν≈Aaμνapplies, these contributions are additive. In the nematic phase, and ...

Chapter 17 Tensor Dynamics Abstract This chapter presents examples for dynamical phenomena involving ten- sors. Firstly, linear ...