Classical Ericksen-Leslie theory

WOF+WE+WM

dx+Z

∂Ω

WPds, (2.6)

whereWOF is the Oseen-Frank energy,WE andWM are the bulk energy density due to electric and magnetic fields, respectively, andWP is the penalty term. A nice account for these effects is given in [Aur15]. One looks for equilibrium solutions for the director fieldnby looking for stationary points of the functional (2.6). This amounts to considering variations of the director configuration of the form

n=n+φ,

for a smooth vector φand a small parameter. This vector is chosen in such a way, that if we have strong anchoring, n satisfies the boundary condition for any . For a given φ, n yields a path of configurations of the director parameterized by. We saynis an equilibrium configuration with respect to the functional if the first variation vanishes for all smooth vectors φ. That is, if

∂

∂Wˆ[n+φ]|=0= 0, for allφ.

2.1.2 Classical Ericksen-Leslie theory

It is not necessary to use the full machinery of the Ericksen-Leslie formalism [WXL13] to derive the equation of interest in this thesis. However it is enlightening to see under what assumptions the Ericksen-Leslie continuum description reduces to the setting we consider here. We thus give a quick overview of the theory, without digging into the rather complicated constitute relations.

As done in [Les92] and [Ste14] one can derive required constitute relations for the governing dynamical equations by considering the typical balance equations of continuum physics: mass conservation, balance of linear momentum, and balance of angular momentum. We introduce the material derivative or so-called Lagrange derivative by

D Dt = ∂

∂t+v· ∇.

Then we can state the three balance laws governing the dynamics of nematic liquid crystals as D

Dtρ= 0, (2.7a)

ρD

Dtv+qD²

Dt²n· ∇n=ρf+g· ∇n+∇ ·T, (2.7b) qD²

Dt²n+λn=g+ ˆg−∂W

∂n +∇ · ∂W

∂(∇n). (2.7c)

Here v ∈ R³ is the velocity field of the liquid crystal flow, f ∈R³ is the vector of external forces, whileg∈R³ is the vector of generalized forces. q >0 is the inertial material constant.

Moreoverλis a Langrange multiplier associated with the constraint (2.8b). T is the full stress tensor consisting of a term involving the pressure pof the liquid and also a viscous stress tensor Tˆ ∈R^3×3. To determine an explicit expression for the stress tensor and the viscous stress tensor, Leslie introduced in [Les92] a rate of work hypothesis. One considers the rate at which forces and moments do work on a sample volumeV of a nematic liquid crystal, and postulate that this is absorbed into changes in the the stored internal energy or the kinetic energy, or that the work will be lost due to viscous dissipation. One considers the resulting strong form which reads

T =−pI−ρ ∂W

∂(∇n)∇n+ ˆT ,

where ˆT consists of 6 terms, all proportional to different viscosity coefficientsµ_ifori∈ {1, . . . ,6}. There are constraints on these coefficients, but we will not go into that here. Finally ˆgis also a vector dependent on the viscosity of the fluid. We assume incompressible fluid flow in the nematic liquid crystal. Otherwise the free energy density representing the internally stored energy of deformations of the director field would not only be a function of n and ∇n, as postulated by Frank-Oseen in (2.5). If we had considered a compressible fluid, we would have to include the densityρof the fluid and possibly variations in density, i.e.,w=w(n,∇n, ρ,∇ρ).

We will assume constant densityρin the nematic liquid crystal. Thus the mass conservation reduces to the typical incompressiblity condition. Hence leading to two constraints

∇ ·v= 0 in Ω, (2.8a)

|n|= 1. (2.8b)

The second constraint results fromnbeing a map to the unit sphere.

Now the two equations (2.7b) - (2.7c) plus the two constraints (2.8a)-(2.8b) yield 8 equations in 8 unknowns. The unknowns are the 3 components ofv andn, in addition to the two Lagrange multipliersλandp. As noted in [HZ95] there are two extreme cases of time-dependent solutions.

• q= 0: Here viscous effects dominate the inertia. The evolution of the director field is governed by a gradient-flow parabolic PDE, and this is typically the most physically significant regime.

• The opposite borderline case is letting inertia dominate viscosity (we neglect viscosity), this amounts to setting ˆT = 0 and ˆg= 0. The director field will in such a case satisfy a hyperbolic PDE, which can be derived from a constrained variational principle.

We will now introduce three simplifying assumptions: we specialize to stationary flow, assume vanishing viscosity, and that there are no external forces acting on the liquid crystal. Postulating stationary flow means that the material derivatives reduce to usual partial derivatives with respect to time. This way we exclusively focuses on the dynamics of the director field. The equations for linear and angular momentum reduce to

qntt· ∇n+∇ ·(pI−ρ ∂W

∂(∇n)∇n) = 0, (2.9a)

qntt+λn+∂W

∂n − ∇ · ∂W

∂(∇n) = 0. (2.9b)

Now we can take the inner product withnin (2.9b) and use the constraint|n|²= 1 to eliminate the Lagrange multiplierλ. In particular we obtain by an integration by parts in time for the first term thatλis given by

λ=−qnt·nt+n·∂w

Equations (2.9a) and (2.10) describe the governing equations for the dynamics of the director field n=n(~x, t). If we keep our assumptions, but allow for a nonzero viscosity, the vector ˆg will also yield a contribution, which manifest itself as a term ˆγnt in (2.9b). This is a lower order damping term, which is not expected to smooth out singularities. If one sets the term qn_tt· ∇= 0 one obtains precisely the same equations as considered in [HS91], and the reason we have this term in the first place is that we have already done some additional manipulations to recast the conservation of linear momentum on the more convenient formulation (2.7b).

We will now apply Hamiltonian’s principle of classical mechanics [Chp2, [GSP14]], which describe the motion of a conservative system. Let the system be described byN generalized coordinates q1, ..., qN with Lagrangian

L=L(q1, .., qN, q1,t, ..., qN,t, t),

depending explicitly on the generalized coordinates, their time derivatives, and timet. Here by generalized coordinates we mean coordinates where we have incorporated holonomic con-straints, i.e., used the holonomic constraint to remove dependent variables. Therefore {qj} are independent coordinates. The Hamiltonian principle states that the evolution of the system between timest₀ tot₁ is given by a path in the configuration spaces, which is such that the time integral of the Lagrangian is stationary. That is,

δ Z t₁

t0

L(q1, .., qn, q1,t, ..., qn,t, t)dt= 0,

whereδdenotes the first variation operator. Then we observe that (2.9b) can be written as the Euler-Lagrange equation of the following action functional (the action functional is defined as the time integral of the Lagrangian functionL)

A[n] =Z t₁

That is, considering variations of the director field of the form n =n+φ, with φhaving compact support, a computation of the first variation yields

δ boundary terms vanish. For this to vanish for all such functions, the integrand must vanish, which leads to (2.9b).