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Director Equation

The director equation describes the evolution of the director field, which represents the average orientation of the liquid crystal molecules, driven by interfacial energies.
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The statement of the theorem

Let n(x,t)\mathbf{n}(\mathbf{x}, t) be the unit director vector field, and let K1,K2,K3K_1, K_2, K_3 be the splay, twist, and bend elastic constants, respectively. The evolution of n\mathbf{n} is governed by the torque balance equation:\nn×(∂n∂t+csn×∂n∂t)=−δFelδn\mathbf{n} \times \left( \frac{\partial \mathbf{n}}{\partial t} + c_s \mathbf{n} \times \frac{\partial \mathbf{n}}{\partial t} \right) = -\frac{\delta F_{el}}{\delta \mathbf{n}}\nwhere the elastic free energy density felf_{el} is:\nfel=12[K1(nx⋅∇nx)2+K2(ny⋅∇ny)2+K3(nz⋅∇nz)2]f_{el} = \frac{1}{2} [K_1 (n_x \cdot \nabla n_x)^2 + K_2 (n_y \cdot \nabla n_y)^2 + K_3 (n_z \cdot \nabla n_z)^2]\n(Note: The full form involves derivatives of n\mathbf{n} and is simplified here for clarity of the governing principle.)
Source: Wikipedia