Can a minimum principle describe shear alignment of liquid crystals?

When a static perturbation is applied to a liquid crystal, the director configuration changes to minimize the free energy. When a shear flow is applied to a liquid crystal, one might ask: Does the director configuration change to minimize any effective potential? To address that question, we derive the Leslie-Ericksen equations for dissipative dynamics, and determine whether they can be expressed as relaxation toward a minimum. The answer may be yes or no, depending on the number of degrees of freedom. Using theory and simulations, we consider two specific examples, reverse tilt domains under simple shear flow and dowser configurations [1] under plane Poiseuille flow, and determine whether each example shows relaxation toward the minimum of an effective potential.
[1] P. Pieranski et al., Phys. Rev. E 94, 042706 (2016); Eur. Phys. J. E 39, 121 (2016).