Beltrami flows equipped with the same crystallographic symmetries as blue phases of cholesteric liquid crystals

Abstract

Abstract The Arnold–Beltrami–Childress (ABC) flow is the best-known example of a Beltrami flow (or a force-free field in plasma physics) periodic in three orthogonal directions. When its three parameters are equalised, its set of streamlines has the same crystallographic symmetry as a model of a blue phase (BP) of cholesteric liquid crystals. This commonality appears in the similarity between the arrangements of invariant tori in the ABC flow and of double-twist cylinders in the BP model. In this paper, three Beltrami flows whose sets of streamlines have the same crystallographic symmetries as three other BP models are derived by using cubic space groups. The first and the third Beltrami flows have invariant tori arranged similarly to double-twist cylinders in two of the BP models. The second Beltrami flow has invariant tori whose arrangement has been neither theoretically nor experimentally known in liquid crystals. Heteroclinic orbits in the three flows are also studied.