Abstract
We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group mathrm{SO}(3) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.
Highlights
Nematic liquid crystals, regarded as fluids in which the high aspect ratio, rigid, rod molecules require descriptive variables for orientation as well as position, are observed to exhibit a wide range of prolonged unsteady dynamical responses to steady shear flow
Extensive theoretical and numerical investigations ([6,21,22,47,54,55,62,64,65,66] to cite only a few) of these and related nematic director or orientation tensor models in 2D or 3D reveal a wide range of periodic molecular dynamical regimes with evocative names [47] logrolling, tumbling, wagging and kayaking according to the behaviour of the principal axis of molecular orientation relative to the shear plane and vorticity axis
For which the principal axis of molecular orientation rotates periodically in the shear plane, are seen to be stable at low shear rates, but become unstable to out-of-plane perturbations and give way to kayaking orbits, for which the principal molecular axis is transverse to the shear plane, and rotates around the vorticity axis, reminiscent of the motion of the paddles propelling a kayak along the shear flow of a calm stream
Summary
Nematic liquid crystals, regarded as fluids in which the high aspect ratio, rigid, rod molecules require descriptive variables for orientation as well as position, are observed to exhibit a wide range of prolonged unsteady dynamical responses to steady shear flow. The approach we take is to regard β as a small parameter and view (1.2) as a perturbation of the co-rotational case This enables us to use tools from equivariant bifurcation theory [15,33,43,69,70] and in particular Lyapunov–Schmidt reduction over the group orbit O to obtain criteria for the persistence or otherwise of the periodic orbits of the co-rotational case after perturbation, and to determine the stability or otherwise of the resulting logrolling, tumbling and kayaking dynamics. Following a brief concluding section there are Appendices giving some technical results arising from symmetries that simplify the main calculations, as well as a discussion of how a fully general form of the molecular alignment term L(Q)D fits into the framework of our analysis
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