Newton-Krylov continuation of amplitude-modulated rotating waves in sheared annular electroconvection

We present an approach for studying the primary, secondary, and tertiary flow transitions in sheared annular electroconvection. In particular, we describe a Newton-Krylov method based on time integration for the computation of rotating waves and amplitude-modulated rotating waves, and for the continuation of these flows as a parameter of the system is varied. The method exploits the rotational nature of the flows and requires only a time-stepping code of the model differential equations, i.e., it does not require an explicit code for the discretization of the linearized equations. The linear stability of the solutions is computed to identify the parameter values at which the transitions occur. We apply the method to a model of electroconvection that simulates the flow of a liquid crystal film in the smectic A phase suspended between two annular electrodes and subjected to an electric potential difference and a radial shear. Due to the layered structure of the smectic A phase, the fluid can be treated as two-dimensional (2D) and is modeled using the 2D incompressible Navier-Stokes equations coupled with an equation for charge continuity. The system is a close analog to laboratory-scale geophysical fluid experiments and thus represents an ideal system in which to apply the method before its application to these other systems that exhibit similar flow transitions. In the model for electroconvection, we identify the parameter values at which the primary transition from steady axisymmetric flow to rotating waves occurs, as well as at which the secondary transition from the rotating waves to amplitude-modulated rotating waves occurs. In addition, we locate the tertiary transition, which corresponds to a transition from the amplitude-modulated waves to a three-frequency flow. Of particular interest is that the method also finds a period-doubling bifurcation from the amplitude-modulated rotating waves and a subsequent transition from the flow resulting from this bifurcation.