Newton-Krylov continuation of amplitude-modulated rotating waves in sheared annular electroconvection Submitted to Physical Review E, (2024). Gregory M. Lewis (1), Jamil Jabbour (1), Mary C. Pugh (2) and Stephen W. Morris (3) (1) Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario, Canada, L1G 0C5 (2) Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada, M5S 2E4 (3) Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7.

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 and is modeled using the 2-D incompressible Navier-Stokes equations coupled with an equation for charge continuity. The system is a close analogue 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 transistion 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.

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See the movies:

• R=560 rotating wave.
• R=620 Amplitude modulated wave.
• R=629 Three frequency modulated wave.
• R=632 Amplitude modulated wave.
• R=637 Amplitude modulated vascillating wave.

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