Dr Rahil Valani

Abstract:

The spontaneous (so-called Quincke) rotation of an uncharged, solid, dielectric, spherical particle under a steady uniform electric field is analyzed, accounting for the inertia of the particle and the transient fluid inertia, or “hydrodynamic memory,” due to the unsteady Stokes flow around the particle. The dynamics of the particle are encapsulated in three coupled nonlinear integro-differential equations for the evolution of the angular velocity of the particle, and the components of the induced dipole of the particle that are parallel and transverse to the applied field. These equations represent a generalization of the celebrated Lorenz system. A numerical solution of these ‘modified Lorenz equations’ (MLE) shows that hydrodynamic memory leads to an increase in the threshold field strength for chaotic particle rotation, which is in qualitative agreement with experimental observations. Furthermore, hydrodynamic memory leads to an increase in the range of field strengths where multistability between steady and chaotic rotation occurs. At large field strengths, chaos ceases, and the particle is predicted to execute periodic rotational motion.

Abstract:

We consider a nonlinear oscillator with state-dependent time-delay that displays a countably infinite number of nested limit cycle attractors, i.e. megastability. In the low-memory regime, the equation reduces to a self-excited nonlinear oscillator and we use averaging methods to analytically show quasilinear increasing amplitude of the megastable spectrum of quantized quasicircular orbits. We further assign a mechanical energy to each orbit using the Lyapunov energy function and obtain a quadratically increasing energy spectrum and (almost) constant frequency spectrum. We demonstrate transitions between different quantized orbits, i.e. different energy levels, by subjecting the system to an external finite-time harmonic driving. In the absence of external driving force, the oscillator asymptotes towards one of the megastable quantized orbits having a fixed average energy. For a large driving amplitude with frequency close to the limit cycle frequency, resonance drives transitions to higher energy levels. Alternatively, for large driving amplitude with frequency slightly detuned from limit-cycle frequency, beating effects can lead to transitions to lower energy levels. Such driven transitions between quantized orbits form a classical analog of quantum jumps. For excitations to higher energy levels, we show amplitude locking where nearby values of driving amplitudes result in the same response amplitude, i.e. the same final higher energy level. We rationalize this effect based on the basins of different limit cycles in phase space. From a practical viewpoint, our work might find applications in physical and engineering system where controlled transitions between several limit cycles of a multistable dynamical system is desired.

Abstract:

We show that a suspension of noninteracting deformable particles subjected to an oscillatory shear flow leads to development of nematic order that arises from the phenomenon of phase synchronization. The synchronized state corresponds to a unique, stable limit cycle confined in the toroidal state space. The limit cycle exists since, unlike rigid particles, deformable particles can modulate aspect ratio, adjust their tumbling rate, and thus achieve phase synchronization. These synchronized regions emerge as Arnold tongues in the parameter space of the driving amplitude and frequency. Considering the rheological implications of ordering dynamics in soft and active matter, our results motivate oscillatory shear flow experiments with deformable particles.