Phase-locking flows between orthogonally stretching parallel plates

In this paper we explore the stability and dynamical relevance of a wide variety of steady,time-periodic, quasiperiodic and chaotic flows arising between orthogonally stretching par-allel plates. We first explore the stability of all the steady flow solution families formerlyidentified by Ayats et al. [Ayats, R., Marques, F., Meseguer, A. and Weidman, P., Flowsbetween orthogonally stretching parallel plates, Phys. Fluids, 33, 024103 (2021)], con-cluding that only the one that originates from the Stokesian approximation is actually sta-ble. When both plates are shrinking at identical or nearly the same deceleration rates,this Stokesian flow exhibits a Hopf bifurcation that leads to stable time-periodic regimes.The resulting time-periodic orbits or flows are tracked for different Reynolds numbers andstretching rates, whilst monitoring their Floquet exponents in order to identify secondaryinstabilities. It is found that these time-periodic flows also exhibit Neimark-Sacker bifur-cations, generating stable quasiperiodic flows (tori) that may sometimes give rise to chaoticdynamics through a Ruelle-Takens-Newhouse scenario. However, chaotic dynamics is un-usually observed, as the quasiperiodic flows generally become phase-locked through a res-onance mechanism before a strange attractor may arise, thus restoring the time-periodicityof the flow. In this work we have identified and tracked four different resonance regions,also known as Arnold tongues or horns. In particular, the 1 : 4 strong resonance regionis explored in great detail, where the identified scenarios are in very good agreement withnormal form theory.