Instabilities, bifurcations and transition to chaos in electrophoresis of charge-selective microparticle
Résumé
Electro-hydrodynamic instabilities near a cation-exchange microgranule in an electrolyte solution under an external electric field are studiednumerically. Despite the smallness of the particle and practically zero Reynolds numbers, in the vicinity of the particle, several sophisticatedflow regimes can be realized, including chaotic ones. The obtained results are analyzed from the viewpoint of hydrodynamic stability andbifurcation theory. It is shown that a steady-state uniform solution is a non-unique one; an extra solution with a characteristic microvortex,caused by non-linear coupling of the hydrodynamics and electrostatics, in the region of incoming ions is found. Implementation of oneof these solutions is subject to the initial conditions. For sufficiently strong fields, the steady-state solutions lose their stability via the Hopfbifurcation and limit cycles are born: a system of waves grows and propagates from the left pole,θ= 180○, toward the angleθ=θ0≈60○. Furtherbifurcations for these solutions are different. With the increase in the amplitude of the external field, the first cycle undergoes multiple perioddoubling bifurcation, which leads to the chaotic behavior. The second cycle transforms into a homoclinic orbit with the eventual chaotic modevia Shilnikov’s bifurcation. Santiago’s instability [Chenet al., “Convective and absolute electrokinetic instability with conductivity gradients,”J. Fluid Mech.524, 263 (2005)], the third kind of instability, was then highlighted: an electroneutral extended jet of high salt concentrationis formed at the right pole (region of outgoing ions,θ= 0○). For a large enough electric field, this jet becomes unstable; the perturbations areregular for a small supercriticality, and they acquire a chaotic character for a large supercriticality. The loss of stability of the concentrationjet significantly affects the hydrodynamics in this area. In particular, the Dukhin–Mishchuk vortex, anchored to the microgranule atθ≈60○,under the influence of the jet oscillations loses its stationarity and separates from the microgranule, forming a chain of vortices moving off thegranule. This phenomenon strongly reminds the Kármán vortices behind a sphere but has another physical mechanism to implement. Besidesthe fundamental importance of the results, the instabilities found in the present work can be a key factor limiting the robust performance ofcomplex electrokinetic bio-analytical systems. On the other hand, these instabilities can be exploited for rapid mixing and flow control ofnanoscale and microscale devices