Ferrofluid annulus in crossed magnetic fields.

We study the dynamics and pattern formation of a ferrofluid annulus enveloped by two nonmagnetic fluids in a Hele-Shaw cell, subjected to an in-plane crossed magnetic field configuration involving the combination of radial and azimuthal magnetic fields. A perturbative, second-order mode-coupling analysis is employed to investigate how the ferrofluid annulus responds to variations in the relative strength of the radial and azimuthal magnetic field components, as well as in the thickness of magnetic fluid ring. By tuning the magnetic field components and the annulus' thickness, we have found the development of several stationary annular shapes, presenting polygon-shaped structures typically having skewed, peaked fingers. Such fingered structures may vary their skewness, sharpness, and number and arise on the inner, outer, or even both boundaries of the annulus. In addition to controlling the morphologies of the ferrofluid annuli, the external field can be used to put the annulus into a rotational motion, with an angular velocity having prescribed magnitude, and direction. Our second-order theory is utilized to obtain a correction to the linear stability analysis prediction of such angular velocity, usually resulting in a decreased weakly nonlinear value as compared with the magnitude predicted by purely linear theory. These theoretical results suggest the use of magnetic-field-controlled ferrofluid annuli in Hele-Shaw cells as a potential laboratory for microscale applications related to the manipulation of shape-programmable magnetic fluid objects and tunable fluidic-mixing devices in confined environments.

Magnetically induced interfacial instabilities in a ferrofluid annulus.

We investigate the flow of a viscous ferrofluid annulus surrounded by two nonmagnetic fluids in a Hele-Shaw cell when subjected to an external radial magnetic field. The interfacial pattern formation dynamics of the system is determined by the interplay of magnetic and surface tension forces acting on the inner and outer boundaries of the annulus, favoring the coupling of the disjoint interfaces. Mode-coupling analysis is employed to examine both linear and weakly nonlinear stages of the flow. Linear stability analysis indicates that the trailing and leading annular boundaries are coupled already at the linear regime, revealing that perturbations arising in the outer interface may induce the emergence of deformed structures in the inner boundary. Moreover, second-order weakly nonlinear analysis is utilized to identify key nonlinear morphological features of the ferrofluid annulus. Our theoretical results show that linear, n-fold symmetric annular patterns having rounded edges are replaced by nonlinear polygonal-like shapes, presenting fairly sharp fingers. It is found that, as opposed to the linear patterns, the nonlinear peaky structures reach a stationary state, characterized by a growth saturation process induced by nonlinear effects. Furthermore, the response of the ferrofluid ring to changes in the thickness of the annulus, in the relative strength of magnetic and surface tensions forces, as well as in the magnetic susceptibility of the ferrofluid material, are also discussed.