Shape, rheology, and breakup of ferrofluidic droplets in linear flow fields

The deformation and breakup of a viscous ferrofluidic droplet is studied when subject to a uniform magnetic field along with a linear background flow. The susceptibility of the ferrofluidic droplet is considered uniform and linear magnetization is assumed.Under the condition of weak flow ($Ca \ll 1$), the method of domain perturbation is applied to obtain the $O(Ca^2)$ dynamical equations for the ellipsoidal corrections to the droplet radius. Droplet deformation and breakup depends strongly on the direction as well as the strength of the magnetic field. Steady-state bifurcation analysis is performed to determine droplet breakup and critical capillary number ($Ca_c$) under different linear flow fields and magnetic field orientation. Finally, emulsion rheology is studied by applying an analysis similar to (G.K. Batchelor, Journal of Fluid Mechanics, 1970) to compute the extra stress due to the emulsion and its associated rheometric functions (e.g., shear viscosity, extensional viscosity). The existence of an asymmetric stress tensor, due to the presence of magnetic field, is demonstrated and the extra torque in the suspension is calculated. This asymmetry necessitates the introduction of novel rheological properties to fully characterize the suspension, properties that are inherently absent when the droplet is subject solely to hydrodynamic flow or a magnetic field individually. These findings underscore the significance of combined magnetic and flow effects in accurately describing ferrofluidic suspension dynamics and rheology