Deformation, break-up and rheology of dilute ferrofluid emulsion

Ferrofluidic emulsions – i.e., droplets enclosing a magnetic suspension -- have been garnering significant interest due to their performance in applications to be strongly affected by a magnetic field. With proper design, such droplets can align, deform in a particular manner, or break on demand. We develop an analytical theory to describe the deformation of a ferrofluid droplet subject simultaneously to a uniform magnetic field and a linear flow field. Using domain perturbation in the limit of small capillary number (Ca<<1), we derive the ellipsoidal correction to the droplet shape, accurate to O($Ca^2$), and demonstrate the non-linear nature of the physics due to Maxwell stress coupling with viscous forces. A steady state bifurcation analysis is performed to identify the critical capillary numbers in the presence of magnetic and viscous flow fields. Finally, volume-averaging of the stress is performed to get closed-form expressions for several viscometric functions (e.g., extensional viscosities) in the presence of planar hyperbolic and uniaxial extensional flows. We also demonstrate the existence of an asymmetric stress tensor and spin viscosity, which is absent when either field or flow acts alone.