Small deformation theory for shape, rheology, and breakup of ferrofluid droplets in linear flow fields

An analytical theory is developed for the deformation of a viscous ferrofluid droplet subject to a uniform magnetic field together with a linear background flow. Under weak flow conditions (𝐶⁢𝑎≪1), we apply the method of domain perturbation to compute ellipsoidal corrections to the droplet shape up to 𝑂⁡(𝐶⁢𝑎2) in the capillary number. The resulting small-deformation equations show that droplet deformation and breakup depend strongly on both the direction and the strength of the imposed magnetic field as demonstrated in existing computational studies. Steady-state bifurcation analysis is used to determine droplet breakup and the critical capillary number as functions of the flow type and magnetic-field orientation. We then analyze the rheology of a dilute ferrofluid emulsion by extending the classical approach of Batchelor~ to compute the emulsion extra stress and the associated rheological material functions (e.g., shear and extensional viscosities). The existence of a non-symmetric stress tensor, reported by previous computational studies, is demonstrated and the extra torque in the suspension is calculated. Analytical expressions for rheological properties of bulk emulsions are derived up to 𝑂⁡(𝐶⁢𝑎).