Magnetic field-mediated ferrofluid droplet deformation in extensional flow

Extensional flow is vital in droplet dynamics, influencing their formation, size, stability, and functionality across diverse applications from industrial processes to biomedical technology. Ferrofluid droplets are pivotal in many such applications, where magnetic fields enable non-contact manipulation without undesirable heating effects. However, controlling ferrofluid droplet dynamics in magnetically influenced extensional flows is challenging due to the complex interplay of induced magnetization, intrinsic magnetic properties, and flow kinematics. Here, we present a first-principle-based theory delving into the morphology of a ferrofluid droplet under the combined influence of an external magnetic field and extensional flow. Unlike previous studies, we employ an asymptotic analysis that delves on the shape alterations by considering local magnetization as dependent on magnetic field intensity. Additionally, we develop a numerical model based on phase-field hydrodynamics to establish the practical applicability of the asymptotic solution and to explore large droplet-deformation regimes. The study demonstrates that increasing the magnetic field intensity, the saturation magnetization of the ferrofluid, and the initial magnetic susceptibility each independently improve droplet deformation. Additionally, we found that in a uniform magnetic field, the extensional viscosity of a ferrofluid emulsion is influenced by the strain rate, leading to strain-thickening behavior in the dilute emulsion. Our findings offer new insights into field-assisted manipulation of ferrofluid droplets, emphasizing their potential in applications ranging from process engineering to biomedical technology.