Deformation dynamics of ferrofluid drops with field-dependent local magnetisation

Achieving precise control over the dynamic manipulation of a drop using an external magnetic field may face challenges due to the intricate relationship between the induced magnetisation and the inherent magnetic properties of the drop. Here, we put forward a fundamental theory that elucidates the morphology and behaviour of a ferrofluid droplet immersed in a different, viscous fluid when subjected to a uniform external magnetic field. Unlike previous studies, we introduce an asymptotic model that investigates the dynamic evolution of the drop by examining the local magnetisation as a function of the magnetic field itself. This leads to an additional contribution to the interfacial energy, resulting in an excess normal traction at the interface. Our analytical findings highlight the significant impact of saturation magnetisation and initial susceptibility of the ferrofluid on the resulting dynamic characteristics, which are further explored through comprehensive numerical simulations to address deformations beyond the scope of the asymptotic theory. Supported by benchmark numerical and experimental results, our study suggests that higher magnetic fields and/or greater saturation magnetisation can enhance drop elongation and accelerate its settling process. We develop a regime map illustrating various dynamic events based on the magnetic properties, which could have fundamental implications for the design and control of micro-encapsulations across a wide range of applications, including thermal processing, chemical synthesis, analysis and medical diagnostics.