Numerical Simulation of Maximum Spreading of an Impacting Ferrofluid Droplet under a Vertical Magnetic Field

Theoretical modeling is proposed to predict the maximum spreading of water-based ferrofluid droplets impacting upon dry surfaces influenced by a vertical magnetic field. Constructed on the principle of energy balance, this model demonstrates excellent agreement with numerical findings across various impact velocities, contact angles, and magnetic strengths. Notably, as magnetic field strength escalates, magnetic forces prevail over viscous and capillary forces, exerting a significant influence on spreading dynamics and diminishing the maximum spreading diameter of ferrofluid droplets if the impacting shape is spherical. However, for freely falling droplets, the shape becomes prolate before impacting and the promoted surface energy balances the magnetic inhibitory effect on droplet spreading, thus resulting in an almost unchanged maximum spreading diameter. By postulating complete conversion of initial kinetic energy into magnetic energy, a scaling law is derived for maximum spreading diameter under extremely high magnetic fields. Further interpolation with viscous dissipation and capillary effects enables universal rescaling under diverse impact conditions. Through comparison with numerical outcomes, the validity of our theoretical model is affirmed, establishing a balanced formula between distinct energy components for predicting maximum spreading diameter of ferrofluid droplets accurately.