We investigate the impact of a magnetic field on surfactant concentration and interfacial forces across droplet surfaces within shear flows. Our analysis centers on a single two-dimensional ferrofluid droplet covered with surfactants, suspended in an immiscible, non-magnetizable liquid. The model combines incompressible Navier-Stokes equations and Maxwell's equations in the superparamagnetic limit in the single-fluid formulation, augmented by terms accounting for Marangoni, capillary, and magnetic forces at the droplet interface. We solve the surfactant convection-diffusion equation at the surface, while a non-linear Langmuir equation of state relates surfactant concentration to surface tension. The model is numerically solved using finite differences, a level-set method for multiphase flow computation, and the closest-point method for concentration equation. Our findings reveal that even though the surfactant is magnetically neutral, the presence of a magnetic field significantly modifies its distribution at the interface. A magnetic field perpendicular to the primary flow direction shifts the maximum concentration zone from the droplet tips toward the flow vorticity direction, while a parallel field produces the opposite effect. Alterations in surfactant distribution directly impact the surface tension field, offering a potential wireless means of controlling droplet dynamics.
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