The breakup phenomenon of a ferrofluid droplet in a simple shear flow under a uniform magnetic field is numerically investigated in this paper. The numerical simulation, based on the finite element method, uses a level set method to capture the dynamic evolution of the droplet interface between the two phases. Focusing on small Reynolds numbers (i.e., Re <= 0.03), systematic numerical simulations are carried out to analyze the effects of magnetic field strength, direction, and viscosity ratio on the breakup phenomenon of the ferrofluid droplet. The results suggest that applying a magnetic field along alpha = 45 degrees and 90 degrees relative to the flow direction initiates breakup in a ferrofluid droplet at a low capillary number in the Stokes flow regime, where the droplet usually does not break up in a shear flow alone. At alpha = 0 degrees and 135 degrees, the magnetic field suppresses breakup. Also, there exists a critical magnetic bond number, Bo(cr), below which the droplet does not rupture, which is also dependent on the direction of the magnetic field. Additionally, the effect of the viscosity ratio on droplet breakup is examined at variable magnetic bond numbers. The results indicate a decrease in the critical magnetic bond number Bo(cr) values for more viscous droplets. Furthermore, more satellite droplets are observed at alpha = 45 degrees compared to alpha = 90 degrees, not only at higher magnetic field strengths but also at larger viscosity ratios.
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