The effect of a spatially uniform magnetic field on the shear rheology of a dilute emulsion of monodispersed ferrofluid droplets, immersed in a non-magnetizable immiscible fluid, is investigated using direct numerical simulations. The direction of the applied magnetic field is normal to the shear flow direction. The droplets extra stress tensor arising from the presence of interfacial forces of magnetic nature is modeled on the basis of the seminal work of G. K. Batchelor, J. Fluid Mech., 41.3 (1970) under the assumptions of a linearly magnetizable ferrofluid phase and negligible inertia. The results show that even relatively small magnetic fields can have significant consequences on the rheological properties of the emulsion due to the magnetic forces that contribute to deform and orient the droplets towards the direction of the applied magnetic vector. In particular, we have observed an increase of the effective (bulk) viscosity and a reversal of the sign of the two normal stress differences with respect to the case without magnetic field for those conditions where the magnetic force prevails over the shearing force. Comparisons between the results of our model with a direct integration of the viscous stress have provided an indication of its reliability to predict the effective viscosity of the suspension. Moreover, this latter quantity has been found to behave as a monotonic increasing function of the applied magnetic field for constant shearing flows (magneto-thickening behaviour), which allowed us to infer a simple constitutive equation describing the emulsion viscosity.
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