  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, it has been observed an increment 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, in addition
to the significant enhancement of the normal stresses. Comparisons between the
results of our model with a direct integration of the viscous stress provided
an indication of its reliability to predict the effective viscosity of the
suspension. Moreover, this latter quantity was found to be a monotonic
increasing function of the applied magnetic field for constant shearing flows
("magneto-thickening" behaviour), which allowed us to derive a simple
constitutive equation describing the emulsion viscosity. Additionally,
approximate equations describing the normal stress differences as function of
the magnetic Bond and capillary numbers have been derived through interpolation
of the data provided by our stress model.
