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 th
e 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.
In this work we consider theoretically the problem of a Newtonian droplet moving in an otherwise quiescent infinite viscoelastic fluid under the influence of an externally applied temperature gradient. The outer fluid is modelled by the Oldroyd-B equ
ation, and the problem is solved for small Weissenberg and Capillary numbers in terms of a double perturbation expansion. We assume microgravity conditions and neglect the convective transport of energy and momentum. We derive expressions for the droplet migration speed and its shape in terms of the properties of both fluids. In the absence of shape deformation, the droplet speed decreases monotonically for sufficiently viscous inner fluids, while for fluids with a smaller inner-to-outer viscosity ratio, the droplet speed first increases and then decreases as a function of the Weissenberg number. For small but finite values of the Capillary number, the droplet speed behaves monotonically as a function of the applied temperature gradient for a fixed ratio of the Capillary and Weissenberg numbers. We demonstrate that this behaviour is related to the polymeric stresses deforming the droplet in the direction of its migration, while the associated changes in its speed are Newtonian in nature, being related to a change in the droplets hydrodynamic resistance and its internal temperature distribution. When compared to the results of numerical simulations, our theory exhibits a good predictive power for sufficiently small values of the Capillary and Weissenberg numbers.
