Revisiting the governing equations of a magnetic suspension of polar particles: From microhydrodynamics analysis to rheological response

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In this review, we describe a formulation for the stress tensor of a monodisperse magnetic suspension of polarized neutrally buoyant spheroidal particles suspended in a non-magnetic liquid. A magnetic suspension affords a rare example of a material for which the stress tensor is non-symmetric. The present formulation is based on a microhydrodynamics description of a spherical particle suspended in a Newtonian fluid subjected to magnetic forces and torques. The magnetic suspension is considered statistically homogeneous and treated as being a homogeneous equivalent fluid. Under this condition, a volume average over all particles in the carrier fluid is used in order to obtain the magnetization equation evolution and the constitutive equation for the stress tensor of the magnetic suspension, in particular the magnetic stress contribution. The average effects on the homogeneous continuum fluid due to particle pressure, particle dipole, and the applied magnetic field on each particle are computed by our constitutive equation. In this approach, the particles are not considered force or torque free since their permanent magnetization allows them to experience the effects of an applied magnetic field. The calculated stress tension can be used for modeling common flows of symmetric or non-symmetric magnetic fluids flowing in arbitrary geometries and in rheological applications for determination of important properties such as the rotational viscosity of non-symmetric magnetic fluids. The final expression of the constitutive equation for the stress tensor based on a particle scale approach presents some difference as compared with current constitutive models proposed in the current literature. Our constitutive equation considers the effect of a magnetic particle pressure, the average particle stresslet contribution in terms of an effective viscosity, the average particle rotlet in terms of a rotational viscosity, and a configurational tensor associated with dipole–dipole interactions. In addition, we discuss the situation in which the dipole moment of the particle is not frozen on it which leads to the necessity of an internal balance of angular momentum in a fluid element to close the governing equations of the model. An extension of the model for emulsions of polar deformable droplets is also proposed.