Rotational diffusion may govern the rheology of magnetic suspensions

Abstract

SynopsisThis paper is focused on the theoretical modeling of the rheological properties of the magnetic suspensions in shear flows under an external magnetic field aligned with the streamlines. The conventional theory postulates that the field-induced aggregates of magnetic particles are highly anisotropic and aligned with the flow direction. Therefore, no substantial variation in suspension viscosity would be expected in the presence of field. However, experiments reveal a strong Bingham rheological behavior of the suspensions with a dynamic yield stress of the same order of magnitude that the one measured in the magnetic fields perpendicular to the flow. We explain the high level of shear stress, generated in longitudinal magnetic fields, by stochastic rotary oscillations of the aggregates caused by many-body magnetic interactions with neighboring aggregates. The inter-aggregate interactions are accounted for by an effective rotational diffusion process with a diffusion constant proportional to the mean square interaction torque a net magnetic torque exerted to a given aggregate by all the neighboring aggregates. Using the equations for the orientation tensors coupled with the balance of forces acting on aggregates, we find the orientation distribution and size of the aggregates as function of the magnetic field intensity and shear rate. Our theory, developed for semi-dilute regime, reproduces the Bingham behavior observed experimentally and fits the 2 experimental data reasonably well in a wide range of particle concentrations. We find that the yield stress increases quadratically with the magnetic field strength and exhibits a cubic growth with respect to the particle volume fraction. A part from resolving a particular rheological problem, the new concept of magnetically induced diffusion reveals the importance of long-range non-hydrodynamic interactions in the rotary diffusion process and could probably contribute to the understanding of this process in other concentrated systems subject to non-hydrodynamic interactions.

Magnetorheological effect in the magnetic field oriented along the vorticity

“…Qualitatively, we adopt the same explanation for the MR effect in the magnetic field aligned with the vorticity as the one used by Kuzhir et al (2011) for the magnetic field aligned with the velocity. If, for some reason, the particle aggregates get misaligned with the vorticity, they will generate substantial hydrodynamic stresses proportional to the square of their length-to-diameter ratio, r e , a parameter called aspect ratio: r / g 0 _ cr 2 e .…”

“…Alternatively, the aggregate orientation cay be described by a unit vector, e, oriented along its major axis. The projections of this vector onto the three basis axes read: e 1 ¼ sin h cos /, e 2 ¼ sin h sin /, e 3 ¼ cos h. Applied to the geometry considered in the present work, the instantaneous force balance [derived in details in Kuzhir et al (2011)] reads…”

“…The magnetic force f m and the aggregate magnetic susceptibility v a , intervening into Eqs. (3) and (4), are calculated as a function of the magnetic field H using finite element simulations, as explained in details in Kuzhir et al (2011).…”

“…The aggregate orientation is commonly described by the second and the fourth rang tensors, he i e k i and he i e k e l e m i called, respectively, the second and the fourth statistical moments. The equation describing the temporal evolution of the second statistical moment comes from the average of the torque balance and, being applied to the case of long aggregates possessing an induced magnetic moment, reads [Pokrovskiy (1978); Kuzhir et al (2011)] dhe i e k i dt ¼ x il he l e k i À he i e l ix lk ½ þ c il he l e k i þ he i e l ic lk ½ À 2he i e k e l e m ic lm…”

“…Misaligned aggregates could generate substantial viscous stresses depending on the aggregate orientation distribution and length, which depend, in their turn, on the interplay between the magnetic and hydrodynamic interactions. Recently, Kuzhir et al (2011) have theoretically studied the MR fluid flow in the longitudinal field and shown that stochastic aggregate misalignments from the flow direction (caused by magnetic interactions between aggregates) could result in a Bingham behavior with an apparent (dynamic) yield stress comparable to that measured in the transverse field. We expect therefore that the same physics may govern the MR fluid rheology in the magnetic field parallel to the fluid vorticity.…”