Effect of elongational flow on ferrofuids under a magnetic field

To set up a mathematical model for the flow of complex magnetic fluids, noninteracting magnetic particles with a small volume or an even point size are typically assumed. Real ferrofluids, however, consist of a suspension of particles with a finite size in an almost ellipsoid shape as well as with particle-particle interactions that tend to form chains of various lengths. To come close to the realistic situation for ferrofluids, we investigate the effect of elongational flow incorporated by the symmetric part of the velocity gradient field tensor, which could be scaled by a so-called transport coefficient λ₂. Based on the hybrid finite-difference and Galerkin scheme, we study the flow of a ferrofluid in the gap between two concentric rotating cylinders subjected to either a transverse or an axial magnetic field with the transport coefficient. Under the influence of a transverse magnetic field with λ₂=0, we show that basic state and centrifugal unstable flows are modified and are inherently three-dimensional helical flows that are either left-winding or right-winding in the sense of the azimuthal mode-2, which is in contrast to the generic cases. That is, classical modulated rotating waves rotate, but these flows do not. We find that under elongational flow (λ₂≠0), the flow structure from basic state and centrifugal instability flows is modified and their azimuthal vorticity is linearly changed. In addition, we also show that the bifurcation threshold of the supercritical centrifugal unstable flows under a magnetic field depends linearly on the transport coefficient, but it does not affect the general stabilization effect of any magnetic field.