We investigate numerically the influence of a homogeneous magnetic field on a ferrofluid in the gap between two concentric, independently rotating cylinders. The full Navier-Stokes equations are solved with a combination of a finite difference method and a Galerkin method. Structure, dynamics, symmetry properties, bifurcation, and stability behavior of different vortex structures are investigated for axial and transversal magnetic fields, as well as combinations of them. We show that a transversal magnetic field modulates the Taylor vortex flow and the spiral vortex flow. Thus, a transversal magnetic field induces wavy structures: wavy Taylor vortex flow (wTVF) and wavy spiral vortex flow. In contrast to the classic wTVF, which is a secondarily bifurcating structure, these magnetically generated wavy Taylor vortices are pinned by the magnetic field, i.e., they are stationary and they appear via a primary forward bifurcation out of the basic state of circular Couette flow.

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 λ(2). 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 λ(2)=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 (λ(2)≠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.

The influence of a magnetic field on the dynamics of the flow of a ferrofluid in the gap between two concentric, independently rotating cylinders is investigated numerically. The Navier-Stokes equations are solved using a hybrid finite difference and Galerkin method. We show that the frequently used assumption that the internal magnetic field within a ferrofluid is equal to the external applied field is only a leading-order approximation. By accounting for the ferrofluid's magnetic susceptibility, we show that a uniform externally imposed magnetic field is modified by the presence of the ferrofluid within the annulus. The modification to the magnetic field has an r(-2) radial dependence and a magnitude that scales with the susceptibility. For ferrofluids typically used in laboratory experiments of the type simulated in this paper, the modification to the imposed magnetic field can be substantial. This has significant consequences on the structure and stability of the basic states, as well as on the bifurcating solutions.

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