Flow control of magnetic fluids exposed to magnetic fields

The description of flow in ferrohydrodynamics (Rosensweig, 1985) is based on a combination of equations, namely the continuity equation, the Navier-Stokes equation, the Maxwell equations and particular equations for the magnetization. Since the different models to describe the relaxation of magnetization differ, the adequate one has yet to be identified. By comparing experimental and simulation data of a model system, this goal may get achieved. As a model system, a Taylor-Couette apparatus was chosen. In this paper, experimental results concerning the transition form circular Couette flow to Taylor vortex flow at different field strengths of an axial magnetic field are compared to a linear stability analysis. The relaxation equation established by Shliomis (Shliomis, 1972) and the Debye-Model with a field dependent relaxation time showed to give qualitative accordance with the experimental data. 1. Motivation The magneto-rotational effect as a cause for an increase of viscosity of ferrofluids exposed to magnetic fields has been the subject of many investigations [1-6]. In a ferrofluid being subject to shear flow, the particles are forced to rotate with their axis of rotation parallel to the orientation of vorticity. When a magnetic field is applied, the suspended particles try to align themselves with their magnetic moment along the orientation of the magnetic field. Assuming magnetically hard particles (i. e. the Brownian relaxation time is shorter than the Néel relaxation time), a (magnetic) torque is exerted on them in case the vorticity of the shear flow and the magnetic field are not collinear. This magnetic torque counteracts the viscous torque exerted by the carrier liquid, causing an increase of the fluid’s viscosity. If vorticity and magnetic field are collinear, no increase of the fluid’s viscosity will be observable. A flow field, with a not vanishing component of vorticity perpendicular to the magnetic field, turns the magnetic dipoles of magnetically hard particles out of the magnetic field direction. The magnetization of the fluid is now not only dependent on the magnetic field, but also on the flow field. It is thus not describable by the equations for equilibrium magnetization anymore. Models to describe the magnetization outside equilibrium are necessary. Shliomis [2] was the first to describe the magnetization of a ferrofluid considering a not vanishing component of vorticity. Since then, other equations for the magnetization where developed, all of which are mainly based on the relaxation of magnetization against equilibrium, which is determined by a relaxation time. The existing equations to describe the relaxation of magnetization are different in their physical concepts and the way they predict experimental results. In order to determine the right 11th Conference on Electrorheological Fluids and Magnetorheological Suspensions IOP Publishing Journal of Physics: Conference Series 149 (2009) 012109 doi:10.1088/1742-6596/149/1/012109 c © 2009 IOP Publishing Ltd 1 one, a comparison of analytical and experimental data of a model system is needed. In Section 2 some equations for the relaxation of magnetization are introduced, in section 3 the experimental apparatus is outlined shortly and in section 4 some results are presented. 2. Theoretical The simplest model to calculate the magnetization outside the equilibrium is the Debye model, denoted here as Debye: ( ) M Ω M M M × + − − = eq t d τ 1 (1) This model implies a relaxation of the magnetization M towards the equilibrium magnetization M at a given external magnetic field H with a relaxation time τ in a frame rotating with the vorticity u Ω × ∇ = 2 1 of the flow u. Shliomis [2] developed the relaxation equation (denoted as S'72) ( ) ( ) M H M M Ω M M M × × + × + − − = κ τ eq t d 1 (2) With an additional constant ( ) 1 ~ 6 − Φ = η κ , where Φ is the volume fraction of the magnetic material and η~ the dynamic viscosity of the carrier liquid. Niklas et al. [4] considered stationary magnetizations (dtM = 0) near the equilibrium (M-M eq << 1), which appear at not too high rotation rates (|Ω|τ << 1). In this case the magnetization equations presented above can be simplified with (denoted as Niklas approach) H Ω M M × = − N eq c (3)