^{1}, Isaac Torres-Diaz

^{2}and Carlos Rinaldi

^{2,a)}

### Abstract

An asymptotic solution is obtained for flow of a ferrofluid between two stationary coaxial cylinders of infinite extent, including the effects of spin viscosity. This solution takes into account the nonuniformity of the magnetic field in the annular space that arises as a consequence of the demagnetizing field of the inner cylinder. In the limit of zero spin viscosity the analysis predicts there is no flow in the annular gap. For nonzero spin viscosity the analysis predicts flow in the annulus with corotation of field and fluid close to the outer cylinder wall and counter-rotation close to the inner cylinder wall. The asymptotic predictions for the translational velocity are compared to experimental measurements obtained using the ultrasound velocity profile method for ferrofluid in an annular gap. The observed experimental velocity profiles are in qualitative agreement with the predictions of the asymptotic model, and transition between corotation to counter-rotation of fluid and field was observed at an intermediate radial position. These observations provide further evidence of the existence of couple stresses in ferrofluids and the importance of spin viscosity in describing some ferrofluidflows.

This work was supported by the U.S. National Science Foundation (Grant No. CBET-0547150).

I. INTRODUCTION

II. ANALYSIS OF FLOW AND TORQUE WITH NONZERO SPIN VISCOSITY

A. Definition of the problem

B. Equations of the problem

C. Scaled governing equations for

1. Magnetic field and magnetization

2. Magnetic body force and body couple

3. Linear and spin velocity fields

III. THEORETICAL PREDICTIONS FOR SPIN AND TRANSLATIONAL VELOCITY

IV. EXPERIMENTAL RESULTS

A. Ferrofluid characterization

B. Velocity profile measurements for ferrofluid in annular gap

V. CONCLUSIONS

### Key Topics

- Ferrofluids
- 47.0
- Magnetic fields
- 38.0
- Viscosity
- 18.0
- Rotating flows
- 17.0
- Velocity measurement
- 14.0

## Figures

Schematic illustration of the coupled magnetic-hydrodynamic problem for ferrofluid in the annular gap between two coaxial cylinders. The radii of the external and internal cylinders are and , respectively. In this figure corresponds to the thickness of the external cylinder or the space between the cylinder and the surface current distribution. In the analysis this thickness is assumed negligible . The azimuthal velocity component and axial directed spin velocity are obtained for an infinite annulus of ferrofluid between subjected to a rotating magnetic field perpendicular to the axis of the cylinder. The magnetic field source is modeled as a -directed surface current distribution . which is backed by a material of infinite magnetic permeability, .

Schematic illustration of the coupled magnetic-hydrodynamic problem for ferrofluid in the annular gap between two coaxial cylinders. The radii of the external and internal cylinders are and , respectively. In this figure corresponds to the thickness of the external cylinder or the space between the cylinder and the surface current distribution. In the analysis this thickness is assumed negligible . The azimuthal velocity component and axial directed spin velocity are obtained for an infinite annulus of ferrofluid between subjected to a rotating magnetic field perpendicular to the axis of the cylinder. The magnetic field source is modeled as a -directed surface current distribution . which is backed by a material of infinite magnetic permeability, .

(a) Calculated spin and (b) translational velocity profiles for ferrofluid in the annular gap of two coaxial cylinders for 85 Hz and 5 mT rms of applied magnetic field. These were obtained using the physical and magnetic properties of the EMG900-3 ferrofluid (Table I) and values of the dimensionless parameter of 20, 33, and 50.

(a) Calculated spin and (b) translational velocity profiles for ferrofluid in the annular gap of two coaxial cylinders for 85 Hz and 5 mT rms of applied magnetic field. These were obtained using the physical and magnetic properties of the EMG900-3 ferrofluid (Table I) and values of the dimensionless parameter of 20, 33, and 50.

(a) Calculated velocity profiles for ferrofluid in the annular gap of two coaxial cylinders as a function of (a) field frequency and 2 mT rms of applied magnetic field and (b) as a function of magnetic field intensity and field frequency of 100 Hz. These were obtained using the physical and magnetic properties of the EMG900-1 ferrofluid (Table I) and values of the dimensionless parameter of 20, 33, and 50.

(a) Calculated velocity profiles for ferrofluid in the annular gap of two coaxial cylinders as a function of (a) field frequency and 2 mT rms of applied magnetic field and (b) as a function of magnetic field intensity and field frequency of 100 Hz. These were obtained using the physical and magnetic properties of the EMG900-1 ferrofluid (Table I) and values of the dimensionless parameter of 20, 33, and 50.

Calculated translational velocity profile showing the effect of on flow magnitude for an 85 Hz and 5 mT rms applied magnetic field. These were obtained using the physical and magnetic properties of EMG900-3 (Table I) and .

Calculated translational velocity profile showing the effect of on flow magnitude for an 85 Hz and 5 mT rms applied magnetic field. These were obtained using the physical and magnetic properties of EMG900-3 (Table I) and .

Velocity profiles for EMG900-3 ferrofluid filling the annular gap between stationary coaxial cylinders, obtained with four transducers at different angles with respect to the diagonal. The measurements were obtained at a field frequency of 100 Hz and a magnetic field intensity of 12.7 mT rms. The inner cylinder wall is located at 9.35 mm and the outer cylinder wall is located at 24.7 mm.

Velocity profiles for EMG900-3 ferrofluid filling the annular gap between stationary coaxial cylinders, obtained with four transducers at different angles with respect to the diagonal. The measurements were obtained at a field frequency of 100 Hz and a magnetic field intensity of 12.7 mT rms. The inner cylinder wall is located at 9.35 mm and the outer cylinder wall is located at 24.7 mm.

(a) Velocity profile dependence on magnetic field frequency with constant amplitude of 15.5 mT rms and (b) dependence on field amplitude for frequency of 100 Hz for EMG900-3 ferrofluid.

(a) Velocity profile dependence on magnetic field frequency with constant amplitude of 15.5 mT rms and (b) dependence on field amplitude for frequency of 100 Hz for EMG900-3 ferrofluid.

Velocity profiles for EMG900-3 ferrofluid filling the annular gap between stationary coaxial cylinders, obtained with four transducers at different angles with respect to the diagonal. The measurements were obtained at a field frequency of 100 Hz and a magnetic field intensity of 12.7 mT. The inner cylinder wall is located at 9.35 mm and the outer cylinder wall is located at 24.7 mm.

Velocity profiles for EMG900-3 ferrofluid filling the annular gap between stationary coaxial cylinders, obtained with four transducers at different angles with respect to the diagonal. The measurements were obtained at a field frequency of 100 Hz and a magnetic field intensity of 12.7 mT. The inner cylinder wall is located at 9.35 mm and the outer cylinder wall is located at 24.7 mm.

## Tables

Physical and magnetic properties at room temperature for oil based ferrofluid.

Physical and magnetic properties at room temperature for oil based ferrofluid.

Parameters used in the DOP2000 ultrasound velocimeter in order to obtain velocity profile measurements of the EMG900-3 ferrofluid.

Parameters used in the DOP2000 ultrasound velocimeter in order to obtain velocity profile measurements of the EMG900-3 ferrofluid.

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