^{1,a)}, C. Gabriel

^{1}and Chr. Kieburg

^{1}

### Abstract

A twin gap magnetorheometer—based on a modified Anton Paar magnetocell MRD180/1T—using plate-plate gaps on both sides of the rotor plate, all plates consisting of ferromagnetic steel, is presented and compared to corresponding single plate-plate magnetorheometry. The twin gap arrangement uses a nonmagnetic housing at the rim, thus allowing nominal shear rates up to without sample loss due to centrifugal forces. Shear stresses in a range of 0.001–105 kPa are accessible. Finite element modeling (FEM) using MAXWELL^{®}2D verified the homogeneity of the flux density field in both shear gaps. The magnetic flux density in the magnetorheological fluid reaches up to 1.5 T and is obtained from an online Hall probe signal using a calibration factor from FEM. The radially constant flux density prevents radial carbonyl iron powder migration. Normal forces acting on both sides of the rotor compensate each other. Gap opening due to large normal forces as observed for the single gap geometry—requiring additional corrections for the effective flux density and shear stress—is not relevant. The apparent shear stress versus flux density characteristics from the twin gap and single gap magnetorheometry agree over the full range investigated (0–1.1 T), as shown for samples containing 90 and iron. Flow curves measured in a range of are presented as true shear stresses versus shear rate. The twin gap design (patent pending) has been licensed to the magnetocell manufacturer.

The authors thank Peter Schuler for the construction of the TG and SG tools. We are indebted to Gerhard Schmidt for FEM simulations and magnetorheometry measurements. Dr. Günter Oetter is thanked for providing the samples. The support of Anton Paar in providing a thermostated prototype upper yoke as well as geometric details of the magnetocell for the FEM simulations is gratefully acknowledged.

I. INTRODUCTION

II. EXPERIMENTAL

III. THE TG MAGNETOCELL

A. Design of the TG magnetocell

B. Magnitude and homogeneity of magnetic flux density

C. Shear rate and shear stress evaluation

1. SG plate-plate rheometry

2. TG plate-plate rheometry

3. Drag torque by eddy currents

D. Normal force compensation

E. Transient shear stress after flux density step

F. Shear stress versus flux density characteristic

IV. RELATED SG MAGNETOCELL

A. Design of the SG magnetocell

B. Magnitude and homogeneity of magnetic flux density

C. Transient shear stress after flux density step

D. Gap opening due to high normal forces

1. Air bearing compliance

2. Shear stress and flux density changes caused by the gap opening

E. Shear stress versus flux density characteristic

V. DIRECT COMPARISON OF TG AND SG DATA

A. Overview on the corrections applied

B. Apparent shear stress versus flux density characteristic

VI. TG FLOW CURVES

A. Typical behavior and data evaluation

B. Flow curves at various flux densities

C. Dissipative heating

VII. CONCLUSIONS

### Key Topics

- Ferromagnetism
- 27.0
- Magnetic flux
- 26.0
- Viscosity
- 16.0
- Torque
- 15.0
- Yield stress
- 15.0

## Figures

Schematic cross section of the magnetocell MRD180/1T (rotor and inserts removed). Circle indicates the location of the specific inserts (TG or SG, see Figs. 2 and 11).

Schematic cross section of the magnetocell MRD180/1T (rotor and inserts removed). Circle indicates the location of the specific inserts (TG or SG, see Figs. 2 and 11).

Schematic of the TG magnetocell insert (compared to the horizontal dimensions, the vertical dimensions are blown up by a factor of 2). For details of the geometry, see text.

Schematic of the TG magnetocell insert (compared to the horizontal dimensions, the vertical dimensions are blown up by a factor of 2). For details of the geometry, see text.

Magnetic flux density versus solenoid current as measured for the TG magnetocell (radial position of the probe’s sensitive area ). Unfilled squares: both gaps empty; full circles: MRF I in both gaps; full triangles: MRF II in both gaps. MAXWELL^{®}2D predictions for empty gaps are denoted by stars, for the MRF I filled gaps by crosses, respectively.

Magnetic flux density versus solenoid current as measured for the TG magnetocell (radial position of the probe’s sensitive area ). Unfilled squares: both gaps empty; full circles: MRF I in both gaps; full triangles: MRF II in both gaps. MAXWELL^{®}2D predictions for empty gaps are denoted by stars, for the MRF I filled gaps by crosses, respectively.

Simulated radial flux density profiles in the middle of the shear gaps (upper gap: squares; bottom gap: circles) filled by MRF I at a solenoid currents of 1 and 3 A, respectively. Radial dimensions of rotor shaft, upper yoke, and rotor plate are indicated by the hatched areas. Broken lines represent calculated flux densities in the middle of the Hall probe channel, while the arrow indicates the radial location of the probe’s sensitive area.

Simulated radial flux density profiles in the middle of the shear gaps (upper gap: squares; bottom gap: circles) filled by MRF I at a solenoid currents of 1 and 3 A, respectively. Radial dimensions of rotor shaft, upper yoke, and rotor plate are indicated by the hatched areas. Broken lines represent calculated flux densities in the middle of the Hall probe channel, while the arrow indicates the radial location of the probe’s sensitive area.

Rheology-relevant average flux density versus flux density at the Hall probe position. Data points (full circles) stem from MAXWELL^{®}2D simulations for solenoid currents 1–5 A and MRF I loaded into the TG magnetocell. Full line has slope .

Rheology-relevant average flux density versus flux density at the Hall probe position. Data points (full circles) stem from MAXWELL^{®}2D simulations for solenoid currents 1–5 A and MRF I loaded into the TG magnetocell. Full line has slope .

TG flux density correction factor versus weight percentage CIP in the MRF as determined by FEM simulations for various solenoid currents (compare Fig. 5). FEM data are represented by circles; straight line depicts Eq. (2).

TG flux density correction factor versus weight percentage CIP in the MRF as determined by FEM simulations for various solenoid currents (compare Fig. 5). FEM data are represented by circles; straight line depicts Eq. (2).

Shear stress overestimation factor versus nominal shear stress as given by Eq. (16). The pre-factor 0.05 causes at .

Shear stress overestimation factor versus nominal shear stress as given by Eq. (16). The pre-factor 0.05 causes at .

Comparison of measured normal forces versus magnetic flux density measured by the Hall probe at a shear rate of for the TG magnetorheometer (TG, full symbols) and the corresponding SG magnetorheometer (SG, unfilled symbols), respectively, loaded by MRF I or MRF II. Data are average values from three independent runs.

Comparison of measured normal forces versus magnetic flux density measured by the Hall probe at a shear rate of for the TG magnetorheometer (TG, full symbols) and the corresponding SG magnetorheometer (SG, unfilled symbols), respectively, loaded by MRF I or MRF II. Data are average values from three independent runs.

Time dependence of the apparent (rim) shear stress in the TG magnetorheometer for MRF I and MRF II, respectively, at constant shear rate after a step from zero magnetic flux density to 0.55 T (flux density correction applied) at time . Data points are averages from three independent runs.

Time dependence of the apparent (rim) shear stress in the TG magnetorheometer for MRF I and MRF II, respectively, at constant shear rate after a step from zero magnetic flux density to 0.55 T (flux density correction applied) at time . Data points are averages from three independent runs.

Apparent (rim) shear stress versus magnetic flux density for MRF I and MRF II, respectively, measured at constant shear rate in the TG magnetorheometer (various symbols indicate independent runs). Both the flux density correction Eq. (2) and the torque correction Eqs. (16) and (17) have been applied. For comparison, the average uncorrected data for each sample are represented by broken lines.

Apparent (rim) shear stress versus magnetic flux density for MRF I and MRF II, respectively, measured at constant shear rate in the TG magnetorheometer (various symbols indicate independent runs). Both the flux density correction Eq. (2) and the torque correction Eqs. (16) and (17) have been applied. For comparison, the average uncorrected data for each sample are represented by broken lines.

Schematic of the SG magnetocell (compared to the horizontal dimensions, the vertical dimensions are blown up by a factor of 2). For dimensions, see text.

Schematic of the SG magnetocell (compared to the horizontal dimensions, the vertical dimensions are blown up by a factor of 2). For dimensions, see text.

Simulated radial flux density profiles in the middle of the shear gap (symbols) filled by MRF I at solenoid currents of 1, 2, and 3 A. Broken lines represent calculated flux densities in the middle of the Hall probe channel (arrows indicate radial location of the probe’s sensitive area).

Simulated radial flux density profiles in the middle of the shear gap (symbols) filled by MRF I at solenoid currents of 1, 2, and 3 A. Broken lines represent calculated flux densities in the middle of the Hall probe channel (arrows indicate radial location of the probe’s sensitive area).

Rheology-relevant average flux density from FEM versus flux density calculated for the Hall probe position. Data points (full circles) stem from MAXWELL^{®}2D simulations for solenoid currents 1–5 A with MRF I loaded into the SG magnetocell. Full line has slope .

Rheology-relevant average flux density from FEM versus flux density calculated for the Hall probe position. Data points (full circles) stem from MAXWELL^{®}2D simulations for solenoid currents 1–5 A with MRF I loaded into the SG magnetocell. Full line has slope .

Time dependence of the apparent (rim) shear stress in the SG magnetorheometer for MRF I and MRF II, respectively, at constant shear rate after a step from zero magnetic flux density to 0.55 T at time . Data points are averages from three independent runs.

Time dependence of the apparent (rim) shear stress in the SG magnetorheometer for MRF I and MRF II, respectively, at constant shear rate after a step from zero magnetic flux density to 0.55 T at time . Data points are averages from three independent runs.

Measured gap height of the SG (full symbols) and of the bottom gap of the TG magnetocell (unfilled symbols) versus normal force . Data for MRF I are represented by circles, those for MRF II by triangles. Straight line represents an axial rheometer compliance of .

Measured gap height of the SG (full symbols) and of the bottom gap of the TG magnetocell (unfilled symbols) versus normal force . Data for MRF I are represented by circles, those for MRF II by triangles. Straight line represents an axial rheometer compliance of .

Apparent (rim) shear stress versus magnetic flux density for MRF I and MRF II, respectively, measured at constant shear rate in the SG magnetorheometer (various symbols indicate independent runs). Flux density correction Eq. (19) and gap opening corrections Eqs. (21) and (22) have been applied. For comparison, the average uncorrected data for each sample are represented by broken lines.

Apparent (rim) shear stress versus magnetic flux density for MRF I and MRF II, respectively, measured at constant shear rate in the SG magnetorheometer (various symbols indicate independent runs). Flux density correction Eq. (19) and gap opening corrections Eqs. (21) and (22) have been applied. For comparison, the average uncorrected data for each sample are represented by broken lines.

Direct comparison of the apparent shear stress versus flux density data at shear rate from the TG (full symbols) and SG (unfilled symbol) for MRF I and MRF II (all corrections listed in Table I applied). Drawn lines represent the average of the TG data.

Direct comparison of the apparent shear stress versus flux density data at shear rate from the TG (full symbols) and SG (unfilled symbol) for MRF I and MRF II (all corrections listed in Table I applied). Drawn lines represent the average of the TG data.

Same data as in Fig. 17 but with logarithmic scale for the apparent shear stress to provide an improved shear stress resolution for the low flux density regime.

Same data as in Fig. 17 but with logarithmic scale for the apparent shear stress to provide an improved shear stress resolution for the low flux density regime.

Shear rate sweeps on (a) MRF I and (b) MRF II at over the full range of shear rates. Stars (first up-sweep), diamonds (down-sweep), and circles (second up-sweep) represent apparent shear stress versus rim shear rate. Dotted line: fit by Eq. (24) to gray circles. Full squares: true rim shear stress from Eq. (5). Full line: fit by Casson law Eq. (26).

Shear rate sweeps on (a) MRF I and (b) MRF II at over the full range of shear rates. Stars (first up-sweep), diamonds (down-sweep), and circles (second up-sweep) represent apparent shear stress versus rim shear rate. Dotted line: fit by Eq. (24) to gray circles. Full squares: true rim shear stress from Eq. (5). Full line: fit by Casson law Eq. (26).

Apparent shear stresses of MRF I versus shear rate from successive up-sweeps with stepwise increase of the flux density. Unfilled symbols: measured data points; drawn line: fit by Eq. (24).

Apparent shear stresses of MRF I versus shear rate from successive up-sweeps with stepwise increase of the flux density. Unfilled symbols: measured data points; drawn line: fit by Eq. (24).

True shear stresses of MRF I versus shear rate from successive up-sweeps with stepwise increase of the flux density. Full symbols: measured data points; drawn line: fit by Casson model Eq. (26). Arrow indicates the true stroke of 3970 for and 0.84 T.

True shear stresses of MRF I versus shear rate from successive up-sweeps with stepwise increase of the flux density. Full symbols: measured data points; drawn line: fit by Casson model Eq. (26). Arrow indicates the true stroke of 3970 for and 0.84 T.

Full symbols: true rim shear stress of MRF I versus rim shear rate (data from Fig. 21) at (squares) and (diamonds), respectively. Unfilled symbols: apparent rim shear stress for a calibration oil. Thin straight line: expected response from Eq. (28). Arrow indicates the onset of the oil viscosity decrease due to dissipative heating.

Full symbols: true rim shear stress of MRF I versus rim shear rate (data from Fig. 21) at (squares) and (diamonds), respectively. Unfilled symbols: apparent rim shear stress for a calibration oil. Thin straight line: expected response from Eq. (28). Arrow indicates the onset of the oil viscosity decrease due to dissipative heating.

Magnetic flux lines and magnetic flux density field simulated by MAXWELL^{®}2D for the TG magnetocell filled with MRF I at 1 A solenoid current (nonmagnetic housing at the rim not shown).

Magnetic flux lines and magnetic flux density field simulated by MAXWELL^{®}2D for the TG magnetocell filled with MRF I at 1 A solenoid current (nonmagnetic housing at the rim not shown).

Magnetic flux lines and magnetic flux density field simulated by MAXWELL^{®}2D for the SG magnetocell filled with MRF I at 1 A solenoid current.

Magnetic flux lines and magnetic flux density field simulated by MAXWELL^{®}2D for the SG magnetocell filled with MRF I at 1 A solenoid current.

Magnetic flux density versus radial position (SG filled with MRF I, solenoid current 2 A). Symbols: middle plane of the MRF; lines: vertical position of Hall probe. Nominal gap 0.3 mm: squares and dotted line, gap 0.315 mm: circles and broken line, gap 0.33 mm: triangles and full line. Sensitive area of the Hall probe is located at . Vertical bars denote .

Magnetic flux density versus radial position (SG filled with MRF I, solenoid current 2 A). Symbols: middle plane of the MRF; lines: vertical position of Hall probe. Nominal gap 0.3 mm: squares and dotted line, gap 0.315 mm: circles and broken line, gap 0.33 mm: triangles and full line. Sensitive area of the Hall probe is located at . Vertical bars denote .

Effective flux density factor versus effective gap height for the SG filled with MRF I at 2 A solenoid current. Unfilled squares calculated from Eq. (A2), full square stems from Fig. 13. Full line: Eq. (A3) using .

Effective flux density factor versus effective gap height for the SG filled with MRF I at 2 A solenoid current. Unfilled squares calculated from Eq. (A2), full square stems from Fig. 13. Full line: Eq. (A3) using .

## Tables

Overview on the applied corrections (example MRF I, data from Fig. 17).

Overview on the applied corrections (example MRF I, data from Fig. 17).

Parameters of the Casson model used for Fig. 21 (MRF I at ).

Parameters of the Casson model used for Fig. 21 (MRF I at ).

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