Research Papers: Fundamental Issues and Canonical Flows

Formulas for Calibration of Rheological Parameters of Bingham Fluid in Couette Rheometer

[+] Author and Article Information
Ying-Hsin Wu

Project Center on Natural Disaster Reduction,
Shimane University,
Matsue 690-8504, Japan
e-mail: yhwu@riko.shimane-u.ac.jp

Ko-Fei Liu

Department of Civil Engineering
and Hydrotech Research Institute,
National Taiwan University,
Taipei 10617, Taiwan

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received July 16, 2014; final manuscript received October 10, 2014; published online December 11, 2014. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 137(4), 041202 (Apr 01, 2015) (11 pages) Paper No: FE-14-1385; doi: 10.1115/1.4028813 History: Received July 16, 2014; Revised October 10, 2014; Online December 11, 2014

We propose simple formulas for calibration of rheological parameters of Bingham fluid using a co-axial cylinder rheometer, in which the inner cylinder rotates at a constant speed, and the outer cylinder is stationary. A critical rotational speed exists due to existence of yield stress. If rotation speed exceeds the critical value, all fluid is fully sheared between the concentric cylinders. Plug flow exists only when rotation speed is less than the critical value. The effects of radius ratio and Bingham number are discussed. The rheometer with radius ratio very close to unity is discussed as a limiting case, and the result confirms previous research. Formulas for calibration are derived using Least Squares and perturbation methods for all values of radius ratio, based on measured rotation speed and torque. Two sets of experimental data are used for verification. The validation shows that the formulas derived here yield reasonable and accurate estimates of rheological parameters.

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Grahic Jump Location
Fig. 1

Definition sketch of two cylinders and the coordinate system. The clockwise direction of θ is defined as positive. The radii of the inner and outer cylinders are R1 and R2, respectively; δ is the shear layer thickness, ur and uθ are radial and angular velocities, respectively. The inner cylinder rotates counterclockwise at a constant speed Ω, and the outer cylinder is stationary.

Grahic Jump Location
Fig. 2

Schematic sketch of flow conditions and normalized BCs: (a) flow with plug layer and (b) flow without plug layer. B is Bingham number; α is radius ratio; β is normalized shear layer thickness and Δ=1+β is the normalized location of the interface between the plug and shear layers.

Grahic Jump Location
Fig. 3

The normalized shear layer thickness β: (a) as a function of Bingham number B and (b) as a function of rotational speed Ω for different Bingham material (different values of τ0/μ).

Grahic Jump Location
Fig. 4

The critical Bingham number Bcr = τ0/μΩcr, where Ωcr is the critical rotational speed, as a function of radius ratio α. The upper and lower zones represent the flow conditions with and without plug layer, respectively.

Grahic Jump Location
Fig. 5

The absolute value of the ratio of inner cylinder wall shear stress Γ to yield stress τ0 as a function of radius ratio α and different values of Bingham number B

Grahic Jump Location
Fig. 6

The absolute value of the ratio of inner cylinder wall shear stress Γ to yield stress τ0: (a) as a function of rotational speed Ω for a given Bingham material (fixed τ0/μ) in the gaps with different values of radius ratio α = R2/R1 and (b) as a function of Ω for different material (different τ0/μ) in the gap with fixed α = 1.5

Grahic Jump Location
Fig. 7

Diagram of ln(η)/η as a function of η, which η = Γi/τ0 represents the ratio of measured value of shear stress Γi to yield stress τ0, and η must be greater than unity



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