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An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in concentric cylinder viscometers

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Abstract

An analytical solution is presented for the calculation of the flow field in a concentric cylinder viscometer of non-ideal Bingham-fluids, described by the Worrall-Tuliani rheological model. The obtained shear rate distribution is a function of the a priori unknown rheological parameters. It is shown that by applying an iterative procedure experimental data can be processed in order to obtain the proper shear rate correction and the four rheological parameters of the Worrall-Tuliani model as well as the yield surface radius. A comparison with Krieger's correction method is made. Rheometrical data for dense cohesive sediment suspensions have been reviewed in the light of this new method. For these suspensions velocity profiles over the gap are computed and the shear layer thicknesses were found to be comparable to visual observations. It can be concluded that at low rotation speeds the actually sheared layer is too narrow to fullfill the gap width requirement for granular suspensions and slip appears to be unavoidable, even when the material is sheared within itself. The only way to obtain meaningfull measurements in a concentric cylinder viscometer at low shear rates seems to be by increasing the radii of the viscometer. Some dimensioning criteria are presented.

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Abbreviations

A, B :

Integration constants

C :

Dimensionless rotation speed = µ∞Φ/τy

c :

= 2µ∞

d :

= τ0 2−2cτy

f(τ) :

= (τ−τ0)2+2c(τ−τy)

r :

Radius

r b :

Bob radius

r c :

Cup radius

r y :

Yield radius

r 0 :

Stationary surface radius

r Φ :

Rotating Stationary radius

Y 0 :

Shear rate parameter = Δτ/Δµ

\(\dot \gamma \) :

Shear rate

ɛ:

= (r y /r b )2− 1

µ∞ :

Bingham viscosity

µ0 :

Initial differential viscosity

Δµ:

µ0-µ∞

Φ:

Rotation speed

ω:

Angular velocity

Ω:

Shear stress

τ b :

Bob shear stress

τ B :

Bingham stress

τ y :

(True) yield stress

τ0 :

Stress parameter = τ B +µ∞ Y 0

Δτ:

τ B -τ y

References

  • Beyer WH (ed) (1978) CRC standard mathematical tables (25th ed). CRC Press, West Palm Beach

    Google Scholar 

  • Bingham EC (1922) Fluidity and plasticity. McGraw-Hill, New York

    Google Scholar 

  • Cross MM (1965) Rheology of non-Newtonian fluids: a new flow equation for pseudo-plastic systems. J Coll Sci 20:417–437

    Google Scholar 

  • Darby R (1985) Couette viscometer data reduction for materials with a yield stress. J Rheol 29:369–378

    Google Scholar 

  • Hunter RJ (1989) Foundations of colloid science, Vol II. Oxford Science Publications, Clarendon Press, Oxford, p 1042

    Google Scholar 

  • Jones TER, Golden K (1990) Rheological behaviour of River Parrett mud. Report for HR Wallingford, Polytechnic South-West, Plymouth

    Google Scholar 

  • Krieger IM, Elrod H (1953) Direct determination of the flow curves of non-Newtonian fluids. II. Shearing rate in the concentric cylinder viscometer. J Appl Phys 24:134–136

    Google Scholar 

  • Krieger IM, Maron SH (1954) Direct determination of the flow curves of non-Newtonian fluids. III. Standardized treatment of viscometric data. J Appl Phys 25:72–75

    Google Scholar 

  • Krieger IM (1968) Shear rate in the Couette viscometer. Trans Soc Rheol 12:5–11

    Google Scholar 

  • Moore F (1959) The theology of ceramic slips and bodies. Trans Brit Ceramic Soc 58:470–492

    Google Scholar 

  • Nguyen QD, Boger DV (1987) Characterisation of yield stress fluids using the concentric cylinder viscometer. Rheol Acta 26:508–515

    Google Scholar 

  • Nguyen QD, Boger DV (1992) Measuring the flow properties of yield stress fluids. Ann Rev Fluid Mech 24:47–88

    Google Scholar 

  • O'Brien JS (1986) Physical properties, rheology and modeling of mud flows. PhD Thesis, Civil Eng Dept, Colorado State University, Fort Collins

    Google Scholar 

  • Reiner M (1929) The theory of plastic flow in the rotation viscometer. J Rheol 1:5–10

    Google Scholar 

  • Pazwash H, Robertson JM (1971) Fluid-dynamic considerations of bottom materials. ASCE J Hydr Div 97:1317

    Google Scholar 

  • Toorman EA, Berlamont JE (1992) Numerical simulation of high concentration suspension flow with free surface. In: Theoretical and applied rheology. Proc XIth Congr Rheol, Elsevier, pp 292–294

  • Williamson RV (1929) Ind Eng Chem 21:108 ff

    Google Scholar 

  • Worrall WE, Tuliani S (1964) Viscosity changes during the ageing of clay-water suspensions. Trans Brit Ceramic Soc 63:167–185

    Google Scholar 

  • Yang TMT, Krieger IM (1978) Comparison of methods for calculating shear rates in coaxial viscometers. J Rheol 22:413–421

    Google Scholar 

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Authors and Affiliations

  1. Hydraulics Laboratory, Civil Engineering Department, Katholieke Universiteit Leuven, Belgium

    Erik A. Toorman

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  1. Erik A. Toorman
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Toorman, E.A. An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in concentric cylinder viscometers. Rheola Acta 33, 193–202 (1994). https://doi.org/10.1007/BF00437304

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  • DOI: https://doi.org/10.1007/BF00437304

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