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An extension of the theory of oscillating cup viscometers

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Abstract

The theory of the fluid motion in the interior of an oscillating or rotating cup is reexamined. The quantity of interest in viscometry is the torque exerted by the fluis on the sides and rims of the cup. In this paper expressions for the torque are obtained for geometries for which the cup height approaches a fluid boundary layer thickness. Interest in such geometries is due to viscosity measurements made in mixtures in the critical region where cups of small height are used in order to minimize gravity effects.

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Abbreviations

D(ζ) :

Torque on the cup, Eq. (5)

E(ζ) :

Truncation error term, Eq. (24)

h :

Internal half-height of a filled cup or the height of the liquid in a partially filled cup

I :

Moment of inertia of cup and suspension system

I′ :

Moment of inertia of fluid inside cup

I n :

Modified Bessel function of order n

J i, n :

Defined in Eq. (13)

R :

Radius of the cup

S n :

Defined in Eq. (7)

S′ n :

Defined in Eq. (10)

x :

Variable 2η 0/π

z :

Variable 2η 0ζ1/2

α(τ) :

Angular displacement of the cup

δ :

Boundary layer thickness

Δ :

Logrithmic decrement

ζ :

Laplace transform variable

η 0 :

Dimensionless height h/δ

θ :

Frequency ratio ω/ω 0

ν :

Kinematic viscosity

ξ 0 :

Dimensionless radius R/δ

ρ :

Density of liquid

τ :

Dimensionless time ω 0 t

φ :

Phase angle of oscillation

ω :

Angular frequency of oscillation with liquid present in cup

ω 0 :

Angular frequency of oscillation in a vacuum

References

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Southern Mississippi, 39406, Hattiesburg, Mississippi, USA

    J. C. Nieuwoudt

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  1. J. C. Nieuwoudt
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Nieuwoudt, J.C. An extension of the theory of oscillating cup viscometers. Int J Thermophys 11, 525–535 (1990). https://doi.org/10.1007/BF00500844

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