Abstract
Rotational viscometry is usually used to measure the properties of fluids in flows which approximate a simple shearing motion (Fig. 9.l).1–4 Such a flow can be effected in a fluid contained between two infinite parallel plates by fixing one and moving the second parallel to the first at a fixed speed. The shear rate is defined to be the plate velocity, V, divided by the gap spacing, h. Of the six components of the symmetric stress tensor, four are nonzero,1–4 with σxz and σyz being zero. We assume that the materials under consideration are incompressible.
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Abbreviations
- a c :
-
fracture crack radius
- Br:
-
Brinkman number, eqn (9.88)
- C 1,C 2 :
-
constants of integration, eqn (9.58)
- C M1—C M5 :
-
Moore model parameters, eqn (9.100)
- e :
-
linear distance between cylinder axes
- f 1,f 2 :
-
arbitrary functions arising from solution to eqn (9.59)
- f insiT :
-
instrumental frequency response
- 𝒯 :
-
total normal thrust, eqns (9.25) and (9.45)
- 𝒯 i :
-
inertial contribution to total normal thrust, eqn (9.80)
- h :
-
gap in simple shearing flow, Fig. 9.1;gap between plates in parallel plate viscometer, Fig. 9.2.
- h c :
-
gap in extended cone and plate geometry, Fig. 9.5(b)
- h R :
-
gap at edge of cone and plate or parallel plate geometries
- k E :
-
constant, eqn (9.92)
- k F :
-
constant, eqn (9.79)
- k θ :
-
thermal conductivity, eqn (9.88)
- L :
-
measuring length in concentric cylinder geometry
- L B :
-
length between pressure taps in helical screw rheometer
- L TS, L BS :
-
distances between vanes and sample free surface and container bottom surface, respectively
- L v :
-
length of vane rheometer
- N li :
-
inertial contribution to N 1 eqn (9.81)
- p :
-
hydrostatic pressure
- p a :
-
atmospheric pressure
- R :
-
radius of cone and plate and parallel plate geometries, Fig. 9.2
- R B :
-
barrel radius of helical screw rheometer
- R cyl :
-
radius of cylinder containing sample for vane rheometer
- Re:
-
Reynolds number
- R i, R 0 :
-
radii of inner and outer cylinders, respectively, for concentric cylinder geometry, Fig. 9.4.
- R in :
-
radius at which edge effects are important in cone and plate geometry
- R t :
-
radius of truncated region of truncated cone and plate geometry, Fig. 9.5(a)
- R v :
-
radius of vane rheometer
- R 1 —R 4 :
-
radii of double concentric cylinder geometry, Fig. 9.6(c)
- s :
-
value of viscometric shear stress function at a particular shear rate
- t appl :
-
duration of applied torque, Fig. 9.12
- t max :
-
time to reach stress overshoot maximum
- 𝒯 :
-
torque, eqns (9.20) and (9.41)
- 𝒯 act, 𝒯 o :
-
actual and ideal torques, eqn (9.83)
- 𝒯 cp, 𝒯 pp, 𝒯 cc :
-
transient torques for cone and plate, parallel plate and concentric cylinder geometries, respectively
- 𝒯 m :
-
maximum torque in a vane rheometer experiment
- 𝒯^ :
-
Torque per unit length in concentric cylinder geometry, eqn (9.61)
- V :
-
velocity of moving plate in simple shearing flow, Fig. 9.1
- Y:
-
eqn (9.66)
- α:
-
cone angle
- ς :
-
(Ri/Ro)2
- γ trans :
-
strain at which stress overshoot is observed
- γ̇ :
-
viscometric shear rate
- γ̇ :
-
shear rate at inner cylinder of Couette geometry, eqn (9.69)
- γ̇ CF, γ̇ CC :
-
critical shear rates for fracture and centrifugal expulsion, respectively
- γ̇ EC :
-
shear rate in extended cone and plate geometry, eqn (9.102)
- Γ:
-
surface tension
- δ :
-
helix angle of screw in helical screw rheometer
- Δ :
-
thickness over which edge effects are important for cone and plate geometry, eqn (9.85)
- θ O, θ :
-
reference and actual temperatures
- θ max :
-
maximum temperature rise in cone and plate viscometer, eqns (9.89) and (9.90)
- λ :
-
Moore model function, eqn (9.101)
- μ :
-
Newtonian viscosity
- ξ :
-
coefficent appearing in viscosity-temperature relationship, eqn (9.91)
- ρ :
-
density
- σ^ :
-
constitutive stress, eqn (9.1)
- σ e :
-
shear stress acting on end of vanes, eqn (9.106)
- σ y :
-
yield stress
- τ i :
-
inertial time scale
- ψ :
-
eqn (9.77)
- ω :
-
angular velocity of fluid, Table 9.1
- Ω:
-
angular velocity of rotating member for cone and plate and parallel plate flows, Fig. 9.2
- Ωi, Ωo :
-
angular velocities of inner and outer cylinders, respectively, Fig. 9.4
- ΩCF, ΩCC :
-
critical angular velocities for fracture and centrifugal expulsion, respectively
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Powell, R.L. (1993). Rotational Viscometry. In: Collyer, A.A., Clegg, D.W. (eds) Rheological Measurement. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2898-0_9
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