Rotational rheometers are the most commonly used devices to investigate the rheological behavior of liquids in shear flows. These devices are used to measure rheological properties of both Newtonian and non-Newtonian, or complex, fluids. Two of the most widely used geometries are flow between parallel plates and flow between a cone and a plate. A time-dependent rotation of the plate or cone is often used to study the time-dependent response of the fluid. In practice, the time dependence of the flow field is ignored, that is, a steady-state velocity field is assumed to exist throughout the measurement. In this study, we examine the dynamics of the velocity field for parallel-plate and cone–plate flows of Newtonian fluids by finding analytical solutions of the Navier–Stokes equation in the creeping flow limit. The time-dependent solution for parallel-plate flow is relatively simple as it requires the velocity to have a linear dependence on radial position. Interestingly, the time-dependent solution for cone–plate flow does not allow the velocity to have a linear dependence on radial position, which it must have at the steady state. Here, we examine the time-dependent velocity fields for these two flows, and we present results showing the time dependence of the torque exerted on both the stationary and rotating fixtures. We also examine the time dependence of spatial non-homogeneities of the strain rate. Finally, we speculate on the possible implications of our results in the context of shear banding, which is often observed in parallel-plate and cone–plate flows of complex fluids.

1.
K.
Walters
,
Rheometry
(
Wiley
,
New York
,
1975
).
2.
C. W.
Macosko
,
Rheology: Principles, Measurements and Applications
(
VCH Publishers
,
New York
,
1994
).
3.
R. B.
Bird
,
R. C.
Armstrong
, and
O.
Hassager
,
Dynamics of Polymeric Liquids
, 2nd ed., Fluid Mechanics Vol. 1 (
Wiley
,
New York
,
1987
).
4.
N.
Adams
and
A. S.
Lodge
, “
Rheological properties of concentrated polymer solutions II. A cone-and-plate and parallel-plate pressure distribution apparatus for determining normal stress differences in steady shear flow
,”
Philos. Trans. R. Soc. London Ser. A
256
,
149
184
(
1964
).
5.
A. J.
Giacomin
and
P. H.
Gilbert
, “
Exact-solution for cone-plate viscometry
,”
J. Appl. Phys.
122
,
175101
(
2017
).
6.
J. C.
Slattery
, “
Analysis of the cone-plate viscometer
,”
J. Colloid Sci.
16
,
431
437
(
1961
).
7.
D. H.
McCoy
and
M. M.
Denn
, “
Secondary flow in a parallel-disk viscometer
,”
Rheol. Acta
10
,
408
411
(
1971
).
8.
R. M.
Turian
, “
Perturbation solution of the steady Newtonian flow in the cone and plate and parallel plate systems
,”
Ind. Eng. Chem. Fundam.
11
,
361
368
(
1972
).
9.
M. E.
Fewell
and
J. D.
Hellums
, “
The secondary flow of Newtonian fluids in cone‐and‐plate viscometers
,”
Trans. Soc. Rheol.
21
,
535
565
(
1977
).
10.
G. H.
McKinley
,
J. A.
Byars
,
R. A.
Brown
, and
R. C.
Armstrong
, “
Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid
,”
J. Non-Newtonian Fluid Mech.
40
,
201
229
(
1991
).
11.
D. F.
Griffiths
and
K.
Walters
, “
On edge effects in rheometry
,”
J. Fluid Mech.
42
,
379
399
(
1970
).
12.
D. O.
Olagunju
, “
Effect of free surface and inertia on viscoelastic parallel plate flow
,”
J. Rheol.
38
,
151
168
(
1994
).
13.
J. S.
Vrentas
,
D. C.
Venerus
, and
C. M.
Vrentas
, “
An exact analysis of reservoir effects for rotational viscometers
,”
Chem. Eng. Sci.
46
,
33
37
(
1991
).
14.
D. C.
Venerus
, “
Free surface effects on normal stress measurements in cone and plate flow
,”
Appl. Rheol.
17
,
36494
(
2007
).
15.
A.
Yoshimura
and
R. K.
Prud'homme
, “
Wall slip corrections for Couette and parallel disk viscometers
,”
J. Rheol.
32
,
53
67
(
1988
).
16.
R. B.
Bird
and
R. M.
Turian
, “
Viscous heating effects in a cone and plate viscometer
,”
Chem. Eng. Sci.
17
,
331
334
(
1962
).
17.
D. O.
Olagunju
,
A. B.
Vyas
, and
S.
Zhang
, “
Analytical and numerical solutions for torsional flow between coaxial discs with heat transfer
,”
SIAM J. Appl. Math.
68
,
1404
1422
(
2008
).
18.
M. G.
Hansen
and
F.
Nazem
, “
Transient normal force transducer response in a modified Weissenberg rheogoniometer
,”
J. Rheol.
19
,
21
36
(
1975
).
19.
T.
Schweizer
and
A.
Bardow
, “
The role of instrument compliance in normal force measurements of polymer melts
,”
Rheol. Acta
45
,
393
402
(
2006
).
20.
C. S.
Dutcher
and
D. C.
Venerus
, “
Compliance effects on the torsional flow of a viscoelastic fluid
,”
J. Non-Newtonian Fluid Mech.
150
,
154
161
(
2008
).
21.
R. B.
Bird
and
C. F.
Curtiss
, “
Tangential Newtonian flow in annuli-I: Unsteady state velocity profiles
,”
Chem. Eng. Sci.
11
,
108
113
(
1959
).
22.
J.
Passard
,
R.
Kouitat Njiwa
, and
P.
Perré
, “
Unsteady flow in cone and plate geometry: How computation can help rheometry
,”
Eur. Phys. J.: Appl. Phys.
3
,
321
342
(
1998
).
23.
D. C.
Venerus
and
H. C.
Öttinger
,
A Modern Course in Transport Phenomena
(
Cambridge University Press
,
Cambridge
,
2018
).
24.
H. S.
Carslaw
and
J. C.
Jaeger
,
Conduction of Heat in Solids
, 2nd ed. (
Oxford University Press
,
New York
,
1959
).
25.
J. C.
Slattery
,
Advanced Transport Phenomena
, Cambridge Series in Chemical Engineering (
Cambridge University Press
,
New York
,
1999
).
26.
E. W.
Hobson
,
The Theory of Spherical and Ellipsoidal Harmonics
(
Cambridge University Press
,
Cambridge
,
1931
).
27.
M.
Abramowitz
and
E. I. A.
Stegun
,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
(
Dover
,
New York
,
1965
).
28.

These functions are related to the usual Bessel functions by the formulas jn(z)=π2zJn+1/2(z) and yn(z)=π2zYn+1/2(z).

29.
C. F.
Dewey
,
S. R.
Bussolari
,
M. A.
Gimbrone
, and
P. F.
Davies
, “
The dynamic response of vascular endothelial cells to fluid shear stress
,”
J. Biomech. Eng.
103
,
177
185
(
1981
).
30.
T.
Nagel
,
N.
Resnick
,
C. F.
Dewey
, and
M. A.
Gimbrone
, “
Vascular endothelial cells respond to spatial gradients in fluid shear stress by enhanced activation of transcription factors
,”
Arterioscler., Thromb., Vasc. Biol.
19
,
1825
1834
(
1999
).
31.
M. H.
Buschmann
,
P.
Dieterich
,
N. A.
Adams
, and
H.-J.
Schnittler
, “
Analysis of flow in a cone-and-plate apparatus with respect to spatial and temporal effects on endothelial cells
,”
Biotech. Bioeng.
89
,
493
502
(
2005
).
32.
P.
Tapadia
and
S.-Q.
Wang
, “
Direct visualization of continuous simple shear in non-Newtonian polymeric fluids
,”
Phys. Rev. Lett.
96
,
016001
(
2006
).
33.
S.
Ravindranath
,
S.-Q.
Wang
,
M.
Olechnowicz
, and
R. P.
Quirk
, “
Banding in simple steady shear of entangled polymer solutions
,”
Macromolecules
41
,
2663
2670
(
2008
).
34.
P. D.
Olmsted
, “
Perspectives on shear banding in complex fluids
,”
Rheol. Acta
47
,
283
300
(
2008
).
35.
G.
Ovarlez
,
S.
Rodts
,
X.
Chateau
, and
P.
Coussot
, “
Phenomenology and physical origin of shear localization and shear banding in complex fluids
,”
Rheol. Acta
48
,
831
844
(
2009
).
36.
Y.
Li
,
M.
Hu
,
G. B.
McKenna
,
C. J.
Dimitriou
,
G. H.
McKinley
,
R. M.
Mick
,
D. C.
Venerus
, and
L. A.
Archer
, “
Flow field visualization of entangled polybutadiene solutions under nonlinear viscoelastic flow conditions
,”
J. Rheol.
57
,
1411
(
2013
).
37.
R. L.
Moorcroft
and
S. M.
Fielding
, “
Criteria for shear banding in time-dependent flows of complex fluids
,”
Phys. Rev. Lett.
110
,
086001
(
2013
).
38.
M.
Mohagheghi
and
B.
Khomami
, “
Elucidating the flow-microstructure coupling in entangled polymer melts. Part II: Molecular mechanism of shear banding
,”
J. Rheol.
60
,
861
872
(
2016
).
39.
T.
Divoux
,
M. A.
Fardin
,
S.
Manneville
, and
S.
Lerouge
, “
Shear banding of complex fluids
,”
Annu. Rev. Fluid Mech.
48
,
81
103
(
2016
).
40.
E. J.
Hemingway
and
S. M.
Fielding
, “
Interplay of edge fracture and shear banding in complex fluids
,”
J. Rheol.
64
,
1147
1159
(
2020
).
41.
M.
Boudaghi-Khajehnobar
,
B. J.
Edwards
, and
B.
Khomami
, “
Effects of chain length and polydispersity on shear banding in simple shear flow of polymeric melts
,”
Soft Matter
16
,
6468
6483
(
2020
).
42.
See https://dlmf.nist.gov/14.3#E5 for a formula for the associated Legendre function of the first kind.
43.
See https://dlmf.nist.gov/14.3#E12 for a formula for the associated Legendre function of the second kind.
44.
G. N.
Watson
, “
A note on gamma functions
,”
Proc. Edinburgh Math. Soc.
42
,
7
9
(
1959
).
You do not currently have access to this content.