The dynamics of a two-layered rotating liquid at low angular velocity in a cylindrical container is explored, both experimentally and theoretically, using an oil-water system. In the two-layered liquid, a transient concave down paraboloid interface, containing undulations, is observed before the final, stable concave up paraboloid interface develops. A simple yet effective model is developed to explain the observed profile dynamics, in which the predicted maximal height of the paraboloid agrees well with experimental measurements over a fairly large range of rotation speeds.
References
1.
D. J.
Baker
, Jr. , “ Demonstrations of fluid flow in a rotating system II: The ‘Spin-Up’ problem
,” Am. J. Phys.
36
, 980
–986
(1968
).2.
J. M.
Goodman
, “ Paraboloids and vortices in hydrodynamics
,” Am. J. Phys.
37
, 864
–868
(1969
).3.
Z.
Sabatka
and L.
Dvorak
, “ Simple verification of the parabolic shape of a rotating liquid and a boat on its surface
,” Phys. Educ.
45
(5), 462
–468
(2010
).4.
R. R.
Turkington
and D. V.
Osborne
, “ On the influence of surface tension on the surface profile of a rotating liquid
,” Proc. Phys. Soc.
82
, 614
–619
(1963
).5.
V. A.
Lubarda
, “ The shape of a liquid surface in a uniformly rotating cylinder in the presence of surface tension
,” Acta. Mech.
224
, 1365
–1382
(2013
).6.
R.
Bergmann
et al., “ Polygon formation and surface flow on a rotating fluid surface
,” J. Fluid Mech.
679
, 415
–431
(2011
).7.
R. E.
Berg
, “ Rotating liquid mirror
,” Am. J. Phys.
58
, 280
–281
(1990
).8.
E. J.
Hitch
, M. A.
Kelmanson
, and P. D.
Metcalfe
, “ Shock-like free-surface perturbations in low-surface-tension, viscous, thin-film flow exterior to a rotating cylinder
,” Phys. Eng. Sci.
460
, 2975
–2991
(2004
).9.
M.
Piva
and E.
Meiburg
, “ Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall
,” Physics of Fluids
17
, 063603
(2005
).10.
W.
Wan
, Y.
Yu
, and H.
Bai
, “ Investigation on the attenuation of a rotating liquid motion in a cylinder with boundary resistance
,” Chinese Journal of Theoretical and Applied Mechanics
46
(4), 505
–511
(2014
).11.
PHYWE
, “ Surface of rotating liquids
” laboratory experiment, available online at <https://www.phywe.com/en/surface-of-rotating-liquids.html>.12.
M.
Basta
, V.
Picciarelli
, and R.
Stella
, “ A simple experiment to study parabolic surfaces
,” Phys. Educ.
35
(2
), 120
–123
(2000
).13.
H.
Yoshikawa
and J. E.
Wesfreid
, “ Oscillatory Kelvin-Helmholtz instability. Part 2. An experiment in fluids with a large viscosity contrast
,” J. Fluid Mech.
675
, 249
–267
(2011
).14.
F. M.
White
, Fluid Mechanics.
4th ed. (McGraw-Hill
, New York, NY
, 2001
), pp. 93
–95
, 326–342.15.
T.
Fukuchi
, “ Hydraulic Elements chart for pipe flow using new definition of hydraulic radius
,” Hydraulic Eng.
132
(9
), 990
–994
(2006
).16.
P.
Klimas
et al., “ Lunar liquid mirror telescope: structural concepts
,” Proc. SPIE
7732
, 77322U
(2010
).17.
J. R.
Holton
and P. H.
Stone
, “ A note on the spin-up of a stratified fluid
,” J. Fluid Mech.
33
, 127
–129
(1968
).18.
J. B.
Flor
, M.
Ungarish
, and J. W. M.
Bush
, “ Spin-up from rest in a stratified fluid: Boundary flows
,” J. Fluid Mech.
472
, 51
–82
(2002
).© 2017 American Association of Physics Teachers.
2017
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
