Frequency Dependence of the Critical Velocity of a Sphere Oscillating in Superfluid Helium-4

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by  R. Hänninen

Frequency Dependence of the Critical Velocity of a Sphere Oscillating in Superfluid
Helium-4
1 2
R. Ha¨nninen and W. Schoepe
1Low Temperature Laboratory, Helsinki University of Technology, FIN-02015 TKK, Finland
2Institut fu¨r Experimentelle und Angewandte Physik,
Universit¨at Regensburg, D-93040 Regensburg, Germany
(Dated: January 16, 2008)
Itisshownthatthecriticalvelocityofasmallsphereoscillatinginsuperfluidheliumincreaseswith
8 thesquarerootoftheoscillation frequency. Thisbehaviorcanbedescribedbyasimpledimensional
0 argument. Thesizeofthesphereandthetemperatureofthesuperfluidarefoundtohavenooronly
0 verylittleeffect. Surfacepropertiesofthesphereandremanentvorticitymayhaveaninfluencebut
2 havenot been undersystematic investigation in these measurements.

n
PACSnumbers: 67.25.dk,67.25.bf,47.27.Cn
a
J

6
A quantitative description of the critical velocity for must be given by κ/R, whatever the length scale R is.
1
theonsetofturbulentdragofamacroscopicbodymoving The question arises whether the physics will be dif-

] insuperfluidheliumisstillanunsolvedproblemalthough ferent in case of oscillatory motion. The turbulent drag
er there is a large amount of experimental data available forceFD =γv2 F0 (γ =CDρπR2/2,whereρisnowthe
−
h from vibrating wires, grids, quartz tuning forks, and os- superfluid density and C 0.4 for a sphere [6]) differs
D
≈
t cillating spheres. At small driving forces F the velocity from the classical result for steady flow only by the shift
o
. amplitude v grows linearly because the drag force due F0 [2] . This shift implies a critical velocity for the on-
t
a toquasiparticlescatteringisproportionaltothevelocity. set of turbulent drag given by vc = F0/γ. The shift is
m Atacertaincriticalvaluev vorticityisshedbythemov- temperatureindependent[7],andsopisthiscriticalveloc-
c
- ingbodyandthedragincreasesstrongly. Typically,these ity. A theoretical calculation of F0 is not available. (It
d
velocityamplitudesareintherangefrommm/suptoal- should be mentioned, however, that a similar drag force
n
mostm/s[1], andthereforemuchlowerthanthe Landau has been calculated for a body moving through a Bose
o
c critical velocity of about 50 m/s. Above vc the velocity condensate[8].) In this situationit is necessaryto resort
[ amplitude grows approximately with the square root of to dimensional considerations. Because there is a time

the driving force because turbulent drag is proportional scalenowgivenbytheoscillationfrequencyω,wehavea
1
v to the square of the velocity [2]. Critical velocities for new velocity scale for the onset of the turbulent regime
1 vortex production have been calculated, for a review see given by [9]
2 [3]. For example, a vortex ring of radius R is calculated
5
to expand if a critical velocity is exceeded that is given
2 v √κ ω. (2)
. approximately by [4] c ∼ ·
1
Remarkably, there is no length scale present any more.
0
Theremightbe,however,alogarithmictermliketheone
8 κ 8R
0 vc = 2πR ·ln(cid:18)a0 (cid:19), (1) in Eq.(1). This implies that this critical velocity does
: not or only very little depend on the size of the body.
v
where κ 10−7m2/s is the circulation quantum and Inthefollowing,thisscalinglawisconfirmedexperimen-
i
X a0 0.15≈nm is the size of the vortex core or the co- tally with spheres oscillating at various frequencies. In
≈
r herence length. For a sphere of radius R = 100 µm this Appendix 1 a model calculation based on Kelvin waves
a
would give 2.5 mm/s for a ring of the same size. This is presented that supports the scaling law. In the Ap-
is at least an order of magnitude lower than the experi- pendix 2, the same dimensional arguments are applied
mental value, see below. Within this picture, it follows to a classical viscous liquid and shown to be consistent
that smaller vortex rings will be shed, probably because with analytical calculations for spheres.
of the roughness of the surface. Numerical simulations The experiments were performed at Regensburg Uni-
of vortex shedding have been performed using full Biot- versityintheyearsfrom1993to2000. Mostoftheresults
Savartcalculationsand assuminga smoothsphere [1, 5]. havebeenpublishedinearlierwork[10],butasystematic
Numerically, the obtained critical velocities are of same analysisofthefrequencydependenceofthecriticalveloc-
order as in experiments. However, the results are some- ities was only performed recently. The data were taken
what sensitive to the computationalaccuracy and there- with 2 different spheres, having a radius of 100 µm and
forethefrequencydependenceforv isnotextractedyet. 124µm. Thesmalleronewasusedina3He cryostat,the
c
In any case, a simple dimensional argument tells us that other one in a dilution cryostat. The magnetic spheres
a characteristic velocity scale in the turbulent superfluid were placed inside a horizontal parallel plate capacitor
2
consisting of niobium electrodes to which a voltage was thermally activated process for the onset of turbulence
appliedwhilethetemperaturewasdroppedbelowthesu- in these experiments.
perconducting transition temperature of niobium, ca.10 It will be interesting to compare these results for
K.Atheliumtemperaturesthechargedsphereswerelev- spheres with those for vibrating wires. Earlier observa-
itating between the capacitor plates. Frozen flux in the tions by the Osaka group indicated frequency indepen-
electrodes was necessary for a stable vertical oscillatory dent critical velocities below 2 kHz [13]. A more recent
motionwhenasmallacvoltagewasappliedatresonance. analysis by other authors [1] did show an increase scal-
The frequency of the oscillations could not be controlled ing approximatelywith ω1/3. A more complete andvery
quantitatively because it depended on the frozen flux of recent work of the Osaka group is in fact in agreement
the particular levitation status. It was observed, how- with Eq.(2) [14].
ever, that a faster cooling rate from 10 K to 4.2 K in We are very grateful to J. J¨ager, H. Kerscher, M.
general lead to higher resonance frequencies. The criti- Niemetz, and B. Schuderer for their excellent thesis
calvelocities were measuredby increasing the driving ac work in the former Low Temperature Group at Regens-
voltage from the linear flow regime to the turbulent one burg University, from which these data were obtained.
andback. Theresultsofthevariousruns,alltakenbelow W.S. has benefitted very much from his visits to the
1 K, for both spheres are displayed in the Figure 1. helium group lead by M. Krusius of the Low Temper-
ature Laboratory at Helsinki University of Technology.
0.1 R.H.acknowledgesthesupportfromtheAcademyofFin-
land(Grant114887)andappreciatesdiscussionswithM.
Tsubota, H. Yano, M. Kobayashi, and S. Fujiyama. We
both thank H. Yano very much for showing us his new
resultsonthe frequencydependence ofthecriticalveloc-
)
/s ities of his vibrating wires before publication.
m
(
y
t
i
c
o
l APPENDIX 1
e
v
One simple way to derive the scaling relation for v
c
(Eq.(2)) is the following. The dispersion relation for
Kelvin waves with wave number k is given by:
0.01
100 1000 κk2 2
frequency (Hz) ω(k) ln . (3)
≈ 4π (cid:18)ka0(cid:19)
FIG.1: Criticalvelocityfortheonsetofturbulenceasafunc-
If we assume that the external drive at frequecy ω only
tion of the frequency of 2 oscillating spheres. Blue squares:
radius 124µm; red dots: radius 100µm; green solid line: drivesthescalesR=2π/k givenbytheabovedispersion
vc =2.85√κ2πf (m/s); temperature is below 1 K. relation, we find
It is obvious that the predicted √κω variationgives a
fair fit to the data. The errors of the individual values R κπln(2/ka0)/ω. (4)
≈
of v vary from ca. 1% to about 5%, depending on how p
c
closely v was approachedin the experiments. The large Inserting this then in Eq. (1) gives a critical velocity
c
scattervisiblenear200Hzisbyfarlargerthananyerror
bar. Recently, it has been found that remanent vortic-
ity affects the critical velocity of vibrating wires and its ln(16π/ka0)
v √κω. (5)
c
absence greatly increased vc [11]. This might be a rea- ≈ 2π πln(2/ka0)
son for the scatter. Moreover, surface properties might p
affect a numerical prefactor in Eq.(2), although, as indi- The coefficient in front should in principle be calculated
catedabove,onlyonsomelogarithmicscale,otherwisewe selfconsistentlyusingthedispersionrelationbutdepends
would have had much more scatter. The mostly higher onlyveryweaklyonthefrequency. Fortypicalsphereex-
resonancefrequenciesofthesmallersphereareduetoits periments (200Hz) the coefficientshouldhavea value of
smallermassandprobablyalsoduethefastercoolingrate 0.40 which is somewhat smaller than 2.85 obtained from
from 10 K used with the 3He cryostat. Towards higher the data of Fig.1. Of course, there is no real proof that
temperatures only a rather weak increase of v by about Eq.(1) should give the correctvalue in case of oscillating
c
10% was observed up to 2 K [12]. This rules out any flows.
3
APPENDIX 2
The same dimensional arguments are applied now to
[1] Forarecentreviewoncriticalvelocities,seeR.H¨anninen,
a classical viscous liquid. For the stationary case the M. Tsubota, and W.F. Vinen, Phys. Rev. B 75, 064502
velocity scale is set by ν/R, where ν is the kinematic (2007).
viscosity. This is confirmed by comparing the laminar [2] J.J¨ager,B.Schuderer,andW.Schoepe,Phys.Rev.Lett.
drag force of a sphere λv with the turbulent drag force 74, 566 (1995).
γv2. From Stokes’ solution we have λ = 6πνρR. Hence, [3] R.J. Donnelly, Quantized Vortices in Helium II (Cam-
bridge University Press, Cambridge, 1991).
by extrapolating both regimes, the drag forces are equal
[4] R.P.Feynman,ProgressinLowTemperaturePhysics,ed.
at a characteristic velocity
C.J. Gorter, vol.1 (North Holland, Amsterdam, 1955).
[5] R.Goto, S.Fujiyama,H.Yano,Y.Nago, N.Hashimoto,
K. Obara, O. Ishikawa, M. Tsubota, and T. Hata, to
λ 12ν
v = = . (6) appear in Phys. Rev. Lett. (2008), arXiv:0711.1448v1
ch
γ CDR [cond-mat.other].Foraveryinterestingvideosimulating
a sphere oscillating in helium, see:
For the oscillating case we get from dimensional argu-
http://matter.sci.osaka-cu.ac.jp/ fujiyama/contents.
ments a velocity scale √νω. Stokes’ solution for the lin- ∼
html.
eardragisnow[6]λ=6πνρR (R/δ),whereδ = 2ν/ω [6] L.D. Landau and E.M. Lifshitz, Fluid Mechanics, 2nd
·
is the viscous penetration depth, and it is assumped that edn. (Butterworth-Heinemann,Oxford, 1987).
R δ, which usually is the case for macroscopic bod- [7] M. Niemetz and W. Schoepe, J. Low Temp. Phys. 135,
ies≫in liquid helium at or above the superfluid transition 447 (2004).
[8] T.Winiecky,J.F.McCann, andC.S.Adams,Phys.Rev.
temperature. The velocity scale is now given by
Lett. 82, 51861 (1999).
[9] Thedimensionalargumentwaskindlybroughttoourat-
12 tention by S.Mio, M. Kobayashi, and H.Yano.
vch = √νω. (7) [10] An almost complete list of the publications of the Re-
√2C
D gensburg group on oscillating spheres can be found in
a recent publication, W. Schoepe, J. Low Temp. Phys.
Note that there is no length scale in this expressionand,
(2008, doi 10.1007/s10909-007-9608-2).
therefore,v doesnotdependonthe sizeofthespheres.
ch [11] N.Hashimoto,R.Goto,H.Yano,K.Obara,O.Ishikawa,
Comparing these results with those for the superfluid, and T. Hata, Phys. Rev.B 76, 020504 (2007).
we note the similarity, except that the kinematic viscos- [12] Jan J¨ager, Laminare und turbulente Str¨omung von He-
ity ν is replaced by the circulation quantum κ. The ma- lium II um eine oszilliernde Kugel, dissertation (Verlag
jor differences, however, are, firstly, that in the classical Dr. K¨oster, Berlin, 1996).
[13] H. Yano, A. Handa, M. Nakagawa, K. Obara,
liquid there are about three orders of magnitude in ve-
O. Ishikawa, and T. Hata, AIP Conf. Proc. 850, 195
locity between the Stokes regime of linear drag and fully
(2006).
developed turbulence and, secondly, because of the ex-
[14] H. Yano, privatecommunication (unpublished).
trapolation, vch is not a point on the v(F) response of [15] M.Blaˇzkov´a,D.Schmoranzer,andL.Skrbek,Phys.Rev.
the body, whereas in the superfluid we have a true crit- E 75, 025302 (2007). See also Ref. [10] for a recent re-
ical velocity of the oscillator for the onset of turbulent viewontransitiontoturbulenceobservedwithoscillating
drag. Experiments with quartz tuning forks, for which bodies.
analyticalexpressionsofthedragforcesarenotavailable,
confirm this classical scaling [15].