Visualization of two-fluid flows of superfluid helium-4Wei Guoa,b, Marco La Mantiac, Daniel P. Lathropd, and Steven W. Van Scivera,b,1

aMechanical Engineering Department, Florida State University, Tallahassee, FL 32303; bNational High Magnetic Field Laboratory, Florida State University,Tallahassee, FL 32310; cDepartment of Low Temperature Physics, Faculty of Mathematics and Physics, Charles University, 180 00 Prague, Czech Republic;and dDepartments of Physics and Geology, Institute for Research in Electronics and Applied Physics, and Institute for Physics Science and Technology,University of Maryland, College Park, MD 20742

Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved December 13, 2013 (received for review July 17, 2013)

Cryogenic flow visualization techniques have been proved inrecent years to be a very powerful experimental method to studysuperfluid turbulence. Micron-sized solid particles and metastablehelium molecules are specifically being used to investigate indetail the dynamics of quantum flows. These studies belong toa well-established, interdisciplinary line of inquiry that focuses onthe deeper understanding of turbulence, one of the open problemof modern physics, relevant to many research fields, ranging fromfluid mechanics to cosmology. Progress made to date is discussed,to highlight its relevance to a wider scientific community, andfuture directions are outlined. The latter include, e.g., detailedstudies of normal-fluid turbulence, dissipative mechanisms, andunsteady/oscillatory flows.

Quantum fluids have been studied experimentally for manyyears and have by now become a major focus of low-tem-

perature physics (ref. 1 and references therein). Applications ofthe subject are widely ranged, from engineering, where super-fluid 4He is used as a coolant for superconducting magnets andinfrared detectors (2), to astrophysics, where superfluidity is in-voked to explain glitches in the rotation of neutron stars (3, 4)and the formation of cosmic strings (5, 6). More recently, su-perfluidity has been used to describe the collective behavior ofbirds (7) and a cosmological model has been used to obtainresults relevant to superfluid turbulence (8). The latter form ofturbulence, occurring in quantum fluids, is indeed an especiallyinteresting topic because of its quantum peculiarities and itssimilarity to classical turbulence. Superfluids, in which turbu-lence can be directly visualized and studied, include superfluid4He and atomic Bose–Einstein condensates (9). Due to the limitof small sample volumes, the experimental study of turbulence inBose–Einstein condensates has hardly begun. The developmentof visualization techniques applicable to superfluid 4He is thusessential, if our understanding of quantum turbulence is to makesignificant progress in the near future.Superfluid 4He is viewed as consisting of two interpenetrating

fluids. The gas of thermal excitations forms the normal compo-nent, which can be considered as a viscous fluid. The superfluidcomponent is inviscid and its rotational motion is possible only inthe presence of topological defects, in the form of quantizedvortex filaments. Turbulence in the superfluid component there-fore takes the form of a tangle of quantized vortex lines. Turbu-lence in the normal fluid is more conventional, although theinteraction between the normal fluid and the vortices leads to thenonclassical force of mutual friction between the two fluids.Turbulence in such a system can exhibit a behavior that is similarto that found in a classical fluid; but it may take forms that areunknown in classical fluid mechanics: for example, forms rele-vant to a fluid in which there is no viscous dissipation, and thosethat depend on the coexistence of the two fluids. Study of quan-tum turbulence can therefore enrich our knowledge of turbulencein general, as well as being interesting in its own right.

Visualization TechniquesFlow visualization techniques have been developed to a highdegree of precision and speed for classical fluid dynamicsinvestigations. However, for liquid helium, such techniques have

not kept pace, in part due to the extremely low temperature andlow density of the fluid. A number of early efforts were devotedto producing macroscopic particles for qualitative investigations(10–12) and the challenge of producing neutrally buoyant par-ticles that faithfully follow the complex flow fields has been themain impediment to quantitative advancement. In addition, sev-eral attempts have been made to visualize fluid dynamics in su-perfluid helium with microscopic tracers, which include neutronabsorption tomography, using 3He particles (13), and acousticcavitation imaging, using electron bubbles (14). These small par-ticles are expected to follow the fluid motion, because Stokes drag,from the normal-fluid flow, is deemed to dominate other forces.However, these methods have specific challenges. Neutron ab-sorption tomography requires a finely collimated neutron beamand the ability to raster the neutron beam through the region ofinterest. The electron-bubble cavitation method relies on thegeneration of strong ultrasonic sound waves in helium that in-evitably disturb the flow to be studied. Recently, the groups rep-resented by the present authors have successfully developed anumber of liquid helium flow-visualization techniques: particleimage velocimetry (PIV) and particle tracking velocimetry (PTV)techniques, using micron-sized solid particles (15–21), and a laser-induced fluorescence imaging technique, using angstrom-sizedHep2 excited molecules (22, 23).

PIV and PTV Techniques. PIV and PTV are valuable, quantitativetools that have been applied to study many scientific and in-dustrial problems (24). PIV can estimate the fluid velocity ina section of the flow field, by assuming a single, smoothly varyingvelocity field, whereas PTV allows the measurement of La-grangian quantities, i.e., the local velocity and its derivatives.With both techniques the particles are suspended in the fluid andreflect the light from a laser sheet that illuminates the flow fieldof interest. The time-dependent positions of the particles arethus captured and analyzed by a suitable digital imaging system.The particles for liquid helium experimentation can be broadlyclassified into two categories: solid particles, as are often used inclassical fluid dynamics experiments, and solidified particles,produced by injecting gases (usually hydrogen or deuterium) intoliquid helium. Micron-sized solid particles have been successfullyused in conjunction with the PIV technique to observe broad,average properties of the turbulent state of superfluid helium(15, 25, 26). However, such particles have proved to be too denseto explore the detailed structure of quantum turbulence. As aresult, most recent experiments have used solidified hydrogen (ordeuterium) particles (16–21). To produce these particles, a gas-eous mixture of helium and hydrogen in a volume ratio of ∼100:1is injected directly into liquid helium. A cloud of solid particleswith diameters typically of a few microns can be produced.

Author contributions: W.G., M.L.M., D.P.L., and S.W.V.S. designed research;W.G. andM.L.M.performed research; W.G., M.L.M., and D.P.L. analyzed data; and W.G., M.L.M., D.P.L., andS.W.V.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: vnsciver@magnet.fsu.edu.

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Although solids from other gases, such as argon, methane, ni-trogen, and propane, have been tested (27), hydrogen and deu-terium produce particles that are close to neutrally buoyant.

Hep2 Fluorescence Imaging Technique. Recently, a new visualizationtechnique, using excited He*2 triplet molecules, was developed(22, 23). These molecules can be produced in large numbers inliquid helium, following the ionization or excitation of groundstate helium atoms (28, 29). The singlet state molecules radia-tively decay in a few nanoseconds (30), but the triplet statemolecules are metastable with a radiative lifetime of about 13 s(31). These triplet molecules form bubbles in liquid helium witha radius of about 6 Å (32) and can be used as tracers. To imagethe He*2 triplet molecules, a cycling-transition laser-inducedfluorescence technique, first developed by McKinsey et al. (33)and McKinsey and coworkers ( 34), has been used. A laser pulseat 905 nm can excite helium molecules from their triplet groundstate a3Σ+

u to the excited electronic state d3Σ+u . Over 90% of the

molecules in the d state quickly decay to an intermediate b3Πgstate, emitting detectable fluorescent photons at 640 nm (35). Afilter can be used to block unwanted laser light, to achieve lowbackground. From the b3Πg state, molecules quench back to thea3Σ+

u state, and the process can be repeated so that each mole-cule produces many fluorescence photons.

Particle Motion in Superfluid Helium. In superfluid helium-4, vor-ticity is concentrated along the filamentary cores of quantizedvortex lines, and the velocity circulation around any such line isequal to κ= h=m4 ’ 10−3   cm2=s, where h is Planck’s constantand m4 indicates the mass of a 4He atom. A particle positionedon a quantized vortex line in superfluid helium displaces liquidthat has high kinetic energy, and, as a result, there is an energybinding the particle to a vortex line. The micron-sized particlesused in the PIV and PTV experiments can consequently gettrapped on quantized vortex lines, besides tracing normal-fluidflows (to date these experiments have been performed at tem-peratures above about 1.5 K, mainly due to heat load concerns).Moreover, as detailed below, imaging trapped particles allowsthe study of interesting vortex-line dynamics. However, in flowswhere the normal fluid, the superfluid, and the vortices havedifferent velocity fields, the behavior of these particles mightbecome difficult to interpret (19, 36, 37). On the other hand, He*2molecular tracers are entrained solely by the normal-fluid com-ponent above 1 K, due to their small binding energy on vortices(38). This makes them suitable for unambiguously probing var-ious normal-fluid flows; and, as revealed in recent experiments,He*2 molecules can attach to quantized vortex lines below 0.2 K(39), which has the potential to allow for vortex-line imaging atlow temperatures. More generally, it is clear that special careshould be taken when choosing the tracer particles, keeping es-pecially in mind that particle size should always be smaller thanrelevant flow length scales (19, 36).

Progress on Flow Visualization in HeliumVortex-Line Imaging Experiments. Bewley et al. (16) first observedsolid hydrogen particles trapped on quantized vortices, using thePTV facilities at University of Maryland (Fig. 1). The particletracking technique was then further developed to investigatevortex reconnection in superfluid helium. Reconnection was firstpredicted by Feynmann in 1955 as a process by which dissipationcould be allowed even in pure superfluid (40). Experiments,which strongly suggest the occurrence of vortex reconnection,were performed by Bewley et al. (41) and Paoletti et al. (17) in2008, supporting both the existence of this topological changeand the details of the predicted vortex motions. Fig. 2 shows anexample in which two vortices, marked with particles as solidlines, meet and cross, exchanging topology and rapidly retracting.The vortex rapid retraction leads to a high particle velocity v,

which is characterized by the predicted v−3 power-law distribu-tion (17, 42). This clearly distinguishes quantum turbulence fromclassical turbulent flows, as the velocity distribution of the latterhas a nearly Gaussian shape (43). Such a power-law shape of thetails of the velocity distributions was later confirmed in two-fluidflow experiments (20, 21) and in superflow numerical simu-lations (44–47). Note, however, that the reasons why the tails ofthe velocity distributions obtained in two-fluid flow (17, 20, 21,42) are consistent with those computed in the absence of normal-fluid flow (44–47) are still not entirely understood and furtherinvestigations are consequently required to address the issue.Besides visualizing vortex reconnection, PTV has also been

used to study Kelvin waves excited on vortex lines, which havelong been discussed as important in the energetics and dynamicsof the quantum fluid state (48). Recent visualization experimentsat the University of Maryland strongly support the existence ofKelvin waves generated by vortex reconnection (49). Thesewaves exhibit a self-similar, traveling helical perturbation to thevortex lines following a line reconnection, as was theoreticallypredicted by Schwarz (50). Vortex rings, loaded by particles,have also been visualized (51) and the corresponding particledynamics theoretically addressed (36).

Thermal Counterflow Experiments. Thermal counterflow is a uniqueflow mode that exists only in the superfluid phase of helium,even though its study might be also relevant to the understandingof heat transport, e.g., in the form of turbulent convection (52,53). The condition can be easily established by applying heat at

A B C

Fig. 1. Intensity inverted images showing hydrogen ice particles trapped onquantized vortex lines in superfluid helium (16). The concentration of hy-drogen ice can be varied such that (A) only isolated particles are trapped onvortices, (B) multiple particles form dotted lines on vortices, or finally (C)solid hydrogen skeletons perturb the dynamics of the vortices and stabilizebranches and crossings. The natural state is for crossings to reconnect.

-37.5 ms -12.5 ms 12.5 ms 37.5 ms

t=t0 t>t0t<t0

(t)(t)

100 microns

Fig. 2. Images at low ice concentration confirm vortex line reconnection andallow one to quantify the dynamics of the intervortex separation δ(t) (42).

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the closed end of a flow channel containing superfluid helium. Thenormal-fluid component carries the heat and moves away from theheater at a mean velocity vn = q=ρST, where q is the heat flux, ρ isthe helium density, T denotes the temperature, and S representsthe specific entropy of helium (54). The superfluid componentmoves toward the heat source, serving to eliminate any net massflow. It has been known for many years that, above a (small) criticalvalue of heat flux, the superfluid component in counterflowbecomes turbulent. This results in a tangle of quantized vortexlines, whose dynamical behavior is an essential ingredient ofquantum turbulence (55). Counterflow allows a controlled forcingof the superfluid state away from equilibrium.A number of flow visualization experiments have been per-

formed in thermal counterflow. Early PIV experiments indicatedthat average particle velocities were typically less than the nor-mal-fluid velocity vn (26). Subsequently, using PTV techniques,Paoletti et al. showed that, at low relative velocity, the particlevelocity distribution is indeed bimodal (56). Some particles movein the opposite direction to the heat current. These particles areinterpreted as being trapped in the tangle of quantized vorticesgenerated by the counterflow [note that the vortex tangle movestoward the heat source with a velocity that is generally differentfrom the superfluid velocity (48)]. The rest of the particles aremainly influenced by Stokes drag, from the normal-fluid flow,and their velocity agrees with the prediction of Landau andLifshitz (54). Note, however, that particle trapping is generallya dynamical phenomenon; i.e., particles can also escape fromvortices, depending on the experimental conditions. More re-cently, Chagovets and Van Sciver (19) also used PTV to showthat the bimodal velocity distribution occurs only at low relativevelocities and that above a critical velocity, associated with theparticles being untrapped from vortex lines, the velocity distri-bution becomes monovalued, similar to that observed by Zhangand Van Sciver, using PIV (26).

To unambiguously examine the normal-fluid motion in ther-mal counterflow, the He*2 molecule visualization technique wasrecently used (23). The He*2 tracers were produced by a tungstenfield-emission source in a glass counterflow channel. A focusedpump laser pulse at 910 nm was used to tag a line of moleculesacross the channel by driving the molecules to a long-lived vi-brational level a(1) (Fig. 3A). This tagged line was imaged sub-sequently, using a probe laser pulse at 925 nm. Up to 40 imageswere superimposed at each given pump–probe delay time toachieve a good image quality. Typical summed images are shownin Fig. 3B, suggesting a flat averaged normal-fluid velocity profilethat should be expected for turbulent flow, in a long enoughchannel. The observed rapid growth of the averaged line widthwith time further supports the claim that the normal-fluid flow isturbulent (23). Note that, due to the mutual friction between thetwo fluid components, dissipation occurs at all length scales inthe normal fluid, which contrasts with the situation in classicalturbulence, where dissipation is deemed to take place only belowa small length scale, called the Kolmogorov length scale (54).The experiment revealed a unique normal-fluid turbulence incounterflow (57).

Normal-Fluid Turbulence in Counterflow. The unique type of tur-bulence just discussed obviously calls for further attention.Studying it not only will likely broaden our understanding ofturbulence in general, but also might have practical significancebecause the turbulent normal-fluid flow could, e.g., alter ourunderstanding of heat transfer. An experiment has been specif-ically designed at Florida State University to examine the nor-mal-fluid velocity field in counterflow. A thin line of He*2 tracersis created via laser-field ionization in helium. To achieve therequired high electric field for ionizing ground state heliumatoms, laser intensity as high as 1013 W/cm2 is needed (58). Sucha high instantaneous laser intensity can be achieved by focusinga femtosecond laser pulse through a tiny cross-section. Themolecule density so created is high enough to allow high-qualitysingle-shot imaging of the tracer line. Fig. 4 shows fluorescenceimages of He*2 tracer lines that have been successfully generatedand imaged in counterflow, at 1.85 K, with a 35-fs laser pulse, at55 μJ. At low heat fluxes, a straight tracer line deforms intoa parabolic shape, indicating the Poiseuille laminar velocityprofile of the normal fluid. At large heat fluxes, the tracer linedistorts, possibly due to the turbulent eddies in the normal fluid.The local normal-fluid velocity could then be estimated by di-viding the center displacement of a small line segment by thedrift time. Structure functions of the turbulent flow could becomputed based on the derived velocities (59), which shouldallow us to gain information on the turbulent energy spectrum.By creating multiple lines to include crosses or grid tracerstructure, measurements of normal-fluid vorticity and othercomplex velocity derivatives can be made.

3ud +

3ua +

3gc +

3gb

012

01

905 nm905 nm

925 nm925 nm 640 nm

910 nm

1073 nm

1099 nm

01∑

∑

∑

∏t=0 ms t=80mst=40ms

925 nm 5 mm

910 nm

A B

Fig. 3. (A) Schematic diagram showing the optical transitions for imagingthe He*2 triplet molecules. The levels, labeled 0, 1, 2 for each electronic state,are the vibrational levels of the corresponding state. (B) Averaged images ofa line of tagged helium molecules via the tagging fluorescence methodacross a square channel (5 mm side width) in thermal counterflow witha heat flux of 640 mW/cm2.

Laminar flow

Heat flux: 10 mW/cm2

Drift time: 900 ms

Turbulent flow

Heat flux: 215 mW/cm2

Drift time: 15 msNo heat flux

T=1.85K

1 cm

80 um

Fig. 4. Fluorescence images showing the motion of a thin line of He*2 tracers in thermal counterflow. The tracer line is created via laser-field ionization byfocusing a femtosecond laser pulse into superfluid helium. The drift time denotes the time between the creation and the imaging of the tracer line. Thesecond image was taken in steady-state flow, whereas the third image was taken by the time the heater was turned off.

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Particle Acceleration Measurements. The Lagrangian dynamics ofsolid deuterium particles, at length scales comparable to themean distance ℓ between quantized vortices, have been recentlystudied in steady-state thermal counterflow at Charles Universityby using the PTV technique (20, 21). It was unexpectedly foundthat the normalized distribution of the particle accelerationappears to follow a classical-like behavior (21). Fig. 5 displaysthe normalized probability density function (PDF) of the in-stantaneous particle acceleration in the vertical direction, that ofthe imposed counterflow, in different experimental conditions, atlength scales about one order of magnitude smaller than thosereported previously (21). The result was obtained by collectingimages at frame rates that allow the study of the particle dy-namics at scales smaller than ℓ. The latter scales can be quanti-fied by introducing the nondimensional time τ= t1=t2, where t1 isthe time used for the calculation of accelerations along thetracks, t2 = ℓ=Vabs, and Vabs denotes the mean particle velocity. Itcan be seen that the agreement with a log-normal fit used forclassical turbulent flows (60, 61) to hold also for τ ’ 0:1, i.e., atlength scales about one order of magnitude smaller than ℓ. Note,

however, that, as suggested in ref. 21, the classical fit seems notto agree with the experimental data at large accelerations, for thesmallest times. The distribution tails of the particle accelerationin the horizontal direction display indeed more noticeabledepartures from the classical shapes, as the particle dynamics aredominated by the imposed vertical velocities (56). The outcome,whose details will be reported elsewhere, represents unique ex-perimental evidence that the mean distance between quan-tized vortices can be seen as the length scale distinguishingclassical-like from quantum behavior.

Flow-Across-Obstacle Experiments. An important, classical fluiddynamical condition is that of the flow past solid objects, such ascylinders and spheres, with the associated drag crisis (62). Initialstudies using solid particles and the PIV technique showed theexistence of large-scale turbulent eddies, both behind and infront of a cylinder in thermal counterflow (15). Such a behaviorhas no classical analog. More recently, Chagovets and Van Sciverdiscussed the existence of two distinct velocity fields in coun-terflow around a cylinder (63). At low relative velocities, typically

Fig. 5. Probability density function (PDF) of ðaz − ameanz Þ=asdz , where amean

z and asdz are the mean and SD of the instantaneous vertical acceleration az, re-spectively (trajectories with at least five points, total number of points of each dataset larger than 105, and the area below the curves normalized to 1). Solidsquares, τ= 0:13; solid circles, τ= 0:21; solid triangles, τ= 0:28; open squares, τ=0:14; and solid line, log-normal fit, calculated by using equation 1 in ref. 61,with s = 1, the employed variable being the particle normalized acceleration shown on the horizontal axis.

)mm( x)mm( x

y (m

m)

q q

A B

Fig. 6. Particle trajectories around a 2-mm cylinder obtained in counterflow by the PTV technique with hydrogen particles. (A) The particles that move with thenormal fluid velocity. (B) The motion of particles that interact with quantized vortex lines (q = 50 mW/cm2, T = 1.93 K, vn = 0.23 cm/s, Reynolds number Re = 485).

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vn < 1 cm/s, the normal-fluid flow appears as a laminar flow,whereas the particles tracking the superflow are more typicallytrapped on the vortex lines (Fig. 6). At higher relative velocity, thefields of the two fluid components are no longer separable, and theparticle motion begins to display the large-scale eddies first ob-served using solid particles (15). As with channel flow (19), thistransition is deemed to occur as a result of Stokes drag, due to thenormal-fluid flow, being sufficient to dislodge the trapped particlesfrom the superfluid vortex lines. Other related visualization studiesinclude preliminary results on the flow past an oscillating sphere(64) and additional investigations on the occurrence of macroscopicvortices in the proximity of cylinders in counterflow (65).

Forced Flow Experiments. Another basic, classical problem thatcan be used to determine the extent to which superfluid dy-namics display classical behavior is the pressure gradient inturbulent pipe flow. Unlike counterflow, in forced flow of tur-bulent superfluid helium the two fluid components are believedto be coupled together by the mutual friction force such that theflow velocity Uave equals the velocities of the two fluids. Thisleads to a turbulent friction factor that scales with the classicalReynolds number Re=UaveW=νn, where W is a relevant flowscale and νn denotes the kinematic viscosity of the normal-fluidcomponent (66). To display such a behavior, the fluid would beexpected to have a velocity boundary layer similar to that seen inclassical fluids (67). At Florida State University, Xu and VanSciver made PIV measurements of the velocity profile near thewall in forced flow of superfluid helium at Re≈ 105 (68). Theexperimental apparatus used calibrated bellows pumps driven bylinear actuators to produce a known average flow rate in a square

cross-section channel, of side W = 20 mm. The channel is hori-zontal, in a specially designed cryostat that allows optical accessto the flow field. The results from these experiments showeda velocity distribution that is essentially classical in character,with a measurable velocity boundary layer that scales with Re(Fig. 7). Further investigations are consequently needed toclarify whether a parameter range exists where forced flow su-perfluid dynamics are different from classical hydrodynamics andalso to assess in which conditions the influence of the physicalboundaries on quantum flows can be neglected.

Conclusion and Future WorkThe study of quantum turbulence has in recent years benefitedfrom the use of flow visualization techniques, which led researchersto appreciate more clearly similarities and differences betweenclassical and quantum flows (69). These very powerful tools pro-duced, at the same time, results that are posing more questions thangiving clear answers, showing thus that the probed phenomena areindeed worth investigating. Thermal counterflow is a well-knowquantum flow, characterized by a unique form of heat transport,whose links to classical turbulent convection (52, 53) are yet tobe explored in detail. Similarly, the newly discovered normal-fluid turbulence (23) deserves further attention, as it might leadto broadening our comprehension of turbulence in general.Vortex reconnection (41) and Kelvin waves (49) are dissipativemechanisms, crucial to the energy transfer in quantum flows, andshould be extensively investigated, also to assess their relevanceto other research fields, such as magnetohydrodynamics (70).The complex interactions between particles, quantized vortices,and macroscopic eddies in quantum flows represent still an openproblem of cryogenic flow visualization (37) and its relations withthe behavior of particles in classical turbulent flows (71) shouldbe fully exploited (21). The study by visualization of unsteady/oscillating flows is a new line of inquiry (64) that could, for ex-ample, give valuable contributions to the popular study of thetemporal and spatial decay of turbulent flows (1, 72). The in-fluence of the boundaries of the experimental volume on thestudied quantum flows is also an open issue (68) and it might beinteresting to know how vortex tangles behave close to solidwalls. The latter possible, future directions of cryogenic flowvisualization are further evidence that the discussed techniquesare very valuable tools for the analysis of puzzling natural phe-nomena, whose understanding can lead to useful insights into theunderlying physics of many, interconnected research fields.

ACKNOWLEDGMENTS. W.G. acknowledges his collaborators D. N. McKinsey,W. F. Vinen, G. Ihas, A. Golov, P. V. E. McClintock, and S. Fisher. M.L.M.thanks D. Duda, M. Rotter, and L. Skrbek for fruitful discussions and valuablehelp. D.P.L. acknowledges his collaborators G. Bewley, M. Paoletti, K. Gaff,E. Fonda, D. Meichle, and M. E. Fisher. S.W.V.S. acknowledges collaboratorsT. Chagovets, S. Fuzier, T. Xu, and T. Zhang. W.G. acknowledges the startupsupport from Florida State University and the National High Magnetic FieldLaboratory. M.L.M. acknowledges the support from the Czech ScienceFoundation under Grant GA�CR P203/11/0442. D.P.L. acknowledges supportfrom the Center for Nanophysics and Advanced Materials and from the Na-tional Science Foundation (DMR-0906109). S.W.V.S. acknowledges the supportfrom the US Department of Energy under Grant DE-FG02-96ER40952.

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2y/W0.0 0.2 0.4 0.6 0.8 1.0

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4.50x105

n=7n=8.8

Re=9.00x10

Fig. 7. Normalized velocity profile near the wall of a channel containingforced flow superfluid helium at 2.1 K. The lines correspond to the fitðUave=UmaxÞ= ð2y=WÞ1=n, where Uave is the flow velocity at a distance y fromthe channel wall of sideW; Umax represents the maximum velocity, which occursat the center of the channel; and n, specified on the curve, shows the best fitbetween 7 and 8.8, depending on the value of the Reynolds number Re (67).

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