Flow visualization and the motion of small particles in superfluid helium Carlo Barenghi, Daniel Poole, Demos Kivotides and Yuri Sergeev (Newcastle), Joe.

Flow visualization and the motion of small particles in superfluid helium

Carlo Barenghi, Daniel Poole,Demos Kivotides and

Yuri Sergeev (Newcastle),Joe Vinen (Birmingham)

SUMMARY1. Flow visualization near absolute zero2. Equations of motion of small particles in He II3. Simple space-independent flows4. Instability of trajectories in pure superfluid limit5. Trapping of particles on vortex lines

1. FLOW VISUALIZATION

In classical fluids:Ink, smoke, Kalliroscope flakes, hydrogen bubbles,hot wire, Baker’s pH, laser Doppler, ultra-sound,

PIV (particle image velocimetry), etc

In liquid helium:Second sound, ion trapping, temperature andpressure gradients, NMR, Andreev reflection

- only probe averaged quantities- no information about flow patterns

Consider the following classical problem (Taylor-Couette):

For ω>ωcrit, azimuthal Couette flow becomes unstable andTaylor vortex flow appears

Flow visualisation shows that the critical wavenumberis k=π/d. This simple information helped G.I. Taylor’s pioneering stability analysis (1923).

Why is flow visualization useful ?

Consider the same Taylor-Couette problem but for He II(first tackled by Chandrasekhar and Donnelly in the 1950’s):

Experiments gave inconsistent results until it was realized that the wavenumber k decreases with the temperature T (Barenghi & Jones 1988), hence Taylor vortices become elongated axially and care must be taken to avoid end effects (Swanson & Donnelly 1991).

ωcrit vs T(Barenghi 1992)

k vs T

PIV in liquid helium

-Donnelly, Karpetis, Niemela, Sreenivasan and Vinen in He I, buoyant hollow glass spheres, 1 - 5 μm size-VanSciver, Zhang and Celik: in He II, heavier, hollow glass / polymer / solid neon, 0.8 - 50 μm size

PIV visualization of large scale turbulent flow around a cylinder in counterflow superfluid 4Heby Zhang and VanSciver, Nature Physics, 1, 36 (2005)

What do the tracer particles actually trace ?The superfluid ?The normal fluid ?The quantised vortices ?None of the above ?

Assuming:-particles do not disturb fluid-do not interact with / are trapped in superfluid vortices-are smaller than vortex spacing and Kolmogorov length-have small Reynolds number with respect to normal fluid-neglect Basset history force, Faxen drag correction, shear-induced lift and Magnus force

])([])([ ssss

nnnnpnp vv

t

vvv

t

vuv

dt

ud

pp udt

rd

NF inertia SF inertia

sec103

4

2

n

pa

Relaxation time:

Then the equations of motion of a neutrally buoyant particle of radius ap, position rp and velocity up are:

1:

2:

Stokes drag

2. EQUATIONS OF MOTION OF SMALL PARTICLES

3. Simple space-independent flows

Assume )exp( tiVv nn

)exp( tiVv ss

then )exp()/exp(0 tiUtUu ppp

where ])1[()1(

1s

sn

np ViVi

iU

Thus, if ωτ>>1 ss

nn

p VVU

particle moves withmass current J=ρnVn+ρsVs

Viceversa, if ωτ<<1, Up=Vn (particles trace normal fluid)

that is, Up=0 for second sound, Up=Vn at high T, and Up=Vs at low T

4. Space dependent flows,Instability of trajectories in pure superfluid limit

pvvt

v

dt

udss

sp

1

)(

)),(()( ttrvtu ssp

)()( trtr sp

is a formal solution of the equation of motion of the solidparticle, where rs(t) is a Lagrangian trajectory of a superfluidelement. One would thus expect that a small, buoyant, inertial particle, which at t=0 has velocity equal to the local superfluid velocity, will move with up=vs

The particle’s equationreduces to:

after using Euler’s equation. The RHS, the force per unit mass acting on a fluid element,is the force on the solid particle that replaces that fluidelement. Thus

Unfortunately even in the simplest case of the motionaround a single straight vortex, the particle’s trajectoryis UNSTABLE

Using polar coordinates (r,θ), the particle trajectory obeys

32

22

2

2

4 pppp r

rrdt

d

02 pppp dt

drr

dt

d

where ωp=dθp/dt . Perturb the circular orbit rp=R, ωp=Ω=Γ/(2π R2) by letting rp=R+r’,ωp=Ω+ω’with r’<<R, ω’<< Ω. Perturbations obey d3 r’/dr3=0, so

)0(')0('

)]0(')0('[2

)(' 22

rtdt

drtr

RRtr

ω’ is also quadratic in time. Any mismatch betweeninitial fluid/particle velocities and any sensitivity on initialconditions (Aref 1983: for a sufficient number of pointvortices there is chaos) will reinforce the instability.

The instability of the motion around avortex is confirmed by 2-dim and

3-dim numerical simulations

2-dim:motion around 3 point vortices on triangle:AB= fluid particleAC=inertial particle

CONCLUSION: at low T, the PIV particles do not trace a space-dependent superflow

5. Particle and vortex

If particle is trapped onto a vortex, helium’s energy is reduced by the equivalent vortex length lost

Tkaa

E Bpps

)/ln(

4

2

Particle arrives fromfar distance…

… and is trapped

The dynamics of the close approach / trapping to the vortexmay involve Kelvin waves. More information is needed using a microscopic model (eg the GP equation used by Berloff & Roberts 2000 and Winiecki & Adams 2000, or the vortex filament model of Tsubota 2005)

6. Neglect trapping and consider particles’ motion in turbulent counterflow

Superfluidvortex tangle,L=vortex line density

Intervortex spacingℓ ≈1/√L≈0.01 cm is muchlarger than the particle’ssize ap=3 X 10-4 cm

Numerical experimentsat T=1.3 K withL=2450 and 9700 cm-2,and T=2.171K withL=7500 cm-2

3][

)(

4)(

,])('['')('),(

RZ

ZdRZRV

VVVSSVVVSVVtSdt

d

i

isnisnis

L vs tHistogram of particles’ velocityNote vp≈vn=0.0118 cm/sec

Left:

sss vv

)(

t

vss

Contributions to acceleration: /)( pn vv

Middle: Right:

Turbulent counterflow with no trapping

CONCLUSIONS0. Need better flow visualization: eg PIV 1. Equations of motion of small particle in two-fluid hydrodynamics 2. Explicit solution for simple time-dependent, space-independent flows3. Pure superfluid: even the motion of a particle around a single straight vortex is unstable.4. Work is in progress to study trapping into vortices5. Other visualization techniques are being investigated: - shadography (Lucas) - excited states of neutral He molecules (McKinsey & Vinen) - micro sensors (Ihas)

For a typical particle size the relaxation time is smaller than the time to orbit a vortex at the typical intervortex distance ℓ=1/√L in a tangle, so the orbital motion will be damped.The radial motion is governed by

32r

u

dt

du pp

Consider neutrally buoyant particle at distance r0 from straightvortex in the presence of stationary normal fluid.

22 8/ s

The time for the particle to arrive at distance, say, 2ap (sufficiently large that the vortex is not much disturbed) is

40

440

161

8 r

art pa

bc=capture cross sectionℓ²/bc=mean free path for captureTp=ℓ²/bcvL=mean free time vL=typical particle velocity with respect to vorticesAssume particle approaches vortex with velocity vL.Assume capture time from a distance r0 for a particleinitially at rest is is of the order of the previous value

The particle will be probably captured if the time spentat distance r0 is greater than ta: r0/vL>ta

8

40rta

This yields the cross section and the mean free time:

n

psc

ab

3

2 23

3/2

2

)/(2

pp aT

1 msec < Tp <10 sec

so trapping may occur or not

Consider an ABC flow V=(Asin2πz+Ccos2πy, Bsin2πx+Acos2πz, Csin2πy+Bcos2πx)

Trajectories of inertial particles areunstable and concentrate in regionswhere the magnitude of the rate ofstrain tensor Sij=(dVi/dxj+dVj/dxi)/2 is large.

Time scale of instability dependson intensity of ABC flow andparticle’s relaxation time τ

vdt

rd f

Pathlines