Flow visualization and the motion of small particles in superfluid helium Carlo Barenghi, Daniel Poole, Demos Kivotides and Yuri Sergeev (Newcastle), Joe Vinen (Birmingham) SUMMARY 1. Flow visualization near absolute zero 2. Equations of motion of small particles in He II 3. Simple space-independent flows 4. Instability of trajectories in pure superfluid limit 5. 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 and pressure gradients, NMR, Andreev reflection - only probe averaged quantities - no information about flow patterns

Consider the following classical problem (Taylor-Couette): For ω>ω crit, azimuthal Couette flow becomes unstable and Taylor vortex flow appears Flow visualisation shows that the critical wavenumber is 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). ω crit vs T (Barenghi 1992) k vs T

PIV in liquid helium -Donnelly, Karpetis, Niemela, Sreenivasan and Vinen in He I, buoyant hollow glass spheres, μm size -VanSciver, Zhang and Celik: in He II, heavier, hollow glass / polymer / solid neon, μm size

PIV visualization of large scale turbulent flow around a cylinder in counterflow superfluid 4 He by 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 NF inertiaSF inertia Relaxation time: Then the equations of motion of a neutrally buoyant particle of radius a p, position r p and velocity u p are: 1: 2: Stokes drag 2. EQUATIONS OF MOTION OF SMALL PARTICLES

3. Simple space-independent flows Assume then where Thus, if ωτ>>1 particle moves with mass current J=ρ n V n +ρ s V s Viceversa, if ωτ<<1, U p =V n (particles trace normal fluid) that is, U p =0 for second sound, U p =V n at high T, and U p =V s at low T

4. Space dependent flows, Instability of trajectories in pure superfluid limit is a formal solution of the equation of motion of the solid particle, where r s (t) is a Lagrangian trajectory of a superfluid element. 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 u p =v s The particle’s equation reduces 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 fluid element. Thus

Unfortunately even in the simplest case of the motion around a single straight vortex, the particle’s trajectory is UNSTABLE Using polar coordinates (r,θ), the particle trajectory obeys where ω p =dθ p /dt. Perturb the circular orbit r p =R, ω p =Ω=Γ/(2π R 2 ) by letting r p =R+r’,ω p =Ω+ω’ with r’<<R, ω’<< Ω. Perturbations obey d 3 r’/dr 3 =0, so ω’ is also quadratic in time. Any mismatch between initial fluid/particle velocities and any sensitivity on initial conditions (Aref 1983: for a sufficient number of point vortices there is chaos) will reinforce the instability.

The instability of the motion around a vortex is confirmed by 2-dim and 3-dim numerical simulations 2-dim: motion around 3 point vortices on triangle: AB= fluid particle AC=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 Particle arrives from far distance… … and is trapped

The dynamics of the close approach / trapping to the vortex may 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 Superfluid vortex tangle, L=vortex line density Intervortex spacing ℓ ≈1/√L≈0.01 cm is much larger than the particle’s size a p =3 X cm Numerical experiments at T=1.3 K with L=2450 and 9700 cm -2, and T=2.171K with L=7500 cm -2

L vs t Histogram of particles’ velocity Note v p ≈v n = cm/sec Left: Contributions to acceleration: Middle:Right: Turbulent counterflow with no trapping

CONCLUSIONS 0. 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 flows 3. Pure superfluid: even the motion of a particle around a single straight vortex is unstable. 4. Work is in progress to study trapping into vortices 5. 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 Consider neutrally buoyant particle at distance r 0 from straight vortex in the presence of stationary normal fluid. The time for the particle to arrive at distance, say, 2a p (sufficiently large that the vortex is not much disturbed) is

b c =capture cross section ℓ²/b c =mean free path for capture T p =ℓ²/b c v L =mean free time v L =typical particle velocity with respect to vortices Assume particle approaches vortex with velocity v L. Assume capture time from a distance r 0 for a particle initially at rest is is of the order of the previous value The particle will be probably captured if the time spent at distance r 0 is greater than t a : r 0 /v L >t a This yields the cross section and the mean free time: 1 msec < T p <10 sec so trapping may occur or not

Consider an ABC flow V=(Asin2πz+Ccos2πy, Bsin2πx+Acos2πz, Csin2πy+Bcos2πx) Trajectories of inertial particles are unstable and concentrate in regions where the magnitude of the rate of strain tensor S ij =(dV i /dx j +dV j /dx i )/2 is large. Time scale of instability depends on intensity of ABC flow and particle’s relaxation time τ Pathlines