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Flow visualization and the motion of small particles in superfluid helium

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

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Flow visualization and the motion of small particles in superfluid helium

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  1. 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

  2. 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

  3. Why is flow visualization useful ? Consider the following classical problem (Taylor-Couette): For ω>ωcrit, 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).

  4. 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

  5. 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

  6. PIV visualization of large scale turbulent flow around a cylinder in counterflow superfluid 4He 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 ?

  7. 2. EQUATIONS OF MOTION OF SMALL PARTICLES 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 Then the equations of motion of a neutrally buoyant particle of radius ap, position rp and velocity up are: 1: Stokes drag NF inertia SF inertia Relaxation time: 2:

  8. 3. Simple space-independent flows Assume then where particle moves with mass current J=ρnVn+ρsVs Thus, if ωτ>>1 that is, Up=0 for second sound, Up=Vn at high T, and Up=Vs at low T Viceversa, if ωτ<<1, Up=Vn (particles trace normal fluid)

  9. 4. Space dependent flows, Instability of trajectories in pure superfluid limit 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 is a formal solution of the equation of motion of the solid particle, where rs(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 up=vs

  10. 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 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 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.

  11. 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

  12. 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

  13. 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)

  14. 6. Neglect trapping and consider particles’ motion in turbulent counterflow Numerical experiments at T=1.3 K with L=2450 and 9700 cm-2, and T=2.171K with L=7500 cm-2 Intervortex spacing ℓ ≈1/√L≈0.01 cm is much larger than the particle’s size ap=3 X 10-4 cm Superfluid vortex tangle, L=vortex line density

  15. Turbulent counterflow with no trapping L vs t Histogram of particles’ velocity Note vp≈vn=0.0118 cm/sec Left: Middle: Right: Contributions to acceleration:

  16. 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)

  17. Consider neutrally buoyant particle at distance r0 from straight vortex in the presence of stationary normal fluid. 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 The time for the particle to arrive at distance, say, 2ap (sufficiently large that the vortex is not much disturbed) is

  18. bc=capture cross section ℓ²/bc=mean free path for capture Tp=ℓ²/bcvL=mean free time vL=typical particle velocity with respect to vortices Assume particle approaches vortex with velocity vL. Assume capture time from a distance r0 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 r0 is greater than ta: r0/vL>ta This yields the cross section and the mean free time: 1 msec < Tp <10 sec so trapping may occur or not

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

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