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Interpreting Superfluid Spin Up Through the Response of the Container

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Abstract

A recipe is presented for interpreting non-invasively the transport processes at work during relaxation of a cylindrical, superfluid-filled vessel, after it is accelerated impulsively and then allowed to respond to the viscous torque exerted by the contained fluid. The recipe exploits a recently published analytic solution for Ekman pumping in a two-component superfluid, which treats the back-reaction self-consistently in arbitrary geometry for the first time. The applicability of the recipe to He II, 3He, 3He–4He mixtures and Bose-Einstein condensates is assessed, and the effects of turbulence discussed.

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Notes

  1. Adams et al. [17] explained the spin up from rest in rough-walled containers below 1.3 K by modelling pinning as friction between the vortices and the walls.

  2. The tension parameter ν s looks like a kinematic viscosity, but it is non-dissipative and gives rise to Kelvin waves. It takes the form \(\nu_{s}=\left(\Gamma/4\pi\right) \ln(b_{0}/a_{0})\), where b 0=(Γ/2Ω s )1/2 is the inter-vortex spacing.

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Acknowledgements

CAVE acknowledges the financial support of an Australian Postgraduate Award and the Albert Shimmins write-up award. CAVE also thanks the anonymous referees for their constructive suggestions on the manuscript.

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    C. A. van Eysden & A. Melatos

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Appendix

Appendix

For a container of arbitrary axisymmetric shape, the fluid is contained within the volume −h(r)<z<h(r), where (r,z) refer to cylindrical coordinates, scaled by the length-scale of the container L. Equation (13) holds, with the modified definitions

(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)

where \(I_{f}^{*}\) is the scaled moment of inertia of the contained fluid (as if it were rotating as a rigid body) and R is the cylindrical radius of the vessel, scaled to L. In a cylinder, one has h(r)=1, L=h, R=r/h, and we recover (14)–(21).

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van Eysden, C.A., Melatos, A. Interpreting Superfluid Spin Up Through the Response of the Container. J Low Temp Phys 166, 151–170 (2012). https://doi.org/10.1007/s10909-011-0444-z

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