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Simulations of Vortex Evolution and Phase Slip in Oscillatory Potential Flow of the Superfluid Component of 4He Through an Aperture

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Simulations of Vortex Evolution and Phase Slip in Oscillatory Potential Flow of the Superfluid Component of 4He Through an Aperture Simulations of Vortex Evolution and Phase Slip in Oscillatory Potential Flow of the Superfluid Component of 4He Through an Aperture J. A. Flaten,1 C. T. Borden,2 C. A. Lindensmith,3 and W. Zimmermann, Jr.4 1Department of Physics, Luther College, Decorah, IA 52101, USA 2Michigan Technological University, Houghton, MI 49931, USA 3Jet Propulsion Laboratory, Pasadena, CA 91109, USA 4Tate Laboratory of Physics, University of Minnesota, Minneapolis, MN 55455, USA Abstract. The evolution of semicircular quantum vortex loops in oscillating potential flow emerging from an aperture is simulated in some highly symmetrical cases. As the frequency of potential flow oscillation increases, vortex loops that are evolving so as eventually to cross all of the streamlines of potential flow are drawn back toward the aperture when the flow reverses. As a result, the escape size of the vortex loops, and hence the net energy transferred from potential flow to vortex flow in such 2π phase-slip events, decreases as the oscillation frequency increases. Above some aperture-dependent and flow-dependent threshold frequency, vortex loops are drawn back into the aperture. Simulations are preformed using both radial potential flow and oblate-spheroidal potential flow. PACS numbers: 67.40.Vs, 47.15.Ki, 47.32.Cc I. INTRODUCTION When superfluid 4He potential flow through an aperture reaches the critical velocity associated with that aperture, energy begins to be dissipated from the flow. This energy dissipation is believed to be due to the motion of quantum vortices in the vicinity of the aperture. When a vortex core evolves in such a way as to cross streamlines of the potential flow through the aperture, energy is transferred from the potential flow field to the flow field of the vortex itself. The vortex, whose core may be in the shape of a closed ring or a loop with ends at the walls in or near the aperture, may then carry this energy away from the vicinity of the aperture by means of a combination of self-induced velocity and background potential flow.1-4 The amount of energy transferred to a quantum vortex is proportional to the mass current of the potential flow crossed by the vortex core as it evolves. Every vortex that evolves so as to cut all of the streamlines of a given constant potential flow through an aperture will transfer the same amount of energy I from potential flow to vortex flow. Here is the quantum of circulation, κ s κ 4 equal to hm 4 , where h is Planck's constant and m 4 is the mass of the He atom, and I s is the superfluid mass current. Such vortex-crossing events are known as 2π phase slips because they 1 result in a 2π change in the difference of the phase of the superfluid order parameter between the two reservoirs connected by the aperture.4 Various sensitive experiments using fluid-dynamical resonators have shown series of abrupt, nearly identically-sized energy-loss events, consistent with individual, independent 2π phase slips involving vortices created by thermal or quantum 5-14 nucleation. Several candidate geometries have been proposed in which quantum vortices will spontaneously cross all of the streamlines of potential flow through an aperture as the vortices evolve. These proposals involve the evolution of fairly simple vortex configurations in steady, unidirectional potential flow through symmetrical apertures. In particular, computer simulations have shown that vortex rings or loops of appropriate shape nucleated in the high-velocity flow region near a circular aperture can indeed evolve in such a way as to cross all of the lines of 2,3,15-19 potential flow through the aperture. In experiments to study this phenomenon using fluid-filled resonators, the actual potential flow through the aperture has been oscillatory. (See, for example, Refs. 5-14, 20, and 21.) If the period of oscillation is long compared to the time that it takes for a vortex to cross all of the lines of potential flow, the flow may be approximated by steady potential flow. However, if this is not the case, the motion of a vortex and the energy that it carries away will be influenced by the time variation of the potential flow. During intervals of reversed flow, the flow will attempt to pull the vortex back toward the aperture, and at frequencies above some aperture-dependent and flow-dependent threshold value, a vortex may in fact be drawn all the way back into the aperture. Three different experiments have been performed in Minneapolis in which the steady flow approximation may not have been valid, at least under some of the conditions studied. All of these experiments showed effects that may be related to the influence of oscillatory flow on the motion of vortex loops. In two experiments, certain changes in the dissipation involving large energy-loss events arose with increasing frequency of oscillation.14,22-24 In the third experiment, 13 energy losses less than the expected steady-flow values were recorded. In an attempt to begin to understand the vortex dynamics responsible for these effects, we have used computer simulations to explore vortex loop evolution in oscillatory potential flow through an aperture in some very simple situations. This article describes the results of these simulations. In this work, the initial formation of a vortex by thermal nucleation or quantum tunneling is not considered, nor is the early fluid-dynamical growth of the vortex. Rather, the simulations deal with the main period of vortex growth, during which almost all of the energy transfer from potential flow to vortex flow takes place and during which frequency-dependent effects occur. 2 II. SIMULATIONS Two computer simulation programs were used to study vortex loop evolution in steady and oscillatory potential flow through an aperture. Both simulations extend previous studies of vortex evolution in steady potential flow alone.3,15-19 The first program, discussed in Sections II A and II B, simulates the motion of a semicircular vortex loop above a flat surface in radial potential flow emanating from a very small aperture in that surface. The second program, discussed in the Sections II C and II D, simulates the motion of a vortex loop in an oblate- spheroidal potential flow field, the necked boundary of which constitutes the aperture. The following simplifications and idealizations are made in both simulations. A small semicircular vortex loop is started near or within the aperture and allowed to evolve according to the rules of classical fluid dynamics for inviscid, incompressible fluids. The vortex is released when the potential flow velocity through the aperture is at its maximum, the time at which vortex nucleation or depinning is most likely to occur. The potential flow velocity is assumed to vary sinusoidally in time, and vortex evolution is studied as a function of the frequency of oscillation. The amplitude of the potential flow is not adjusted to compensate for the energy transferred to the evolving vortex, as if the amount of energy dissipated in a single 2π phase slip is small compared to the maximum kinetic energy of the potential flow, a condition that was well satisfied in the Minneapolis experiments. In each simulation run, only one vortex loop is present. The ends of the vortex loop are assumed to move freely over the surface of the aperture so that the vortex core meets the surface perpendicularly. No pinning is assumed to occur. Furthermore, no frictional effects are included. Thus the results correspond to vortex motion at temperatures below about 1 K, where there is essentially no normal fluid. In all cases studied, nonadjacent portions of the vortex core remain well separated, and except for the sections at the ends, the vortex core remains well away from the walls. Thus vortex reconnections need not be considered. A. Simulation with Radial Potential Flow In the first simulation, oscillatory potential flow is assumed to radiate into the upper half-space from an origin in a horizontal flat plate, as if the flow were passing through a tiny aperture centered at the origin. Maximum upward flow is assumed to occur at time t = 0. Hence the potential flow velocity v p r,t at position r relative to the origin and time t is given by ( ) Q r v r,t = 0 cos 2πft , (1) p() 3 ( ) 2π r 3 where Q 0 denotes the amplitude of the volume rate of flow away from the origin and f is the frequency of the flow. At time t = 0 a semicircular vortex loop is assumed to be situated above the plate, with its axis lying in the plane of the plate and passing through the aperture, as shown in Fig. 1(a). The vortex loop is oriented so that its self-induced velocity will carry it toward the aperture. When combined with its half-ring image in the plate, such a vortex loop has the same self- induced velocity as that of a complete circular vortex ring of the same radius in an unbounded fluid, which may be written as κ 8R vv ()R = ln , (2) 4πR e1 / 2a 0 where R is the radius of the vortex ring and a is the vortex core radius parameter, equal to 0 -10 1.3 × 10 m.25,26 The net motion of the vortex core is dictated by the vector sum v + v at p v each point along its length. Under these simple conditions, the vortex loop remains a perfect semicircle perpendicular to the plate for its entire evolution, and its center moves along a fixed axis, which we call the x -axis, that coincides with the axis of the initial vortex, as shown in Fig.
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