Karen L. Henderson

The anomalous motion of superfluid helium in a rotating cavity

We numerically solve the nonlinear two-fluid Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations for superfluid helium confined inside a short Couette annulus. The outer cylinder and the ends of the annulus are held fixed whilst the inner cylinder is rotated. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries. It also allows us to investigate further the boundary conditions associated with the HVBK model. The main result of our investigation is the anomalous motion of helium II when compared to a classical fluid. The superfluid Ekman cells always rotate in the opposite sense to a classical Navier-Stokes fluid due to the mutual friction between the two fluids, whilst the sense of rotation of the normal fluid Ekman cells depends on the parameter range considered. We also find that the tension of the vortex lines forces the superfluid to rotate about the inner cylinder almost like a rigid column.

Numerical Methods for Two-Fluid Hydrodynamics: Application to the Taylor Vortex Flow of Superfluid Helium II

We describe the numerical method which we have developed to solve for the first time the fully nonlinear HVBK equations. These equations generalise Landaus two fluid model to take into account the presence of quantised vortices. We apply the method to investigate the flow pattern of helium II between rotating concentric cylinders (Taylor vortex flow) at increasing Reynolds numbers. We compare the results against classical Taylor vortex flow.

The stability of a superfluid rotating jet

We consider the linear stability of a cylindrical rotating jet of pure superfluid held together by surface tension. A necessary and sufficient condition for stability to axisymmetric disturbances is derived which corresponds to that of a classical inviscid fluid. For axisymmetric disturbances we find that the vortex tension does not affect the range of unstable axial wave numbers, only the temporal growth rate and the most critical wave number. A sufficient condition for the stability of a general disturbance is derived which corresponds to that of a classical inviscid fluid. We find for non-axisymmetric disturbances, that the vortex tension increases the range of unstable wave numbers. The temporal growth rates of the unstable azimuthal modes increase with vortex tension.

Transition from Ekman flow to Taylor vortex flow in superfluid helium

By numerically computing the steady axisymmetric flow of helium II confined inside a finite-aspect-ratio Couette annulus, we determine the transition from Ekman flow to Taylor vortex flow as a function of temperature and aspect ratio. We find that the low-Reynolds-number flow is quite different to that of a classical fluid, particularly at lower temperatures. At high aspect ratio our results confirm the existing linear stability theory of the onset of Taylor vortices, which assumes infinitely long cylinders.

End effects in rotating helium II

Abstract We numerically solve the two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations to compute the steady flow of helium II confined inside a unit aspect ratio Couette annulus. The ends and outer wall of the cylinder are held fixed, while the inner cylinder is rotated. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries which are both parallel and perpendicular to the natural axial direction of the vortex lines. The main result of our investigation is the anomalous motion of helium II when compared to the motion of a classical fluid. We also find that the normal fluid and the superfluid components of helium II can move in ways which are very different to each other, due to the tension in the vortex lines.

OPEN ACCESS STATISTICS RESOURCES

In this paper we illustrate an open access statistics resource related to the statistical activities involved in choosing and carrying out an appropriate one sample test for location (mean or median) on a randomly generated data set. The development of the resource was funded through a sigma Resource Development Grant and is freely available from the UK national statstutor site. Five e-Assessment modules are available and these may be accessed independently or can be taken sequentially mimicking the flow of a full statistical analysis using the SPSS software package. On accessing the resource a new statistical data set may be generated or an existing data set used. Each module requires the data to have been downloaded to the SPSS statistical package, relevant analysis output obtained and a few questions answered to demonstrate understanding of the results. On submission, the e-Assessment system marks the responses immediately and provides full bespoke feedback for inappropriate test choices as well as other incorrect analysis. Videos and instruction pamphlets are accessible as links from each e-Assessment, which give clear instructions as to how to carry out the analyses and interpret results using SPSS. These additional resources, together with repeated use of the e-Assessment modules, facilitates learning how to identify and employ the correct test on a variety of data sets.

Torsional Oscillations of a Rotating Column of 3He-B

No HeadingWe have analysed the axisymmetric and non-axisymmetric modes of a continuum of vortices in a rotating superfluid. We have investigated how changing the temperature affects the growth rate of the disturbances. We find that, in the long axial wavelength limit the condition q = α/(1 − α′) = 1, where α and α′ are temperature-dependent mutual friction parameters, is the crossover between damped and propagating Kelvin waves. Thus at temperatures for which q > 1, perturbations on the vortices are unlikely to cause vortex reconnections and turbulence. These results are in agreement with the recent discovery of Finne et al1 of an intrinsic condition for the onset of quantum turbulence in 3He-B.

Using the HVBK Model to Investigate the Couette Flow of Helium II

We review the application of the two-fluid HVBK equations to helium II in Couette geometry, that is flow between concentric, rotating cylinders. This application is particularly interesting as a large number of experiments have been carried out in this geometry and also because Couette flow is an exact solution of the HVBK equations for both the normal fluid and superfluid.

Evolution of coupled Gaussian vortex systems in helium II

Abstract We have investigated the evolution of the normal and superfluid flow in helium II using the two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations. We found that closely matched Gaussian vorticity distributions form in both the superfluid and the normal fluid and decay together with a viscous time scale that is set by the total fluid density and not by the density of the normal fluid alone. This calculation suggests that the normal fluid and superfluid can have velocity fields which are strongly coupled together so that the two fluids behave as a single fluid.

Superfluid Helium Between Rotating Cylinders

We numerically solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations for superfluid helium II confined inside a small aspect ratio Couette annulus. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries. We find that the tension of the vortex lines forces the superfluid to move in a way which is very different from the motion of a classical fluid. We have considered rotation of the inner cylinder only and find that there is a higher density of vortex lines close to the inner cylinder than at the stationary outer cylinder.