## Abstract

Drain vortices are among the most common vortices observed in everyday life, yet their physics is complex due to the competition of vorticity’s transport and diffusion, and the presence of viscous layers and a free surface. Recently, it has become possible to study experimentally drain vortices in liquid helium II, a quantum fluid whose physics is characterised by the absence of viscosity and the quantisation of the circulation in the superfluid component. Using the Gross–Pitaevskii equation, we make a simple model of the problem which captures the essential physics ingredients, showing that the drain vortex of a pure superfluid consists of a bundle of vortex lines which, in the presence of a radial drain, twist, thus strengthening the axial flow into the drain.

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## 1 Introduction

Quantised vorticity is a distinguishing property of superfluids, indeed hundreds of papers have been written on this subject since Vinen’s detection of single quanta of circulation in superfluid helium [1]. It is therefore remarkable that, until recently, very little attention has been dedicated in the superfluid context to *drain vortices* (also called *suction vortices* or *bathtub vortices*). Combining azimuthal motion around an axis with radial/axial inflow into a hole, drain vortices are familiar to everybody because they can be easily created in a kitchen or bathroom sink filled with water. Familiarity is not the same as physical understanding, however: boundaries and a free surface create subtle Ekman layers, drainpipe axial flows and upwellings which, even for water drain vortices, have been studied only recently [2, 3].

We consider liquid helium (\(^4\)He) at temperature \(T < T_{\uplambda }\) where \(T_{\uplambda }=2.17~\mathrm K\) is the critical temperature at saturated vapour pressure. In this low temperature regime, liquid helium consists of two inter-penetrating fluid components, an inviscid superfluid and a viscous normal fluid, whose proportions strongly depend on *T*. The normal fluid fraction tends to one for \(T \rightarrow T_{\uplambda }\) and to zero for \(T \rightarrow 0\); vice versa, the superfluid fraction tends to zero for \(T \rightarrow T_{\uplambda }\) and to one for \(T \rightarrow 0\). (In practice, at \(T=1~\mathrm K\) the superfluid fraction is already more than \(99 \%\).)

As mentioned, the key property of the superfluid component is that any vorticity is concentrated in thin vortex lines of fixed circulation \(\kappa =h/m\) where *h* is Planck’s constant and *m* is the mass of one helium atom. Therefore we expect that, in the configuration of a drain vortex, the flow pattern of the normal fluid should be (in the first approximation) similar to that of water, including viscous layers and a continuous vorticity field which fills the system, whereas the vorticity of the superfluid should be confined to a central cluster of vortex lines. However, since the vortex lines scatter the thermal excitations (phonons and rotons) which make up the normal fluid, there should also be a mutual friction force between normal fluid and superfluid components, whose precise effects are difficult to guess. (The friction depends on the density of the vortex lines and their velocity difference with respect to the normal fluid.) We also expect that the flow pattern should depend on whether the drain vortex is created mechanically (e.g., using a propeller) or thermally (e.g., using a heater), as the induced superfluid and normal fluid velocities will be parallel or antiparallel, respectively. Finally, unless the container is very large, details of any fluid reinjection into the system will be important. In summary, the superfluid drain vortex problem contains many physical ingredients which may combine in a nontrivial way, even before considering the presence of a free surface, which may deepen creating a funnelling drainpipe.

Experimentally, the problem has been tackled recently by Yano and collaborators [4,5,6]. Using a rotor, they created a drain vortex in helium at \(T=1.6~\mathrm K\) (corresponding to a superfluid fraction of \(83\%\)). By measuring the attenuation of second sound, they showed that the drain vortex consists of a cluster of approximately \(10^4\) quantised vortex lines which accumulate in a narrow central region near the axis of symmetry above the drain hole. The experiment was followed up by numerical simulations [7] based on the vortex filament model (VFM). These simulations determined the evolution of seeding vortex lines at nonzero temperatures in the presence of a prescribed normal fluid in the shape of a Rankine vortex with a superimposed constant axial flow into the drain hole.

In this paper we do not try to model Yano’s experiments precisely: rather, we try to isolate the essential physics ingredients, leaving aside finite-temperature effects. We consider the superfluid drain vortex problem in its simplest form: a vortex cluster in the presence of a drain flow in a pure superfluid at \(T=0\), in the absence of a free surface. We model the problem using the Gross–Pitaevskii Equation (GPE) for a weakly-interacting gas of bosons [9, 10]. The GPE is idealized for helium (which is a liquid, not a dilute gas), and, since its numerical solution requires the resolution of length scales smaller than the vortex core, our calculation is necessarily limited to small-scale vortex configurations. Still, our model captures the essential physical ingredients, namely the dynamics of vortex lines in a nontrivial three-dimensional geometry which includes a draining superflow and the presence of boundaries, ingredients which were not accounted in the VFM approach [7].

This paper is articulated as follows: Sect. 2 describes our model and the numerical methods employed and Sect. 3 presents our results.

## 2 Model

### 2.1 Governing Equations

It is convenient to write the GPE in dimensionless form. We use the healing length \(\xi =\hbar /\sqrt{g m n_0}\) and the speed of sound \(c=\sqrt{n_0 g/m}\) as units of length and speed, respectively, \(\tau =\xi /c\) as unit of time, the density of a uniform condensate, \(n_0\), as the unit of density, and express the trapping potential in units of the chemical potential \(g n_0\), *g* being the interaction parameter, *m* the mass of one atom, and \(\hbar =h/(2 \pi )\) the reduced Planck’s constant. The resulting dimensionless GPE is

where \(\psi (\mathbf{{r}},t)\), \(n(\mathbf{{r}},t)=\vert \psi (\mathbf{{r}},t)\vert ^2\), \(V(\mathbf{{r}})\), \(\mathbf{{r}}\) and *t* are the dimensionless wavefunction, density, trapping potential, position and time, respectively. (Hereafter all quantities are meant to be dimensionless.) During the evolution, Eq. (1) conserves the total number of atoms, *N*, in the volume \({{\mathcal {V}}}\) of the system, and the energy, *E*, given, respectively, by

We solve the three-dimensional GPE numerically in the computational domain \(x_\textrm{min} \le x \le x_\textrm{max}\), \(y_\textrm{min} \le y \le y_\textrm{max}\), \(z_\textrm{min} \le z \le z_\textrm{max}\) using the \(4^{\textrm{th}}\) order Runge–Kutta method in time and centred differences in space; the spatial discretization consists of \(N_x\), \(N_y\) and \(N_z\) discretization points in the *x*, *y* and *z* directions, respectively. Convergence is checked by monitoring the conservation laws; for typical time evolutions described in the next section, *N* and *E* are conserved with relative errors \((N-N_0)/N_0 \approx 10^{-5} \, \%\) and \((E-E_0)/E_0 \approx 0.05 \, \%\) where \(N_0\) and \(E_0\) are the initial values.

### 2.2 Geometry of the System

For a system whose geometry is described in Fig. 1, we set \(V(\mathbf{{r}})=V_\textrm{trap}(\mathbf{{r}})\) in Eq. (1), where the trapping potential \(V_\textrm{trap}(\mathbf{{r}})\) is equal to zero inside the following three regions: (i) a cylinder of radius \(r_{ad}\) extending from \(z_1\) to \(z_2\) which represents the experimental cell; (ii) a drain hole in the shape of a truncated cone with top radius \(r'_\textrm{sink}\) and bottom radius \(r_\textrm{sink}\) extending from \(z_\textrm{min}\) to \(z_1\); (iii) an injection annulus of inner radius \(r_\textrm{ad}\) and outer radius \(r_{in}\) extending from \(z_3\) to \(z_4\). Outside these three regions, we set \(V_\textrm{trap}(\mathbf{{r}})=10\) (in units of the chemical potential). This value corresponds to a high potential barrier imposing \(\psi =0\) outside the regions (i) - (iii). In the experiment of Yano and collaborators [4] the injection annulus is at the bottom of the experimental cell, so here we present the results for \(z_3=z_1\) (i.e., the injection annulus is at the bottom of the cylinder).

### 2.3 Drain and Injection

To model the drain hole, we add a negative imaginary part to the potential \(V(\mathbf{{r}})\) in region (ii) (the truncated cone below the cylinder), setting

where \(V_{sink}(\mathbf{{r}})\) is equal to a positive constant \(V_0\) in the drain hole and zero elsewhere. Between *t* and \(t+\Delta t\), this negative imaginary potential removes \(\Delta N(t)\) atoms from the drain hole, depleting the density in that region. To see this, we derive [9] from Eq. (1) the modified Euler equation

and the modified continuity equation

where

is the pressure, \({\textbf{v}}={\textbf{j}}/n\) is the velocity, and the current \({\textbf{j}}\) is defined as

The last term of Eq. (5) is called the quantum pressure term. By integrating Eq. (8) over the whole volume, we obtain that the rate of loss of particles into the system due to drain potential is given by \(\displaystyle - \frac{\textrm{d}N}{\textrm{d}t} = 2 V_0 N_\textrm{sink}(t)\), where \(N_\textrm{sink}(t)\) is the number of atoms in the drain hole. The number of atoms \(\Delta N(t)\) removed from the drain hole in time \(\Delta t\) is hence, in the first approximation, given by \(\Delta N(t) \approx 2 V_0 N_\textrm{sink}(t) \Delta t\). As the quantum fluid described by Eqs. (5) and (6) is barotropic, see Eq. (7), the density difference arising from the atoms removal in the drain creates a pressure difference which drives a flow towards the drain (see Sect. 3). In order to conserve the total number of atoms of the system, we add into the injection annulus the same number of atoms \(\Delta N(t)\) that we have removed from the drain hole. After time-stepping the wavefunction from \(\psi (\mathbf{{r}},t)\) to \(\psi (\mathbf{{r}}, t+\Delta t)\), the naive approach would be to change the density in the injection annulus from \(n(\mathbf{{r}}, t+\Delta t)\) to \(n'(\mathbf{{r}},t+\Delta t) = n(\mathbf{{r}}, t+\Delta t)+\Delta N/{{{\mathcal {V}}}}_{in}\), where \({{{\mathcal {V}}}}_{in}\) is the volume of the injection annulus. However, we have found that the resulting unphysical small radial discontinuity of the density at the edge of the injection annulus (\(r=r_{ad}\)) tends to destabilize the solution even at small values of the drain parameter \(V_0\). A more stable injection is obtained if the injected density profile is continuous. Therefore we set \(n'(\mathbf{{r}},t+\Delta t) = n(\mathbf{{r}}, t+\Delta t) + f(r) \Delta N(t)\), where the distribution function *f*(*r*) is not a step-function but vanishes at \(r=r_{ad}\) and \(r=r_{in}\), and its volume integral is normalized to one. For simplicity we choose

where

Other choices of distribution function which guarantee a continuous density profile at the edge of the annulus give a similar drain flow.

Our particle re-injection protocol can physically be interpreted as the inclusion of a source term \(\sigma _\textrm{in}\) in the continuity equation Eq. (6), i.e.,

where the production term, \(\sigma _\textrm{in}(\mathbf{{r}},t)\), is nonzero only inside the injection annulus. Since \(V_\textrm{sink}(\mathbf{{r}})\) is equal to a constant, \(V_0\), inside the drain hole and vanishes outside it, in order that the number of atoms in the system remains the same, \(\sigma _\textrm{in}(\mathbf{{r}},t)\) must satisfy

## 3 Results

### 3.1 Ground State

To find the ground state of the system, we start from the Thomas-Fermi approximation, imposing that \(\psi _{TF}(\mathbf{{r}})=\sqrt{1 -V_\textrm{trap}(\mathbf{{r}})}\) if \(V_\textrm{trap}(\mathbf{{r}})<1\) and zero otherwise. Without any drain flow or injection, we integrate Eq. (1) in imaginary time, replacing *t* with \(-it\) and we enforce at every time step the condition that the total number of atoms in the system does not change. After few time steps, boundary layers of thickness of the order of the healing length (which is unity in our dimensionless formulation) develop near the boundaries, where the density drops from its bulk value in the interior to zero, and the wavefunction settles down to the desired time-independent state which minimizes the energy.

### 3.2 Steady Drain Flow

We solve the GPE using the ground state as initial condition, imposing drain flow and injection as described in Sect. 2.3. After an initial transient, we obtain a steady drain flow into the drain hole which is exactly compensated by the injection of atoms in the annular region, so that the number of atoms in the system remains constantly equal to the initial atom number.

It is interesting to relate \(V_0\) (the amplitude of the imaginary potential in Eq. (4) generating the drain flow) to the quantity which is controlled in the experiment: the flow rate from the injection annulus into the drain hole, \(Q_{inj}\). To do this, we integrate Eq. (11) over the volume \({{{\mathcal {V}}}}_s\) of the drain hole, and define

the flow rate out of \({{{\mathcal {V}}}}_s\), where \({{\Sigma }_s}\) is the surface which encloses \({{{\mathcal {V}}}}_s\). Because of the box-trap boundary conditions, the only flow in and out of \({{{\mathcal {V}}}}_s\) is across the top surface \({\overline{\Sigma }}_s\) of the truncated cone, i.e., \(\displaystyle Q_\textrm{sink}=\int _{{\overline{\Sigma }}_s} n \mathbf{{v}}\cdot d\mathbf{{S}}\). We find

In the steady state \(Q_\textrm{inj}=-Q_\textrm{sink}\), hence

The steady number of atoms in the drain hole \(N^0_{sink}\) is not constant but depends on the potential \(V_0\), implying that the relation between \(Q_\textrm{inj}\) and \(V_0\), Eq. (15), is not linear. The dependence of \(N^0_\textrm{sink}\) on \(V_0\) can be, at least qualitatively, understood employing the Bernoulli equation which can be derived [9] from Eq. (1) neglecting the quantum pressure effects:

where \(v_s\), \(p_s\) and \(v_c\), \(p_c\) are velocities and pressures, respectively, in the centre of the drain hole just below the cylinder (where the velocity is \(\mathbf{{v}}=-v_s {\hat{\varvec{{z}}}}\), \({\hat{\varvec{{z}}}}\) being the unit vector in the *z*-direction and \(v_s > 0\)) and in the cylinder far away from the drain hole (where \(|\mathbf{{v}}| = v_c \approx 0\)). Employing the Bernoulli equation Eq. (16), we obtain

where the density in the drain hole is approximately \(n_s=N^0_\textrm{sink}/{{{\mathcal {V}}}}_s\). In the first approximation, the flux across the top surface \({\overline{\Sigma }}_s\) of the drain hole is \(Q_\textrm{inj}=n_s v_s A_s\), where \(A_s\) is the area of \({\overline{\Sigma }}_s\). Using this last relation, the continuity equation Eq. (15) and the Bernoulli relation Eq. (17), we obtain the following expression for \(N^0_{sink}\):

leading to

This non-linear relation between \(Q_\textrm{inj}\) and \(V_0\) can be indeed observed in Fig. 2a (solid black line), where we also plot Eq. (19) (solid yellow line) and its linear approximation (red dashed line) at small \(V_0\): the agreement at small values of \(V_0\) is very good. It is clear that our assumptions in deriving Eq. (19) are valid when \(V_0\) is not too large compared to the chemical potential.

This nonlinear effect is also visible in Fig. 2b where we plot the dependence on the potential \(V_0\) of the averaged radial and axial components of the current, \(\overline{j_r}\) and \(\overline{j_z}\), respectively, flowing into the drain hole, averaged over a horizontal disk of radius \(r'_{sink}\) at \(z=(z_3+z_4)/2\).

Figure 3 shows the steady drain flow pattern in a larger geometry plotted on the *xz* plane (a) and on the *xy* plane (b). It is apparent that the radial and axial flows into the drain hole are confined to the bottom of the cylinder, the radial component being stronger than the axial one. This flow pattern is very different from the flow pattern of a viscous drain hole described in Ref. [2], which suffers viscous friction near all boundaries; our drain flow is instead irrotational and without viscosity. The plot of the pressure distribution (see Fig. 4) confirms that the flow is driven by pressure gradients, as expected.

### 3.3 Drain Vortex Flow

In the next numerical experiment, we compute the time evolution of a lattice of vortex lines in the presence of a drain flow. We add \(N_v=10\) parallel vortex lines aligned along the *z*-direction which form a small lattice (bundle) of initial radius \(r_b=25\). This vortex imprinting is done in a standard way by multiplying the ground state wavefunction with the approximate wavefunction of a vortex in a homogeneous condensate and let the system then relax in imaginary time.

By integrating in time the GPE, we follow the evolution of the vortex bundle, which is shown in Fig. 5. We find that, since vortex lines are advected by the superflow (Helmholtz’s theorem) and since near the bottom of the cylinder the inward radial flow is much stronger than near the top, the lower part of the bundle is sucked radially inwards, towards the centre of the drain hole, while the top part is basically unaffected (this effect is also apparent in Fig. 6a which displays the bundle’s radius at two different heights). The quantisation of the superfluid’s angular momentum [9, 11] implies that the angular velocity \(\Omega (z,t)\) of the bundle at height *z* and time *t* is given by the relation \(\Omega (z,t)=N_v\kappa /[2\pi r_b^2(z,t)]\). This implies that the bottom parts of the vortex lines which are near the drain hole rotate at larger angular velocity \(\Omega\) (as \(r_b\) is smaller), moving hence ”ahead” with respect to the top parts of the vortex lines close to the top of the cylinder: the vortex bundle becomes more and more twisted, as shown in Fig. 5b. As this twist develops, the projection of the total vortex length on a plane orthogonal to the cylinder axis, \(\Lambda _\perp\), increases, leading to an increase of the total length, \(\Lambda\), as shown in Fig. 6b. It takes a finite time for this initial twist to propagate upwards (see Fig. (5)) carried by Kelvin waves on vortices. Once the Kelvin waves reach the top of the cylinder, the bundle’s radius, \(r_b\), starts decreasing near the top as well, decreasing \(\Lambda _\perp\). However, given the enduring asymmetry of the radial flow (stronger close to the drain hole, negligible near the top of the cylinder), Kelvin waves persist on the vortices, leading \(\Lambda _\perp\) to settle to a finite, non-zero value. The dynamics of this rotating, twisted bundle is characterised by vortex reconnections which occasionally scramble the vortex lines, leading to a moderately disordered vortex configuration (\(\Lambda _{\perp }/\Lambda \approx 0.23\) at the end of the simulation) which is still strongly polarised in the *z* direction.

An important consequence of the twisted nature of vortex lines is that it induces a downwards axial superflow into the drain hole, stronger than the axial superflow caused by the drain hole without vortex lines. Figure 7a and b shows the current on the *xz* and *xy* planes, respectively. The presence of the strong axial flow towards the drain hole which is induced by the vortices is evident, especially when compared to the axial flow when vortices are absent: compare Figs. 3a and 7a. The azimuthal flow induced by vortices can be observed in Fig. (7)b. As the twist is transported along the *z* direction by Kelvin waves, it increases the downward axial flow in the upper part of the bundle. Probably this is the mechanism which is responsible for the formation of a central drainpipe funnel when the helium has a top free surface, as observed by Yano and collaborators in their experiments [4]. Further numerical simulations with \(N_v=25\) vortices confirm the scenario which we have described.

## 4 Discussion and Conclusion

We have used the Gross–Pitaevskii equation to model the superfluid drain vortex in its simplest form: at \(T=0\) (i.e., without any normal fluid and associated viscous effects) and without complications arising from a free surface. We have found that the superfluid drain vortex consists of a moderately disordered twisted bundle of quantised vortex lines (the twist being quantified by the ratio \(\Lambda _{\perp }/\Lambda\)). The twist is generated by the radial inflow into the drain, which is stronger near the drain which brings the vortex lines closer to each other. We have also found that the twist of the vortex lines induces a strong central axial flow into the drain; in the presence of a free surface at the top, this axial flow is probably the origin of the dimple, or funnel or drainpipe, which has been observed [4].

Clearly the normal fluid must play an important role in the experiments, and we can speculate that our results shed partial light into it. At small velocities (compared to the speed of sound) and at length scales larger than the healing length, the GPE effectively describes the dynamics of a classical fluid, in particular of the normal fluid, provided we leave aside viscous and friction effects. We can argue therefore that if we start with a normal fluid Rankine vortex at the centre of a cylindrical container and then switch on a drain flow at the bottom of the cylinder, the evolution of the Rankine vortex will be similar the evolution of vortex lines showed in Fig. 5. At the bottom, the radius of the Rankine core will decrease, as fluid moves radially into the drain, leading to faster angular velocity compared to the top which will twist the Rankine core, hence induce an axial current inside it. Of course viscous and friction effects will damp the twist depending on temperature, and to estimate this damping an approach like that used in Ref. [7] is necessary.

The twisted vortex bundle which we report is similar to other twisted vortex states which have been observed in superconductors [12], and, more relevantly, in rotating \(^3\)He [13] and large-scale \(^4\)He vortex rings [14,15,16].

Finally, by combining order (the strong axial polarization) with a controlled amount of disorder (induced by the radial drain flow), our results show that this bathtub vortex flow is a promising configuration to develop theoretical tools based on the HVBK equations and Vinen equation [17, 18] to model problems of rotating quantum turbulence [19], ranging from liquid helium counterflow [20, 21] to neutron star glitches [22, 23] to atomic gases [24].

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## Acknowledgments

We are grateful to Sam Patrick and Silke Weinfurtner for discussions and acknowledge the financial support of UKRI grant *Quantum simulators for fundamental physics* (ST/T006900/1). LG acknowledges the support of Istituto Nazionale di Alta Matematica (INdAM).

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Ruffenach, W., Galantucci, L. & Barenghi, C.F. Superfluid Drain Vortex.
*J Low Temp Phys* **212**, 375–388 (2023). https://doi.org/10.1007/s10909-023-02943-9

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DOI: https://doi.org/10.1007/s10909-023-02943-9