Superfluid turbulence in rotating containers: Phenomenological description
of the influence of the wall
M. S. Mongiovì
1
and D. Jou
2
1
Dipartimento di Metodi e Modelli Matematici, Università di Palermo,
c/o Facoltà di Ingegneria, Viale delle Scienze, 90128, Palermo, Italy
2
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain
Received 22 February 2005; revised manuscript received 21 June 2005; published 23 September 2005
In this paper a previous equation for the evolution of vortex line density L in counterflow superfluid
turbulence in rotating containers is generalized, in order to take into account the influence of the walls. This
model incorporates the effects of counterflow velocity V and of angular velocity of the container, and
introduces corrective terms depending on / d, being the intervortex spacing, of the order L
-1/2
, and d the
diameter of the channel. The stability of the solutions for L, for several regimes of averaged counterflow
velocity V and angular velocity , is analyzed. Our mathematical analysis reveals that qualitative consistency
allows us to reduce the four coefficients characterizing the dependence on / d to only one additional indepen-
dent coefficient, linked to the critical angular velocity
c
needed for the appearance of vortex lines in a rotating
superfluid.
DOI: 10.1103/PhysRevB.72.104515 PACS numbers: 67.40.Vs, 47.37.+q, 47.27.-i, 05.70.Ln
I. INTRODUCTION
In recent years there has been growing interest in super-
fluid turbulence,
1–3
because of its similarity with classical
turbulence.
4,5
Among the aspects receiving much attention,
there are the effects of the walls as, for instance, on boundary
layers. Besides its theoretical interest, the influence of walls
on superfluid turbulence may have a practical incidence in
refrigeration of small devices by means of flow of superfluid
helium along narrow tubes: when the tubes become nar-
rower, the relevance of wall effects will increase. In the
analysis we will perform, and following previous literature
on this problem, we emphasize the spatial average of the heat
flux, rather than its detailed local features.
Superfluid turbulence has been much studied in two
physical situations: counterflow experiments and rotating
containers. As is known, thermal counterflow induces a
quasi-isotropic disordered tangle, while rotation creates an
ordered polarized vortex array. In both cases the vortex
tangle is described by introducing a scalar quantity L, the
average vortex line length per unit volume, briefly called the
vortex line density. In pure rotation, the vortex lines are
aligned along the rotation axis, and L depends on the angular
velocity of the sample as
6
L = L
R
2
, 1.1
where is the quantum of vorticity = h / m, with h the
Planck constant, and m the mass of the helium atom:
9.97 10
-4
cm
2
/s. In counterflow experiments, there is a
disordered tangle of vortex lines, with a line density L in
fully developed turbulence given by
6
L = L
H
AV
2
, 1.2
where V = |V| is the modulus of the spatial average of the
counterflow velocity V = v
n
- v
s
, v
n
and v
s
being the veloci-
ties of normal and superfluid components, respectively. This
averaged quantity is related to the absolute value of the heat
flux by q = TSV, q, , T, and S being, respectively, the heat
flux, density, temperature, and entropy of liquid helium II.
Here, following most of the references on this topic, we con-
sider V as homogeneous; one may consider it as an average
value of V over the cross section of the tube. This is so,
because this average value is the easiest one to measure.
Note that in our analysis for slow rotation the wall effects are
not restricted to a narrow zone near the walls but they have
an effect over the whole cross section, because in this case
the average separation between vortices, L
-1/2
, is comparable
to the diameter of the channel. Then, it is compatible, in this
situation, to talk about wall effects and consider a homoge-
neous counterflow velocity across the channel.
The evolution of L in superfluid counterflow turbulence is
described by Vinen’s equation,
7
which, in its original form,
states that
dL
dt
=
1
VL
3/2
- L
2
, 1.3
with
1
and dimensionless constants. The steady-state so-
lution of Eq. 1.3 is L
H
=
1
/
2
V
2
, which has the form
mentioned in Eq. 1.2.
A microscopic derivation of Vinen’s equation has been
obtained by Schwarz
8
using the vortex filament model and
assuming homogeneous and isotropic turbulence. Recently
Lipniacki
9
has modified the Vinen-Schwarz equation, intro-
ducing in it the effects of the anisotropy. He characterizes the
anisotropy by means of a vector I, related to the vortex
tangle structure by I = s' s / |s|, where s , t describes
the vortex lines, with the length along the vortices; the
primes indicate differentiation with respect to . Angular
brackets stand for averages over the total vortex length of the
tangle.
PHYSICAL REVIEW B 72, 104515 2005
1098-0121/2005/7210/1045158/$23.00 ©2005 The American Physical Society 104515-1