Nonequilibrium thermodynamics of unsteady superfluid turbulence in counterflow
and rotating situations
D. Jou
1
and M. S. Mongiovì
2
1
Departament de Física, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia, Spain
2
Dipartimento di Metodi e Modelli Matematici, Università di Palermo c/o Facoltà di Ingegneria, Viale delle Scienze, 90128,
Palermo, Italy
Received 17 December 2004; revised manuscript received 6 July 2005; published 21 October 2005
The methods of nonequilibrium thermodynamics are used in this paper to relate an evolution equation for the
vortex line density L, describing superfluid turbulence in the simultaneous presence of counterflow and rota-
tion, to an evolution equation for the superfluid velocity v
s
, in order to be able to describe the full evolution of
v
s
and L, instead of only L. Two alternative possibilities are analyzed, related to two possible alternative
interpretations of a term coupling the effects of the counterflow and rotation on the vortex tangle, and which
imply some differences between situations where counterflow and rotation vectors are parallel or orthogonal to
each other. One arrives to a modified Gorter-Mellink equation with new terms dependent on the angular speed.
Finally, two proposals to describe the effects of anisotropy of the vortex tangle on the dynamical equations for
v
s
and L are examined.
DOI: 10.1103/PhysRevB.72.144517 PACS numbers: 67.40.Vs, 47.37.+q, 47.27.-i, 05.70.Ln
I. INTRODUCTION
The use of nonequilibrium thermodynamics for the analy-
sis of unsteady superfluid turbulence
1–5
has revealed useful
to explore its evolution equations and to suggest some ex-
periments to discriminate between different microscopic in-
terpretations leading to different macroscopic equations for
the evolution of counterflow superfluid turbulence. A full de-
scription of this well-known phenomenon would require an
evolution equation for the averaged vortex line density L
describing the vortex tangle and another equation for the
evolution of the averaged counterflow velocity V = v
n
- v
s
v
n
and v
s
being the averaged velocities of the normal and
superfluid components, which is related to the averaged
heat flux q as q =
s
TsV,
s
the density of the superfluid
component, T the temperature, s the entropy.
The more subtle characteristics of the process, that differ
from the averaged quantities L, V, v
n
, and v
s
, and which may
establish a link between the rotational of the local velocity
and the vorticity, do not participate in this macroscopic de-
scription. Thus, for instance, an average homogeneous heat
flux or V produces, beyond some critical value, a complex
mesh of vortex lines, whose local detailed description re-
quires a statistical analysis. The macroscopic descriptions of
this problem directly explore the relation between the mac-
roscopic averages of L and V. Though, up to now, most of
the experiments in this field are carried out under a constant
value of the counterflow velocity V, some specific situations
where the simultaneous variation in V and L may arise are,
for instance: a letting V change in a periodic way and
studying the effect of the frequency of this change on the
time variation of L b cutting down suddenly the heat supply
to the superfluid and studying the simultaneous decay of V
which will not be instantaneous and of L. In both situa-
tions, the vortices will not follow the instantaneous value of
V, but the rate of change of variation of V will have an
influence on the instantaneous value of L. An analysis of
such unsteady situations is certainly challenging for a more
complete understanding of the interactions between the
counterflow and the vortex formation and destruction.
Another point studied in the present paper is the interac-
tion between rotation, counterflow and vortex formation. The
most known experiment on simultaneous rotation and coun-
terflow is the apparatus of Swanson et al.
6
in which rotation
and heat flow are parallel to each other. It would be easy to
make them antiparallel, by simply rotating the container in
the opposite sense, and this would reveal features which are
not seen if only the parallel situation is studied. Furthermore,
it would be easy to have a situation where the heat flux and
the rotation vector are neither parallel nor antiparallel: for
instance, one could incorporate a thin heat conductor along
the rotation axis and keep it at a temperature higher than that
of the wall: in this way, one would have a controllable radial
heat flux in addition to the usual longitudinal heat flux. This
would make that the local heat flux were not locally parallel
to the rotation vector. Though here we are interested in av-
eraged values of the vortex line density, rather than in a
detailed local formulation, we could consider the average
over angular sections around the axis, which should exhibit
the consequences of this lack of parallelism between both
vectors. An other experiment in which heat flux is orthogonal
to the rotating axis, which have a very different geometry, is
that performed by Yarmchuk and Glaberson
7
which will be
discussed in Sec. VI.
In this paper, we will carry out an analysis of these situ-
ations, by combining nonequilibrium thermodynamics and a
previous equation
8
we proposed for the interaction between
counterflow and rotation when they are parallel to each
other. Even in this situation, we outline two different pos-
sible extensions of our former equation to the situation where
counterflow and rotation are not parallel to each other. We
explore the restrictions of Onsager-Casimir reciprocity rela-
tions in both cases, and such an analysis let us obtain two
alternative versions for the microscopic force between the
PHYSICAL REVIEW B 72, 144517 2005
1098-0121/2005/7214/14451711/$23.00 ©2005 The American Physical Society 144517-1