Departament de Física, Universitat de les Illes Balears,
07071 Palma de Mallorca, Spain
Department of Physics,
Ben-Gurion University,
Beer-Sheva 84105,
Israel
Institut Mediterrani d'Estudis Avançats (CSIC--UIB),
07071 Palma de Mallorca, Spain
Julyan H. E. Cartwright
[],
Mario Feingold
[], and
Oreste Piro
[]
Physical Review Letters, 75, 3669--72, 1995
PACS numbers: 47.52.+j, 05.45.+b
Dynamical systems that arise in problems of particle diffusion [] in incompressible and nonturbulent fluid flows, apart from being of theoretical interest, hold much relevance for technological applications. Properties of emulsions, dispersion of contaminants in the atmosphere and ocean, sedimentation, and mixing, are just a few examples. In comparison with the present state of knowledge of chaotic advection, mixing and transport in two-dimensional [] or three-dimensional time-independent [] flows, little is known for three-dimensional time-dependent flows.
The strobed dynamics of fluid parcels --- so called passive scalars --- in three-dimensional time-dependent incompressible fluid flow is qualitatively equivalent to the iteration of a three-dimensional volume-preserving map. A few years ago we began to investigate nearly-integrable classes of such maps --- which we have termed Liouvillian maps --- in search of possible generic features []. We found that we could characterize Liouvillian maps in a similar manner to Hamiltonian dynamical systems, by using the number of slow action and fast angle variables that, in the integrable limit, remain invariant and rotate uniformly, respectively. The cases having one or two of such actions were found to be particularly interesting. For the one-action case, which has been shown to be generic for a class of flows [], a KAM-type theorem exists [], and with it the associated barriers to global transport []. On the other hand, the case of two actions, which is also generic for a class of flows, displays in Liouvillian maps a new phenomenon of resonance-induced diffusion leading to global transport throughout phase space []. However, despite this theoretical progress, until now no realistic flow --- in the sense of being at least an approximate solution of the Navier--Stokes equations for a realizable experiment --- has been observed to show the above properties. The purpose of this Letter is to fill the gap for the two-action case by introducing the first engineerable flow demonstrating the presence of resonance-induced diffusion in real fluid flows.
We begin by briefly illustrating the phenomenon for a map, introducing at the same time the adiabatic invariants and resonant surfaces we encounter in two-action flows. The map
represents a small perturbation of an integrable Liouvillian map having two
action and one angle variables []: for vanishing 
,
the two action variables 
and 
remain invariant while the angle
rotates with a constant angular frequency that depends only on the
actions. When 
is nonzero but small the action variables drift
slowly compared to the angle. This separation of scales allows us to average
over 
leading to an adiabatic description of the motion. Before the
actions change appreciably, the angle is able to traverse a large sample of
its domain. Hence the evolution of the actions is sensitive only to the
average value of the angle: the action equations can be averaged over the
angle, and so become decoupled from the angle equation to yield an
area-preserving map
where the bar represents the 
average.
Being a small perturbation of the two-dimensional identity, the map of the
action plane can be well approximated by a two-dimensional autonomous
Hamiltonian flow
One is then led to the conclusion that the dynamics of the map in
Eq. (1) occurs on invariant surfaces that are the product
of the almost uniform angular motion of 
, and the level curves of
the Hamiltonian
However, this analysis breaks down when the angular variable does not
sample its domain of variation uniformly. This failure is bound to occur
when the rotation is resonant; when 
,
where k and n are integers. The family of curves in the action plane
defined by Eq. (4) is generically transversal to the curves on
which these resonances occur. Although the resonances are dense in the
action plane, they have a hierarchical structure in which the relative
strength of different orders 
of resonances is governed by the Fourier
expansion of Eq. (1), such that in general the lowest-order
resonances have the smallest denominators n. Hence one might expect the
adiabatic approximation to provide a good description of the action motion,
except at the intersections with the lowest-order resonances. This is
depicted in Fig. 1(a) for a particular two-action Liouvillian
map. Close to resonances the trajectory oscillates wildly between
different invariant curves 
. In Fig. 1(b) we have
plotted the time evolution of the adiabatic invariant H for this map to
show that it remains nearly constant almost all the time, but jumps
chaotically from one invariant curve to another at each intersection.
Figure 1:
(a) Projection of a trajectory in a two-action Liouvillian map
onto the plane z=0. The map is 
,
, 
.
The diagonal dashed line shows the location of the lowest-order 
resonance.
(b) The time evolution of the corresponding adiabatic invariant
.
The study of two-action properties in a real fluid flow is made difficult
by the fact that very few three-dimensional time-dependent fluid flows are
analytically tractable. In the following we introduce a fluid flow that
can be solved analytically as a perturbation expansion of the Navier--Stokes
equations and that exhibits the phenomenon just illustrated. The flow we
have chosen to consider is incompressible flow at low Reynolds numbers
between two concentric spheres which rotate with different angular velocities
about a common axis that switches in turn between two different directions
separated by an angle 
. In the absence of time dependence introduced
by the axis switching, the flow consists of a primary spherical Couette flow
about the rotation axis, superposed with an orthogonal secondary flow in the
meridian plane 
[]. Owing to axial symmetry, none of
the flow components depend on the azimuthal coordinate 
. The secondary
flow, which is made up of one or two Taylor vortices in each hemisphere,
depending on the flow parameters [], is then two dimensional
and described by a stream function 
that can be computed perturbatively
in the Reynolds number 
. The velocity field up to first order
in the Reynolds number reads
where 
is given by
and 
, 
, 
, 
, 
, 
, are constants dependent on the
ratios of the radii and the angular velocities of the spheres
[]. Notice that for 
the secondary
flow 
is very much slower than the primary flow 
.
The perturbation of this flow by periodic axis switching introduces
time dependence at the same time as true three dimensionality by coupling
together the velocity components of Eq. (5).
Since 
, we can assume that fluid inertia is
not important, such that trajectories of passive scalars will be a piecewise
juxtaposition of the steady flow about each axis. Notice that the r and
coordinates are always coupled by the secondary flow except when the
Reynolds number is precisely zero. Furthermore, adding a second rotation axis
leaves the equation describing the evolution of r unchanged, but introduces
a coupling between 
and 
which is of 
for small axis
separation 
. All this implies that, although we do not have direct
access to the three-dimensional integrable case, we can however intuit that
it should exist close by the case we are considering for sufficiently small
perturbations. We can construct a reference frame 
tilted midway between the two axes [], with the rationale
that each semiperiod has its own adiabatic invariant, Eq. (6), tilted
with respect to the other, and we expect that the composed trajectory will on
average lie on the adiabatic invariant midway between the two. We can show that
this will be the case in the limit when the axis separation and the Reynolds
number are sufficiently small. The change in r and 
over a
period is bounded by an quantity of 
, while 
changes by
an arbitrary amount over the same period. Thus for small axis separation
and low Reynolds numbers, r and 
are action variables,
and 
is an angle. Having this set of approximate action--angle
variables, we are able to calculate the invariant surfaces and the resonant
surfaces associated with a nonintegrable perturbation of a two-action flow.
Figure 2:
Level curves of the adiabatic invariant surfaces (thin lines) are shown
inside the two spheres (thick lines) together with the lowest-order resonant
surfaces intersecting them (thin dashed lines) on the plane
for a particular set of parameter values for small
axis separation. The resonances shown are, from the inner sphere to the outer,
, 
, 
, 
, 
, and 
.
Although we are not able to obtain an exact expression for the adiabatic
invariant of the stroboscopic map for all 
, we can obtain
perturbatively in 
an expression
valid for small axis separations, where 
, and 
represent
the constants 
and 
evaluated for the two semiperiods j=1, 2.
We can similarly obtain the resonant surfaces: in the tilted coordinate
system 
, 
correspondingly strobed is
the fast angular variable. The resonant rotation of 
, as in
Liouvillian maps, occurs when 
, where k and n
are integers. For small axis separations, this is true when
where T is the axis-switching period.
Notice that these resonant surfaces are independent of 
, i.e., are
spherical shells of varying radii 
, up to first order in the axis
separation. In Fig. 2 we show the locations in the meridian plane
of both the adiabatic invariant and the resonances for a particular value of
the parameters.
Figure 3:
(a) The projection of a strobed trajectory onto the plane
. The parameter values were chosen to be the same
as in Fig. 2, such that the lowest-order 
resonance
(dashed) is in the middle of the region between the spheres, while leaving
other primary resonances distant. The axis separation angle is
and the Reynolds number 
.
(b) Time evolution of the adiabatic invariant H for this case.
Numerically computed strobed trajectories show that the theoretical picture we painted above is accurate and that the phenomenon of resonance-induced diffusion, previously reported only for Liouvillian maps, is present here with striking resemblance in a real fluid flow. To illustrate this, we choose the flow parameters such that the relative geometry of the adiabatic invariants and the resonances matches that of Fig. 1(a) above: the lowest-order resonance in Fig. 2 passes near the middle of the family of adiabatic invariants. In Fig. 3(a) we show a projection on the meridian plane of the stroboscopically-sampled trajectory of the flow at these parameter values. The similarity between Fig. 3(a) and Fig. 1(a) is immediately obvious. Furthermore, in Fig. 3(b), we show the strobed time evolution of the invariant given by Eq. (7). Again, its likeness to Fig. 1(b) is undeniable, which highlights both the presence of resonance-induced diffusion here, and the accuracy of our assumptions about the adiabatic invariant.
Figure 4:
A slice of a strobed trajectory between 
for the same parameter
values as Fig. 3. Locations of low-order resonances
are shown as in Fig. 2. Forty thousand points
(corresponding to many times this number of periods) are plotted here,
all from the same initial condition.
Finally, as the slice of a strobed trajectory in Fig. 4
attests, the mechanism of resonance-induced diffusion is responsible for
the existence of asymptotically-globally-space-filling trajectories. This
slice illustrates how effective resonance-induced diffusion is at mixing;
the perturbation, which is of 
, is very small here, yet the only
regions where the trajectory has not yet ventured are close to the poles,
and the middle of the vortices, which lie inside the 
resonance shell.
In time, the trajectory will diffuse into these regions too, through the
action of the higher-order resonances that are dense throughout the space.
Increasing the axis separation 
increases the amount of time the
trajectory is captured into resonance, and with it the size of the jumps
and the diffusion rate. Other quantities relevant to mixing --- the stretching
and folding measured by the Lyapunov exponents of the flow --- also increase
as powers of 
. Experimental control of the mixing rate can then be
achieved through adjustment of the strength and density of the resonances
with the axis separation 
, the angular velocity ratio of the spheres,
and the period of the motion []. Further quantitative studies
of mixing efficiency, as well as the effects of the interplay of space-filling
trajectories and molecular diffusion on mixing and transport, are now in
progress.
In summary, we have shown that the properties of nearly-integrable two-action Liouvillian maps are highly relevant to the transport features of a large class of real fluid flows. We expect to see in the stroboscopic maps of such flows the resonance-induced diffusion characteristic of two-action maps, consisting of motion on invariant surfaces interspersed with periods of motion on resonant surfaces. We have also shown that our action--angle classification of Liouvillian maps has good predictive capability. This finding opens up an avenue for the experimental verification of the existence of space-filling trajectories in this and other similar flow geometries in the presence of small periodic modulations. There are many technological implications of this novel diffusion phenomenon, and we aim to have excited the reader about the new possibilities for enhanced mixing that may arise.
MF acknowledges the support of the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities, and OP and JHEC that of the Spanish Dirección General de Investigación Científica y Técnica, contract numbers PB94-1167 and PB94-1172 and European Union Human Capital and Mobility contract number ERBCHBICT920200.
