Control of mixing via entropy tracking
Citation for published version (APA):Hoeijmakers, P. G. M., Fontenele Araujo, F., Heijst, van, G. J. F., Nijmeijer, H., & Trieling, R. R. (2010). Control of mixing via entropy tracking. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 81(6), 066302-1/7. [066302]. https://doi.org/10.1103/PhysRevE.81.066302
DOI:
10.1103/PhysRevE.81.066302 Document status and date: Published: 01/01/2010
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Control of mixing via entropy tracking
Maarten Hoeijmakers,1 Francisco Fontenele Araujo,2GertJan van Heijst,2Henk Nijmeijer,1and Ruben Trieling2 1
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 2
Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
共Received 6 January 2010; revised manuscript received 26 March 2010; published 4 June 2010兲 We study mixing of isothermal fluids by controlling the global hydrodynamic entropy具s典. In particular, based on the statistical coupling between the evolution of具s典 and the global viscous dissipation 具⑀典, we analyze stirring protocols such that 具s典⬃t␣⇔具⑀典⬃t␣−1, with 0⬍␣ⱕ1. For a model array of vortices 关Fukuta and Murakami, Phys. Rev. E 57, 449共1998兲兴, we show that: 共i兲 feedback control can be achieved via input-output linearization,共ii兲 mixing is monotonically enhanced for increasing entropy production, and 共iii兲 the mixing time tmscales as tm⬃具⑀典−1/2.
DOI:10.1103/PhysRevE.81.066302 PACS number共s兲: 47.51.⫹a
I. INTRODUCTION
Since the seminal paper by Maxwell 关1兴, control theory
has played an important role in science and technology. Be-sides the numerous applications in industrial motion systems—often using proportional/differential controllers— other problems range from the control of chaos in dynamical systems 关2兴 to the design of engineering devices for ship
maneuvering关3兴.
In fluid dynamics, control theory has stimulated many ad-vances in thermal convection 关4,5兴, drag reduction 关6–8兴,
mixing 关9–11兴, turbulence 关12,13兴, and flow over bluff
bod-ies关14兴. A common feature in such contexts is the feedback
actuation, which usually involves blowing/suction through specific boundary points关7,11兴 or modulation of
electromag-netic forces in the neighborhood of walls关6,9兴. Whatever the
mechanism, successful control crucially depends on the flow regime.
Stokes flows, for instance, can be controlled and even optimized via linear methods关9兴. However, beyond such
re-gime, hydrodynamic nonlinearities pose major challenges for control theory 关10,12兴. To circumvent part of the technical
issues that arise, one often resorts to simplified nonlinear models with the aim of extracting insight on how to enhance 共or suppress兲 a particular flow response 关5兴. In this spirit, we
focus on mixing—an essential process in industrial applica-tions and an inspiring subject for basic research. Inspiring but nontrivial, nontrivial in the sense of its quantification, feedback design, and experimental calibration. Despite many attempts to characterize mixing in terms of statistical prop-erties of flow snapshots关9,15–17兴, there is still no consensus
on which mixing measure m共兲 best represents the process at time. Moreover, from a control standpoint, feedback design requires substantial knowledge of m as explicit function of the velocity field U. But at present, such analytic relation is inaccessible from fundamental principles as well as realistic coordination of sensors with actuators remains a challenge for experiments.
Here, we do not address such experimental issues. In-stead, we focus on mixing in a simplified flow model: an array of vortices 关18兴 whose mathematical structure is
sus-ceptible to classical control theory 关19,20兴. In particular, we
wish to reveal how mixing depends on hydrodynamic quan-tities such as the global viscous dissipation,
具典 ⬅ 2
冓
兺
␣兺
冉
U␣ X + U X␣冊
2冔
, 共1兲wheredenotes the kinematic viscosity of the fluid, U␣ ve-locity components, X␣Cartesian coordinates, and具¯典 space average over the flow domain.
To accomplish that, the present paper is organized as fol-lows. Section II sketches the general lines of our approach. After pointing out the gap between the purely statistical and the purely kinematical descriptions of mixing, we adopt the hydrodynamic entropy 具S典 as intermediate between the two descriptions. The benefit of such approach is threefold:共i兲 it relates mixing with velocity gradients.共ii兲 It allows feedback control in terms of global flow quantities共具典 and 具S典兲 rather than local measurements关7,11兴. 共iii兲 It facilitates the control
actuation via adjustments in the driving force of the flow instead of blowing and/or suction through ad hoc boundary points关7,11兴. In this spirit, we choose 具S典 as a control target
such that具典 is a statistically stationary/decaying function of time. The stirring protocol is then dynamically adjusted and the status of the mixture further characterized in terms of the mixing measure proposed by Stone and Stone关15兴.
Sections IIIandIVare devoted to the application of the above strategy to a simplified flow model. We begin by in-troducing the array of vortices derived by Fukuta and Mu-rakami关18兴, which is inspired by experiments on a shallow
layer of fluid driven by electromagnetic forcing关21,22兴. The
governing equations for the stream function are given in Sec.
III. In terms of them, we compute the global viscous dissi-pation of the flow, discussing its geometrical representation in state space and its relation to entropy production. Then, in Sec. IV, we show that the vortex model, although nonlinear, is amenable to feedback control via input-output lineariza-tion 关19,20兴. In this framework, we prescribe stirring
proto-cols such that the dimensionless viscous dissipation 具典 evolves as具典⬃−1+␣, with 0⬍␣ⱕ1. In particular, for sta-tistically stationary具典, we define a mixing timemand show
thatm⬃具典−1/2. For the statistically decaying case, we show
that mixing is monotonically enhanced for increasing en-tropy production.
Finally, Sec. V provides a summary of results, conclu-sions, and open questions.
II. MIXING, ENTROPY, AND VISCOUS DISSIPATION Mixing is traditionally described at two levels: 共i兲 kine-matically, in terms of stretching/folding of fluid elements 关16,17兴 and 共ii兲 statistically, by taking snapshots of the
mix-ture and computing average properties of each image 关15–17兴. The former is related to velocity gradients; the latter
to entropy关23–25兴. But how to connect these two pictures in
a consistent way? Answers to this question could be formu-lated in terms of vorticity, persistence of strain 关26,27兴, or
viscous dissipation, to cite just a few possibilities. Among those, the global viscous dissipation 共1兲 provides a conve-nient alternative since it is related to the hydrodynamic en-tropy via the differential equation共see Landau and Lifshitz 关关28兴, p. 195, Eq. 共49.6兲兴兲, d d具S典 =
冓
共ⵜT兲2 T2冔
+冓
T冔
+冓
T共ⵜ · U兲 2冔
,wheredenotes the density of the fluid, T the temperature, the thermal conductivity, andthe second viscosity. In par-ticular, for incompressible and isothermal flow, the evolution of the global entropy具S典 is simplified to
d d具S典 =
1
T具典.
Here, it is convenient to introduce dimensionless variables such that X␣= Lx␣, =L2t, U␣=Lu␣, =L34⑀, and S =
2
TL2s,
where L is a typical length scale of the flow. Thus
d
dt具s典 = 具⑀典. 共2兲
Physically, Eq. 共2兲 establishes an interesting connection
be-tween average velocity gradients and entropy. For instance, if 具s典 evolves as a power law, so does 具⑀典. From the control standpoint, this suggests the specification of具s典 as a control target. Thus, the study of stirring protocols based on statisti-cal properties of 具s典 and 具⑀典 may contribute to a better un-derstanding of the mixing dynamics. That is the main point of the present paper.
But how to assess the plausibility of such argument? To answer this question, we should somehow quantify mixing. We do so by adopting the mixing number m proposed by Stone and Stone 关15兴, since its sensitivity on image
reso-lution is considerably weaker than in other methods. For de-tails, we refer the reader to Ref. 关15兴. Here, we just present
an informal definition of m for mixing between two species in a rectangular domain. The basic idea is as follows. Let a snapshot image I共t兲 discretized in N cells, each of which colored as black or white. Given a cell C␣, consider the set O␣共t兲傺I共t兲 whose color is opposite to that of C␣. Then,
in-troduce the distance between C␣ and O␣ as ⌬共C␣,O␣, t兲 = min兵d共C␣, C兲:C苸O␣共t兲其, where d共C␣, C兲 denotes the Cartesian distance between cells. In this way, the mixing number m共t兲 is defined as 关15兴 m共t兲 ⬅
兺
␣=1 N ⌬2共C ␣,O␣,t兲 N . 共3兲Qualitatively, Eq.共3兲 measures the average distance between
black and white species at time t.
In summary: to bridge the gap between the statistical and kinematical descriptions of mixing, we adopt the hydrody-namic entropy 具s典 as intermediate between them. In particu-lar, Eq. 共2兲 suggests the choice of 具s典 as a control target,
which is presumably related to mixing 关via Eq. 共3兲兴. Next,
we apply this idea to the case of mixing in an array of vor-tices.
III. MIXING IN AN ARRAY OF VORTICES Vortices are pervasive structures in nature. In geophysical flows, for instance, they emerge in a variety of length scales, encompassing extreme events such as tornadoes and snow avalanches 关29兴. In condensed matter physics, they play a
less threatful but important role in superconductivity 关30兴,
Bose-Einstein condensates 关31兴, and fluid dynamics as a
whole.
Under laboratory conditions, vortex distributions in regu-lar lattices offer a convenient platform for research on flow structures. Among the many examples are vortex arrays in soap films 关32兴 and in shallow layers of electrolytes
关21,33,34兴. The latter, in particular, is experimentally
real-ized by setting a row of evenly spaced magnets underneath the flow container and then passing an electric current through the fluid. Such simple setup has inspired theoretical studies on drag reduction 关6兴, three-dimensional resonant
mixing 关34,35兴, and mixing control in two-dimensional
Stokes flow关9兴.
In the present paper, we address neither three-dimensionality issues nor the Stokes regime. Instead, our fo-cus is on feedback control in a two-dimensional nonlinear model 关18兴.
A. Model
Consider a two-dimensional, viscous, and incompressible flow modeled by the streamfunction关18兴,
共x,y,t兲 =0共t兲sin共kx兲sin共y兲 +1共t兲sin共y兲
+2共t兲cos共kx兲sin共2y兲, 共4兲
where the time amplitudes0,1,2共t兲 are governed by
d0 dt = k共k2+ 3兲 2共k2+ 1兲12− k2+ 1 R 0+ 共k2+ 1兲F R , 共5兲 d1 dt = − 3k 4 02− 1 R1, 共6兲 d2 dt = − k3 2共k2+ 4兲01− k2+ 4 R 2. 共7兲
Equations 共4兲–共7兲 mimic the spatiotemporal dynamics of a
shallow layer of fluid driven by a Lorentz force 关18,22兴. In
HOEIJMAKERS et al. PHYSICAL REVIEW E 81, 066302共2010兲
such a context, F is related to the amplitude of the electric current through the fluid, k to the distance between magnets, and R to the viscosity of the electrolyte.
Insight into the basic structure of the model is provided by plotting streamlines of Eq. 共4兲. As shown in Fig. 1, the 0-mode induces an array of counter-rotating vortices关plate
1共a兲兴; 1 a pure shear flow 关plate 1共b兲兴, and 2 a vortex lattice 关plate 1共c兲兴. From this perspective, proper combina-tions of 0,1,2 may be sought in order to achieve a desired
flow response. Since our focus here is on mixing, we shall specify an appropriate hydrodynamic output as follows. First, we relate viscous dissipation and entropy in the vortex system 关Eqs. 共4兲–共7兲兴. Then, we adopt a control strategy
based on input-output linearization and systematically quan-tify mixing for different stirring protocols.
B. Viscous dissipation and entropy
To determine the global viscous dissipation 关Eq. 共1兲兴 in
the vortex system关Eqs. 共4兲–共7兲兴, we write the velocity
com-ponents in terms of the streamfunction as ux=/y and uy
= −/x. Then, we compute the corresponding velocity
gra-dients and take the space average, 具 ¯ 典 = 1 LxLy
冕
0 Lx冕
0 Ly 共 ¯ 兲dxdy,with Lx= 2/k and Ly=. In this way, the dimensionless
viscous dissipation具⑀典 is given by 具⑀典 = A00 2 + A11 2 + A22 2 , 共8兲 where A0=14共1+k2+ k4兲, A 1= 1 2, and A2= 1 + 2k2+ k4 4. Equation
共8兲 has geometrical counterparts in state space. For instance,
具⑀典=r0= constant corresponds to an ellipsoid, as shown in
Fig. 2. Physically, the stirring strength 共r0兲 determines the volume of the ellipsoid and the forcing wave number共via A0
and A2兲 its aspect ratio. Although simplistic, such geometric
picture will provide some insight into the control dynamics 共cf. Fig.3兲.
To conclude this section, we relate the viscous dissipation 关Eq. 共8兲兴 with the dimensionless entropy 具s典 of the flow. On
the basis of Eq.共2兲 one readily finds
d具s典
dt = A00
2+ A
112+ A222. 共9兲
From the theoretical standpoint, Eqs.共8兲 and 共9兲 are
equiva-lent in the sense that either具⑀典 or 具s典 may be chosen as basic hydrodynamic output. Next, we address this possibility from a control perspective.
IV. CONTROL
Once a physically relevant outputis identified, it may be desirable to drive the system to a target response r, so that the difference − r asymptotically converges to zero. Ex-amples of such outputs include measurements of wall-shear stresses in channel flows关7,11兴 and local temperatures 关4,5兴
in thermal convection. In the case of Eqs.共5兲–共7兲, we choose
=具⑀典 and implement feedback control via input-output lin-earization关19,20兴.
x
y
0 1 2 3 4 5 6 7 8 9 0 1 2 3 (a)x
y
0 1 2 3 4 5 6 7 8 9 0 1 2 3 (b)x
y
0 1 2 3 4 5 6 7 8 9 0 1 2 3 (c)FIG. 1. Streamlines of the model 共4兲 for k=1. 共a兲
Counter-rotating vortices generated by the main mode 0. 共b兲 Pure shear
flow induced by1.共c兲 Vortex lattice due to2.
FIG. 2. Geometric structures in state space for k = 1 and R = 10. Ellipsoid: surface for which具⑀典=0.4. Straight lines: stable equilibria for 0ⱕ具⑀典ⱕ1.
FIG. 3. Geometric structures in state space for k = 1 and R = 10. Ellipsoid: surface for which 具⑀典=2. Straight lines 共black兲: stable equilibria for 2ⱕ具⑀典ⱕ2.5. Vectors: internal dynamics. U-shaped line共white兲: typical trajectory of the closed loop system.
A. Input-output linearization Consider a dynamical system of the form
d
dt = a共兲 + Fb共兲, 共10兲
where =共0, . . . ,n−1兲 denotes the state vector, a and b
vector fields, and F a scalar input.
In terms of Eq.共10兲, the time derivative of the output
can be written as
d
dt =·
d
dt =· a + F· b, 共11兲
where =共/0, . . . ,/n−1兲. Since d/dt explicitly
depends on the input F, system关Eq. 共10兲兴 has relative degree
␥= 1关19,20兴. Under such property, a tracking feedback
con-troller may be achieved by choosing F as
F =
−· a −⌫共− r兲 +dr
dt
· b , 共12兲
where⌫⬎0 is the control gain. This so-called input-output linearization is valid for· b⫽0, cf. 关19,20兴.
Thus, we apply feedback control关Eq. 共12兲兴 to the system,
=
冤
0 1 2冥
, 共13兲 a =冤
k共k2+ 3兲 2共k2+ 1兲12− k2+ 1 R 0 −3k 4 02− 1 R1 − k 3 2共k2+ 4兲01− k2+ 4 R 2.冥
, 共14兲 b =冤
k2+ 1 R 0 0冥
, 共15兲 =具⑀典 = A002+ A112+ A222, 共16兲 r = r共t兲, 共17兲where the target r remains to be specified 共we do so in the next subsection兲. In addition, since · b =2A0共k
2+1兲
R 0,
input-output linearization关Eq. 共12兲兴 holds on L=兵苸R3: 0⫽0其.
As shown in Fig. 3, the internal dynamics 关19兴 of system
关Eqs. 共10兲–共16兲兴 for r=r0= constant evolves on a surface T =兵苸L:= r0其.
B. Mixing
Now we consider mixing in the array of vortices 关Eq. 共10兲–共16兲兴. In particular, we study stirring protocols such that
the viscous dissipation具⑀典 is targeted at
r共t兲 = D共t + t0兲␣−1, 共18兲
where the coefficient D is related to the stirring strength,␣is the scaling exponent 共0⬍␣ⱕ1兲, and t0 a time offset
共t0= 1兲. Physically, target 关Eq. 共18兲兴 corresponds to
time-decaying dissipation. But in contrast to the several studies on freely decaying flows关36,37兴, here the driving force F共t兲 is
dynamically adjusted by the controller关Eq. 共12兲兴 so that 具⑀典
converges to the power law 关Eq. 共18兲兴. In this sense, 具⑀典
= D共t+t0兲␣−1is equivalent to
具s典 =D
␣共t + t0兲␣, 共19兲
since Eq.共2兲 enables control of mixing via entropy tracking.
In order to characterize the mixing dynamics in terms of
D and␣, we perform numerical simulations as follows. First,
we fix the model parameters at representative values, namely: k = 1 and R = 10. Then, we discretize the physical space共x,y兲 in an 200⫻100 grid. As initial configuration, we consider a horizontal layer of black fluid occupying the cen-tral third of the grid while the remaining area 共bottom and top兲 is filled with white fluid. In this setting, imaging of the mixture is performed via forward advection and the mixing number关Eq. 共3兲兴 is computed as function of D and␣.
1. Role of the coefficient D
To reveal the dependence of the mixing dynamics on具⑀典, we fix the scaling exponent at␣= 1 and compute the mixing number关Eq. 共3兲兴 for increasing values of D.
As shown in Fig. 4, increasing D leads to faster mixing. This result supports the notion that entropy and mixing are closely related. Furthermore, note that the curves tend to col-lapse on a minimum mixing number m⬇6⫻10−3. In this ultimate regime, the time series become indistinguishable from each other due to the spatial resolution of the mixing number.
Clearly, mixing tends to evolve faster for increasing 具⑀典 = D. But what is the connection between the viscous dissipa-tion rate and the mixing rate? To answer this quesdissipa-tion, we introduce a mixing time tmdefined as the instant at which the
t
m
(t
)/
m
(0)
D = 0.2 D = 1.0 D = 3.0 D = 6.0 10−1 100 101 102 10−2 10−1 100FIG. 4. Normalized mixing number as function of time, for stirring protocols such that 具s典=D共t+t0兲⇔具⑀典=D. The stirring
strength D is increased from 0.2 共solid line, top curve兲 to 6.0 共dashed line, bottom curve兲. The horizontal gray line corresponds
m共t兲/m共0兲=0.25; its intersection with each curve defines a mixing
time tm.
HOEIJMAKERS et al. PHYSICAL REVIEW E 81, 066302共2010兲
average distance between two species of the mixture has halved. In terms of the mixing number this corresponds to
m共tm兲/m共0兲=0.25 共see Fig.4兲.
Figure 5shows that tm⬃具⑀典−0.51. Such scaling is
surpris-ingly robust. Further numerical simulations indicate that the exponent −0.5共1兲 remains basically unchanged for ratios 0.1ⱕm共tm兲/m共0兲ⱕ0.8.
On dimensional grounds, the result of Fig.5may be writ-ten as
m= a
冑
具典, 共20兲
where a is a dimensionless coefficient. This simple analysis supports our numerical result and evidences that the mixing dynamics is related to velocity gradients.
2. Role of the exponent␣
Now we characterize the mixing dynamics in terms of␣. In particular, we fix the stirring strength at D = 1 and compute the mixing number as function of:共i兲 time, 共ii兲 entropy, 共iii兲 viscous dissipation, and共iv兲 control effort.
To begin, note that the larger the exponent␣in Eq.共19兲,
the larger the entropy production d具s典/dt in the flow. This suggests that mixing should be enhanced for increasing ␣.
Indeed, this is the case. Figure6 shows that the time series
m共t兲 decays faster as ␣is increased from 0.2 to 0.8. To further quantify the decaying dynamics, we plot m as function 具s典. As shown in Fig.7, the larger the entropy the better the mixing. But here the␣-dependence is more subtle: for a given value of 具s典, the mixing number is smaller for
decreasing ␣ 共compare, for instance, the curves for ␣= 0.2
and␣= 0.8兲. Such trend reveals that just the magnitude of the entropy is insufficient to assure the mixing quality; the
tem-poral dynamics of具s典 is also crucial for achieving low values
of m.
The findings above may be complemented by plotting m as function of the viscous dissipation具⑀典. As shown in Fig.8, at具⑀典=1 关i.e., at t=0, according to Eq. 共18兲兴, the normalized
mixing number is equal to 1 for all values of␣. Then, as具⑀典 decays 关again, cf. Eq. 共18兲兴, m experiences a transient and
eventually a monotonic decrease. Comparing the curves at 具⑀典=0.5, for instance, one infers that the mixing performance is improved for increasing␣. This trend is physically reason-able and in agreement with the limiting case ␣= 1共cf. Sec.
IV B 1兲. In addition, Fig. 8 shares a common feature with Figs.6and7, namely: a glitch around m共t兲/m共0兲=0.05. Such
lethargic interval reflects suppression of the higher modes 共1and2兲 and dominance of the main flow 共0兲 in the final
stage of the mixing process共see Fig.1兲.
t
m
(t
)/
m
(0
)
α = 0.2 α = 0.4 α = 0.6 α = 0.8 0 10 20 30 40 50 10−2 10−1 100FIG. 6. Normalized mixing number as function of time, for stirring protocols such that具s典⬃t␣⇔具⑀典⬃t␣−1. The exponent␣ is
increased from 0.2共solid line, top curve兲 to 0.8 共dashed line, bottom curve兲.
t
m -0.51 10−2 100 102 10−1 100 101 102FIG. 5. Mixing time tmas a function of the global viscous
dis-sipation 具⑀典. Solid dots: numerical simulations. Line: linear fit tm
= 3.5具⑀典−0.51.
s
m
(t
)/
m
(0
)
α = 0.2 α = 0.4 α = 0.6 α = 0.8 0 10 20 30 10−2 10−1 100FIG. 7. Normalized mixing number as function of entropy, for stirring protocols such that具s典⬃t␣⇔具⑀典⬃t␣−1. The exponent␣ is
increased from 0.2共solid line, bottom curve兲 to 0.8 共dashed line, top curve兲.
m
(t
)/m
(0)
α = 0.2 α = 0.4 α = 0.6 α = 0.8 0 0.2 0.4 0.6 0.8 1 10−2 10−1 100FIG. 8. Normalized mixing number as function of viscous dis-sipation, for stirring protocols such that 具s典⬃t␣⇔具⑀典⬃t␣−1. The
exponent ␣ is increased from 0.2 共solid line, left curve兲 to 0.8 共dashed line, right curve兲.
Finally, let us consider m as function of the cumulative control effort ⌰共t兲⬅兰0tF共t
⬘
兲dt⬘
. Figure 9 shows that for given ⌰ 共say ⌰=50兲, the mixing number m is smaller for decreasing ␣. This suggests that the cost benefit of stirring protocols of the form 关Eq. 共18兲兴 involves a balance betweenthe control effort and the duration of the mixing process. If the priority is to reduce the control effort 共rather than the stirring time兲, satisfactory mixing can be achieved by decay-ing flows such that␣⬇0. On the other hand, if fast mixing is the priority, statistically stationary stirring共␣= 1兲 would suf-fice.
V. CONCLUSIONS
We studied mixing of isothermal fluids by controlling the entropy of the flow. This was accomplished by connecting
hydrodynamic关Eq. 共2兲兴 and statistical 关Eq. 共3兲兴 aspects of the
phenomenon.
For the array of vortices modeled by 关Eqs. 共4兲–共7兲兴, we
succeeded to design a feedback controller via input-output linearization 关Eqs. 共10兲–共16兲兴. Moreover, we found that the
mixing time for statistically stationary viscous dissipation scales as tm⬃具⑀典−1/2. Nevertheless, it remains to be seen
whether this relation is flow specific or somehow more gen-eral. Analysis of mixing time series from other models could clarify this issue.
In any scenario, the main conclusion is that the study of stirring protocols based on statistical properties of具s典 and 具⑀典 may indeed contribute to a better understanding of the mix-ing dynamics. From this perspective, the present work could be extended in several ways. For instance, instead of speci-fying具⑀典 as statistically stationary or as a statistically decay-ing power law, one may consider other functions of time that enhance or suppress mixing.
Finally, notwithstanding the intrinsic aesthetics of fluid mixtures in nature and technology, nonlinear control of mix-ing remains a tremendous challenge from the experimental standpoint.
ACKNOWLEDGMENTS
We thank the referees for the constructive suggestions. This work was supported by the TU/e stimulation program on “Fluid and Solid Mechanics.”
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