Control of mixing via entropy tracking

PHYSICAL REVIEW E 81, 066302 共2010兲

Control of mixing via entropy tracking 1

Maarten Hoeijmakers,1 Francisco Fontenele Araujo,2 GertJan van Heijst,2 Henk Nijmeijer,1 and Ruben Trieling2

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 ⬍ ␣ ⱕ 1. 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 tm scales 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. Besides 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 advances 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 bodies 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 electromagnetic 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 regime, 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 applications 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 properties 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. Instead, we focus on mixing in a simplified flow model: an array of vortices 关18兴 whose mathematical structure is susceptible to classical control theory 关19,20兴. In particular, we wish to reveal how mixing depends on hydrodynamic quantities such as the global viscous dissipation, 1539-3755/2010/81共6兲/066302共7兲

具␧典 ⬅

␯ 2

冓

兺␣ 兺␤

冉

⳵ U␣ ⳵ U␤ + ⳵ X␤ ⳵ X␣

冊冔 2

,

共1兲

where ␯ denotes the kinematic viscosity of the fluid, U␣ velocity components, X␣ Cartesian coordinates, and 具 ¯ 典 space average over the flow domain. To accomplish that, the present paper is organized as follows. 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 III and IV are devoted to the application of the above strategy to a simplified flow model. We begin by introducing the array of vortices derived by Fukuta and Murakami 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 dissipation 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 linearization 关19,20兴. In this framework, we prescribe stirring protocols such that the dimensionless viscous dissipation 具␧典 evolves as 具␧典 ⬃ ␶−1+␣, with 0 ⬍ ␣ ⱕ 1. In particular, for statistically stationary 具␧典, we define a mixing time ␶m and show that ␶m ⬃ 具␧典−1/2. For the statistically decaying case, we show that mixing is monotonically enhanced for increasing entropy production. Finally, Sec. V provides a summary of results, conclusions, and open questions.

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II. MIXING, ENTROPY, AND VISCOUS DISSIPATION

Mixing is traditionally described at two levels: 共i兲 kinematically, in terms of stretching/folding of fluid elements 关16,17兴 and 共ii兲 statistically, by taking snapshots of the mixture 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 formulated 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 convenient alternative since it is related to the hydrodynamic entropy via the differential equation 共see Landau and Lifshitz 关关28兴, p. 195, Eq. 共49.6兲兴兲, d 具␳S典 = d␶

冓 冔冓冔冓 ␬共ⵜT兲2 T2

+

␳ ␧ + T

冔

␨ 共ⵜ · U兲2 , T

where ␳ denotes the density of the fluid, T the temperature, ␬ the thermal conductivity, and ␨ the second viscosity. In particular, for incompressible and isothermal flow, the evolution of the global entropy 具S典 is simplified to 1 d 具S典 = 具␧典. d␶ T Here, it is convenient to introduce dimensionless variables 2 3 ␯2 such that X␣ = Lx␣, ␶ = L␯ t, U␣ = L␯ u␣, ␧ = L␯ 4 ⑀, and S = TL 2 s, where L is a typical length scale of the flow. Thus d 具s典 = 具⑀典. dt

共2兲

Physically, Eq. 共2兲 establishes an interesting connection between 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 statistical properties of 具s典 and 具⑀典 may contribute to a better understanding 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 resolution is considerably weaker than in other methods. For details, 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, introduce 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

⌬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 hydrodynamic entropy 具s典 as intermediate between them. In particular, 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 vortices. 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 regular 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 realized 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 threedimensionality issues nor the Stokes regime. Instead, our focus 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 amplitudes ␺0,1,2共t兲 are governed by k2 + 1 共k2 + 1兲F d␺0 k共k2 + 3兲 ␺ ␺ − ␺ + = , 1 2 0 dt 2共k2 + 1兲 R R

共5兲

3k 1 d␺1 = − ␺ 0␺ 2 − ␺ 1 , dt 4 R

共6兲

k2 + 4 k3 d␺2 =− ␺ ␺ − ␺2 . 0 1 dt 2共k2 + 4兲 R

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

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y

3 2 1 0 0

1

2

3

5

4

6

7

8

9

6

7

8

9

x

(a)

y

3 2 1 0 0

1

2

3

5

4

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 0 ⱕ 具⑀典 ⱕ 1.

x

(b)

y

3 2 1 0 0

1

2

3

5

4

6

7

8

9

x

(c)

FIG. 1. Streamlines of the model 共4兲 for k = 1. 共a兲 Counterrotating vortices generated by the main mode ␺0. 共b兲 Pure shear flow induced by ␺1. 共c兲 Vortex lattice due to ␺2.

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 combinations 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 quantify mixing for different stirring protocols.

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 = A0␺20 + A1␺21 + A2␺22 . dt

共9兲

From the theoretical standpoint, Eqs. 共8兲 and 共9兲 are equivalent 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 output ␰ is identified, it may be desirable to drive the system to a target response r, so that the difference ␰ − r asymptotically converges to zero. Examples 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 linearization 关19,20兴.

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 components in terms of the streamfunction as ux = ⳵␺ / ⳵y and uy = −⳵␺ / ⳵x. Then, we compute the corresponding velocity gradients and take the space average, 具¯典=

1 L xL y

冕冕 Lx

0

Ly

0

共 ¯ 兲dxdy,

with Lx = 2␲ / k and Ly = ␲. In this way, the dimensionless viscous dissipation 具⑀典 is given by 具⑀典 = A0␺20 + A1␺21 + A2␺22 , A0 = 41 共1 + k2 + k4兲,

A1 = 21 ,

4 A2 = 1 + 2k2 + k4 .

共8兲

where and Equation 共8兲 has geometrical counterparts in state space. For instance, 具⑀典 = r0 = constant corresponds to an ellipsoid, as shown in

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 2 ⱕ 具⑀典 ⱕ 2.5. Vectors: internal dynamics. U-shaped line 共white兲: typical trajectory of the closed loop system.

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100

d␺ = a共␺兲 + Fb共␺兲, dt

m(t)/m(0)

Consider a dynamical system of the form 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␰ d␺ = ⵱␰ · = ⵱␰ · a + F ⵱ ␰ · b, dt dt

F=

⵱␰ · b

dr dt

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,

冤冥

␺0 ␺ = ␺1 , ␺2

a=

冤

共13兲

k2 + 1 k共k2 + 3兲 ␺ 1␺ 2 − ␺0 2 2共k + 1兲 R − −

3k 1 ␺ 0␺ 2 − ␺ 1 4 R

k2 + 4 k3 ␺ 0␺ 1 − ␺2 . 2共k + 4兲 R 2

b=

冤 冥 k2 + 1 R 0

,

冥

10

D D D D

−2

10−1

= 0.2 = 1.0 = 3.0 = 6.0

共14兲

共15兲

0

␰ = 具⑀典 = A0␺20 + A1␺21 + A2␺22 ,

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 2A 共k2+1兲 next subsection兲. In addition, since ⵱␰ · b = 0 R ␺0, inputoutput 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

102

t FIG. 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.

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 ⬍ ␣ ⱕ 1兲, and t0 a time offset 共t0 = 1兲. Physically, target 关Eq. 共18兲兴 corresponds to timedecaying 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␣−1 is equivalent to 具s典 =

,

101

100

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 controller may be achieved by choosing F as − ⵱␰ · a − ⌫共␰ − r兲 +

10−1

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 central 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 collapse 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 dissipation rate and the mixing rate? To answer this question, we introduce a mixing time tm defined as the instant at which the

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m(t)/m(0)

100

tm

101 -0.51

100

α = 0.2 α = 0.4 α = 0.6 α = 0.8

10−1

10−2

10−1 −2 10

100

10

0

102



 FIG. 5. Mixing time tm as a function of the global viscous dissipation 具⑀典. Solid dots: numerical simulations. Line: linear fit tm = 3.5具⑀典−0.51.

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 5 shows that tm ⬃ 具⑀典−0.51. Such scaling is surprisingly robust. Further numerical simulations indicate that the exponent −0.5共1兲 remains basically unchanged for ratios 0.1ⱕ m共tm兲 / m共0兲 ⱕ 0.8. On dimensional grounds, the result of Fig. 5 may be written 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. Figure 6 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 temporal 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 reasonable 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. 6 and 7, namely: a glitch around m共t兲 / m共0兲 = 0.05. Such lethargic interval reflects suppression of the higher modes 共␺1 and ␺2兲 and dominance of the main flow 共␺0兲 in the final stage of the mixing process 共see Fig. 1兲. 100

α = 0.2 α = 0.4 α = 0.6 α = 0.8

10−1

10−1 α = 0.2 α = 0.4 α = 0.6 α = 0.8

10−2

10−2 0

10

30

20

40

30

FIG. 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)

m(t)/m(0)

100

20


s

0

50

0.2

0.6

0.4

0.8

1





t FIG. 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.

FIG. 8. Normalized mixing number as function of viscous dissipation, 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.

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100

α = 0.2 α = 0.4 α = 0.6 α = 0.8

10−1

10−2 100

101

Θ FIG. 9. Mixing number as function of the control effort, 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.

Finally, let us consider m as function of the cumulative control effort ⌰共t兲 ⬅ 兰t0F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 between the 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 decaying flows such that ␣ ⬇ 0. On the other hand, if fast mixing is the priority, statistically stationary stirring 共␣ = 1兲 would suffice. V. CONCLUSIONS

We studied mixing of isothermal fluids by controlling the entropy of the flow. This was accomplished by connecting

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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 general. 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 mixing dynamics. From this perspective, the present work could be extended in several ways. For instance, instead of specifying 具⑀典 as statistically stationary or as a statistically decaying 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 mixing 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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