Simon Gerwig

Bilal Sari

Federica Garin

Carlos Canudas-De-Wit

Reduced Model for Control in a Hydroelectric Unit at Off-Design Operation

This paper describes the development of a reduced-order linear mathematical model of a hydroelectric unit subject to pressure perturbations in the draft tube of the hydraulic turbine, as it is the case at off-design operation (partial load or full load). The objective behind this development is to design a control algorithm able to attenuate the effect of these hydraulic pressure oscillations on the hydroelectric unit. First, we develop the reduced-order linear hydraulic model of the turbine and the pipes. Then this hydraulic model is combined with a reduced order model of a single machine infinite bus (SMIB) system. Finally we present simulation results of a controller designed with this model.

I. INTRODUCTION

Hydroelectric power plants contribute to the stability of the electrical network thanks to their fast time response in comparison to other electricity sources. The hydraulic turbine converts the potential energy of water stored in natural or artificial reservoirs into rotating mechanical energy, which is then converted to electricity by the electric generator. Fig. 1 is a drawing of a two-units hydroelectric powerplant showing two reaction type turbines with their draft tubes and the generators.

Typical Francis or pump turbines are designed for an optimal flow and are operated around this point. With the increase use of intermittent energy sources such as solar panels and wind turbines, the operating range of the turbines is extended. At off-design operation of the hydraulic turbine (when the flow through the turbine is smaller or larger than the optimal flow), some hydraulic phenomena set up in the draft tube and produce pressure oscillations. At partial load, for example, a helical vortex rope builds up in the draft tube and induces pressure oscillations in the hydraulic system with a frequency in the range of 0.2 to 0.4 times the turbine rotational frequency [START_REF] Nicolet | Methodology for risk assessment of part load resonance in Francis turbine power plant[END_REF]. These pressure oscillations are converted into mechanical torque oscillations and eventually into electric power oscillations. Therefore the idea is to design a control algorithm able to attenuate the effect of these pressure oscillations on the hydroelectric unit. An important effort towards this aim is to find a suitable model of the physical system, simple enough to make the control design effective, but rich enough to capture the system behavior.

S. Gerwig is a PhD candidate in the NeCS team, Inria, and at Alstom Hydro, France (email: simon.gerwig@inria.fr).

B. Sari is research engineer at Alstom Hydro, Process Control R&D, Grenoble, France (email: bilal.sari@power.alstom.com) F. Garin is a researcher with the NeCS team, Inria, Grenoble, France (email: federica.garin@inria.fr) C. Canudas-de-Wit is Director of Research at CNRS, GIPSA-Lab (NeCS team), Grenoble, France (email: carlos.canudas-de-wit@gipsa-lab.fr).

A. State of the Art

The primary focus of the control literature of hydraulic turbine is to tackle the two main features of the hydraulic system, the non-minium phase characteristic and the nonlinearities of the turbine [START_REF] Chaudhry | Applied Hydraulic Transients[END_REF] [START_REF] Kundur | Power System Stability and Control[END_REF]. Therefore mathematical models have been developed to represent these two aspects and [START_REF]Hydraulic turbine and turbine control models for system dynamic studies[END_REF] is the result of a working group on turbine modeling, where the authors develop several models of nonlinear ideal turbines along with models of pipes taking into account traveling waves or not through the use of hyperbolic functions.

Several strategies have been developed to design a controller able to control the nonlinear turbine, such as PID gain scheduling [START_REF] Orelind | Optimal PID gain schedule for hydrogenerators -design and application[END_REF] which uses linearized characteristic curves for the turbine model, robust control [START_REF] Jiang | Design of an optimal robust governor for hydraulic turbine generating units[END_REF] with the linear ideal model described in [START_REF]Hydraulic turbine and turbine control models for system dynamic studies[END_REF] and adaptive control [START_REF] Malik | Design of a robust adaptive controller for a water turbine governing system[END_REF].

Predictive control has been developed on a six-units power plant in [START_REF] Muoz-Hernandez | MIMO generalized predictive control for a hydroelectric power station[END_REF] where the authors have developed a linear model of the hydraulic turbines sharing a common penstock and a linear model of the electric generator. They compare their proposed algorithm with the classical PI controller in terms of cross-coupling between the units.

One-dimensional dynamic models of pressure perturbations occurring at partial or full load exist in the hydraulic literature [START_REF] Alligne | Cavitation surge modelling in francis turbine draft tube[END_REF], [START_REF] Doerfler | Evaluating 1D models for vortex-induced pulsation in Francis turbines[END_REF] which rely on two parameters characterizing the vortices. A more simpler approach is to represent the pressure oscillations as an external perturbation as in [START_REF] Nicolet | Methodology for risk assessment of part load resonance in Francis turbine power plant[END_REF]. The latter will be used here.

Some authors have considered the partial vortex effects on the active power. In [START_REF] Padoan | Dynamical behavior comparison between variable speed and synchronous machines with PSS[END_REF], they consider torque oscillations at partial load and tune the power system stabilizer (PSS) of a the synchronous machine to damp the active power oscillations. In a similar manner, the author in [START_REF] Mallik | Reducing the effect of vortex rope oscillation on hydro units by power system stabilizer designed using genetic algorithm[END_REF] tunes the PSS of the unit to damp the effect of the vortex rope oscillations considering that there is an external perturbation acting on the mechanical torque.

B. Paper Contribution

In this paper, we develop a reduced order mathematical model of a hydroelectric unit subject to off-design pressure oscillations. We consider that an external perturbation is entering the system in the draft tube of the turbine, with the perturbation modeled as a sine function of known frequency. The hydraulic sub-system is represented by a linear model taking into account travelling pressure waves into the pipes using the finite elements method and the turbine is modeled by a linearization of its characteristic curves. After an order reduction, this hydraulic model is combined with a linear third order model of a synchronous generator connected to an infinite bus. We finally present simulation results of a controller designed with this model.

II. NOTATIONS AND DEFINITIONS

III. SYSTEM DESCRIPTION AND MATHEMATICAL MODEL

A. System Physical Description

We consider a hydroelectric unit where a penstock is providing water from an upstream reservoir to a reaction turbine (Francis for example), from which the turbine produces mechanical torque. This mechanical energy is then converted to electricity by a synchronous generator connected to the electrical network.

Mathematical dynamic models for off-design pressure oscillations in the draft tube have been developed in the hydraulic literature [START_REF] Doerfler | Evaluating 1D models for vortex-induced pulsation in Francis turbines[END_REF]. While these models are useful to understand the phenomena, they are too complicated for control design purposes. This is the reason why we consider a simpler approach where the pressure oscillations induced by off-design hydraulic phenomena are considered to be an external perturbation. This external perturbation h w will be mathematically represented by a sine function: where A h is the amplitude of the perturbation in meters, ω h its frequency in rad/s and t the time in seconds. With laboratory and on-site measurements, it is possible to determine ω h . The amplitude A h is difficult to determine because this model of the perturbation is not directly measurable in reality and because it depends on the operation point, it should be treated as an unknown parameter.

h w = A h sin ω h t, (1)

B. Hydraulic Subsystem Mathematical Model

Turbine: The hydraulic turbine mathematical model is made of characteristic curves [START_REF] Chaudhry | Applied Hydraulic Transients[END_REF]. These charts are built from laboratory measurements and consist of nonlinear mappings:

q t = f 1 (h n , ω r , α) T m = f 2 (h n , ω r , α).
(

) 2
Where q t , h n , ω r , α and T m are given in Table I and Table II. We can linearize these equations around an operating point (q t0 , h n0 , ω r0 , α 0 , t m0 ) and we get

∆q t = α 1 ∆h n + α 2 ∆ω r + α 3 ∆α ∆T m = β 1 ∆h n + β 2 ∆ω r + β 3 ∆α, (3)
where the α i , β i are the tangents of curve at the considered operating point.

Pipes: The dynamics of the conduits is described by the hyperbolic partial differential equations:

a 2 ∂q(x, t) ∂x + gA ∂h(x, t) ∂t = 0 ∂q(x, t) ∂t + gA ∂h(x, t) ∂x + f 2DA q(x, t)|q(x, t)| = 0. (4)
We discretize these equations by using the finite elements method, we now consider that the pipe of length L is made of N b pipe elements of length dx = L/N b . The number of elements N b has to be arbitrarily high to be as close as possible to reality. We can apply (4) on each small element and linearize around the operating point (h 0 , q 0 ). For i = 1, . . . , N b we obtain:

∆q i+1 -∆q i + σ 1 d∆h i+1/2 dt = 0 ∆h i+1/2 -∆h i-1+1/2 + σ 2 d∆q i dt + σ 3 ∆q i = 0, (5)
where

σ 1 = Agdx a 2 , σ 2 = dx Ag , σ 3 = f |q 0 |dx 2gDA 2 . (6)
Finally, the full model of a pipe discretized in N b elements can be expressed by

Λ d dt            ∆q 1 ∆h 1+1/2 ∆q 2 . . . ∆q N b ∆h N b +1/2 ∆q N b +1            + Σ            ∆q 1 ∆h 1+1/2 ∆q 2 . . . ∆q N b ∆h N b +1/2 ∆q N b +1            =            ∆h in 0 0 . . . 0 0 -∆h out            , (7)
where Λ and Σ are defined by

Λ = diag ( 1 2 σ 2 , σ 1 , σ 2 , ..., σ 2 , σ 1 , 1 2 σ 2 ) (8)
Σ =            1 2 σ 3 1 0 -1 0 1 0 0 -1 σ 3 1 0 . . . 0 -1 σ 3 1 0 0 -1 0 1 0 -1 1 2 σ 3            . (9)
Full reduced-order hydraulic subsystem: We use 40 elements for each pipe of the hydraulic system using [START_REF] Malik | Design of a robust adaptive controller for a water turbine governing system[END_REF] to have a good accuracy for the low frequency eigenvalues, and combine these equations with the turbine equations (2). This high order nonlinear model is implemented in the software package Simsen [START_REF] Nicolet | New tool for the simulation of transient phenomena in Francis turbine power plants[END_REF] and will be used later for the numerical simulations.

This model is linearized around the operating point q 0 = 0.5 p.u. where we observe the partial vortex rope. The head deviations at the upstream and downstream reservoirs are neglected and the external perturbation adds a difference in head between the outlet and the inlet of pipe 1 and pipe 2 respectively, see Fig. 2. The hydraulic model obtained has 241 states hence we need a reduced-order model more suitable to control design. We carry out a frequency analysis of the transfer functions from the guide vane opening α and the perturbation h w to the mechanical torque T m . For this purpose numerical values of a physical system are needed, and as an example, the data of a real project developed by Alstom are considered. We also have used the laboratory and on-site measurements at offdesign operation of the turbine necessary to characterize the perturbation. The main specifications of the physical system can be found in Table III.

For the model reduction, we make the assumption that the actuator bandwidth is limited to 5 Hz, and the pressure perturbation induced by the partial load vortex rope has a frequency, ω h = 0.5 Hz in [START_REF] Nicolet | Methodology for risk assessment of part load resonance in Francis turbine power plant[END_REF]. Hence, we keep the poles and zeros of the system in this frequency range. Fig. 3 shows the Bode diagram comparison for the transfer function between the perturbation input h w and the mechanical torque T m .

The resulting state-space hydraulic model of order 7 is represented by

Ẋh = A h X h + B h α + B he ω r + B hw h w Y h = T m = C h X h + D h α ω r , ( 10
)
where X h is the state vector, α the guide vane opening (control input), ω r the rotational frequency, h w the head perturbation from (1) and Y h the output. All these variables are deviations around an operating point.

The matrix entries of A h , B he , B hw , C h and D h can be found in Appendix I.

C. Electro-Mechanical Subsystem Mathematical Model

We model the generator and the electrical network as the well known SMIB (Single Machine Infinite Bus) [START_REF] Kundur | Power System Stability and Control[END_REF]. The third order nonlinear model is linearized around an operating point (P t0 , E t0 , Q t0 ). The resulting linear state-space system is described by

Ẋe = A e X e + B e ∆T m ∆E f d Y e = C e X e . ( 11
)
where the algebraic expressions of the variables are defined as follows: The expressions of the a ij , b ij and c ij constants and of the initial conditions can be found in Appendix II and the numerical values in Appendix I.

X e =   ∆ω r ∆δ ∆Ψ f d   , Y e =     ∆ω r ∆P t ∆Q t ∆E t     , A e =  

D. Full Hydroelectric Mathematical Model

We aggregate the two subsystems together to obtain a state-space model of a hydroelectric unit with a hydraulic perturbation in the draft tube:

ẋ = Ax + Bu + B w h w y = Cx, ( 12
)
where the state vector x is a concatenation of the hydraulic state vector X h and the electric state vector X e , y is the output vector, u is the control input and h w is the hydraulic perturbation input.

x = X h X e , y = Y e , u = α E f d .
The concatenated matrices are defined as follows:

A =          0 0 0 A h B he . . . . . . 0 0 0 b 11 C h 0 • • • 0 Ãe 0 • • • 0          , Ãe = A e +   b 11 D h2 0 0 0 0 0 0 0 0   , B =          0 B h . . . 0 b 11 D h1 0 0 0 0 b 32          , C = 0 4,7 C e .

IV. MODEL SIMULATION

We now compare our developed 10th order model to the full nonlinear model through simulation in closed-loop with the system subject to a pressure perturbation in the draft tube represented by [START_REF] Nicolet | Methodology for risk assessment of part load resonance in Francis turbine power plant[END_REF] with A h = 0.055 p.u. and ω h = π which corresponds to a perturbation frequency of 0.5 Hz. This comparison is done with the classical controllers designed to have both good tracking and regulation performance on the active power for the turbine and the voltage for the generator. The full nonlinear model is implemented in the software package Simsen [START_REF] Nicolet | New tool for the simulation of transient phenomena in Francis turbine power plants[END_REF], which has been validated with on-site measurements to simulate the hydraulic subsystem. The turbine is modeled with the nonlinear hill charts (2) and the hydraulic subsystem is modeled with a high number of pipe elements in each pipe using [START_REF] Malik | Design of a robust adaptive controller for a water turbine governing system[END_REF]. The generator mathematical model is the 7-th order nonlinear model with damper windings along with the second order network model [START_REF] Kundur | Power System Stability and Control[END_REF] with X E = 0.3. The resulting model has an order 130. The same controller topologies are used for both models, the results are displayed in Fig. 4 for the active power and the voltage. The deviations of the reduced model from the full model are displayed in Fig. 5. Wee see that our reduced model is close to the full model in this case.

V. CONCLUSION

We developed a 10th order model of a hydroelectric unit subject to hydraulic perturbations in the draft tube, for example pressure oscillations induced by partial or full load vortex rope. This model captures the dynamics of the water-hammer effet (travelling waves) in the pipes, we used the finite element method in this purpose and the model was reduced by keeping poles and zeros in the frequency region of interest for the control loop. Finally, simulation results were presented of our designed model compared to the full nonlinear model when the system is subject to pressure perturbation in the draft tube. The details of a control algorithm designed with this model will be published in a forthcoming paper.

APPENDIX I NUMERICAL VALUES

A. Hydraulic Subsystem

The hydraulic subsystem is linearized around the operating point T m0 = 0.5 p.u. with the parameters given in Table IV. The numerical values of the matrices of [START_REF] Doerfler | Evaluating 1D models for vortex-induced pulsation in Francis turbines[END_REF] are given here:

       B h =        -0.33 -0.99 0.18 -1.6 1.2 -1.0 -2.8        B he =        0.027 1.1 -0.68 1.9 -1.1 0.92 2.9        B hw =        -0.

B. Electrical Subsystem

The electrical subsystem is linearized around the operating point P t0 = 0.5 p.u., E t0 = 1 p.u. and Q t0 = 0 p.u. which is equivalent to considering the bus voltage E B0 = 1 p.u.. The matrices of ( 11) are expressed below. L adu = 0.918 X ds = 0.277 L ads = 0.796 Laqs = 0.507

L f d = 0.155 R f d = 3.92 × 10 -4 X E = 0.3 R E = 0.03

APPENDIX II ELECTRO-MECHANICAL MODEL

In this section, we define the matrices of the electromechanical state-space model as well as the expressions to calculate the initial conditions of the machine. Table II is the nomenclature of all electro-mechanical variables.

A. Matrix Constants

We develop here the expressions for the matrix constants of (11), [START_REF] Kundur | Power System Stability and Control[END_REF].

a 11 = - K D 2H a 12 = - K 1 2H a 13 = - K 2 2H a 21 = ω 0 = 2πf 0 a 32 = - ω 0 R f d L f d m 1 L ads a 33 = - ω 0 R f d L f d 1 - L ads L f d + m 2 L ads b 11 = 1 2H b 32 = ω 0 R f d L adu c 22 = n 1 (Ψ ad0 + L aqs i d0 -R a i q0 )
-m 1 (Ψ aq0 + L ads i q0 + R a i d0 )

c
X Tq = X E + (L aqs + L l ) = X E + X qs X T d = X E + (L ads + L l ) = X E + X ds D = R 2 T + X Tq X T d R T = R a + R E L ads = 1 1 L ads + 1 L f d

B. Initial Conditions

P t0 , Q t0 and E t0 at the operating point are fixed and the following expressions are used to compute the initial conditions [START_REF] Kundur | Power System Stability and Control[END_REF].

I t = P 2 t + Q 2 t E t Φ = cos -1 P t E t I t δ i = tan -1
X qs I t cos Φ -R a I t sin Φ E t + R a I t cos Φ + X qs I t sin Φ e d0 = E t sin δ i e q0 = E t cos δ i i d0 = I t sin(δ i + Φ) i q0 = I t cos(δ i + Φ)

E Bd0 = e d0 -R E i d0 + X E i q0 E Bq0 = e q0 -R E i q0 -X E i d0 E B0 = E 2 Bd0 + E 2 Bq0 δ 0 = tan -1 E Bd0 E Bq0 i f d0 = e q0 + R a i q0 + L ds i d0 L ads E f d0 = L adu i f d0
Ψ ad0 = L ads (-i d0 + i f d0 ) Ψ aq0 = -L aqs i q0

Fig. 1 .

 1 Fig. 1. Hydroelectric power plant picture with 1) the hydraulic turbine, 2) the turbine draft tube and 3) the synchronous generator.

Fig. 2 .

 2 Fig.2. Hydroelectric system schematic diagram. The hydraulic subsystem consists of an upstream reservoir, a penstock, a pump turbine which draft tube is modeled by two pipes in series with a pressure source (PS), and a downstream reservoir. The turbine provides rotating mechanical energy to the synchronous generator which is connected to the electrical grid through a line impedance.

Fig. 3 .

 3 Fig.3. Bode diagram comparison between the high order and the reduced order models for the transfer function from the perturbation in the draft tube to the mechanical torque.

a 11 a 12 a 13 a 11

 11

Fig. 4 .Fig. 5 .

 45 Fig.[START_REF]Hydraulic turbine and turbine control models for system dynamic studies[END_REF]. Comparison of 10th order model (-) with the 130th order model (--) in closed-loop whith the system subject to a sinusoidal head perturbation of amplitude 4 m and frequency 0.5 Hz from 15s to 55s.

  With the synchronous machine parameter numerical values detailed in TableV.

K 1 =L ads L f d i q0 m 1 =

 11 n 1 (Ψ ad0 + L aqs i d0 ) -m 1 (Ψ aq0 + L ads i q0 ) K 2 = n 2 (Ψ ad0 + L aqs i d0 ) -m 2 (Ψ aq0 + L ads i q0 ) + E B (X Tq sin δ 0 -R T cos δ 0 ) D n 1 = E B (R T sin δ 0 + X T d cos δ 0 ) D m 2 = X Tq D L ads (L ads + L f d ) n 2 = R T D L ads (L ads + L f d )

  23 = n 2 (Ψ ad0 + L aqs i d0 -R a i q0 ) -m 2 (Ψ aq0 + L ads i q0 + R a i d0 ) + L ads L f d i q0 c 32 = m 1 (Ψ ad0 -(L l + L ads )i d0 ) + n 1 (Ψ aq0 -(L l + L aqs )i q0 ) c 33 = m 2 (Ψ ad0 -(L l + L ads )i d0 ) + n 2 (Ψ aq0 -(L l + L aqs )i q0 ) + m 1 + L l n 1 + L aqs n 1 ) + e q0 E t0 (-R a n 1 + L l m 1 + L ads m 1 ) c 43 = e d0 E t0 (-R a m 2 + L l n 2 + L aqs n 2 )

		L ads L f d	i d0
	c 42 =	e d0 E t0 (-R + e q0 E t0 -R

a a n 2 + L l m 2 + L ads 1 L f d -m 2

ACKNOWLEDGMENT

The first author is funded by CIFRE Ph.D. grant n. 2013/1187.

The authors would like to thank Thomas de Colombel and Josselin Guillozet for the enlightening discussions on turbine partial loading pressure oscillations.


published or not. The documents may come


ouverte pluridisciplinaire HAL, est destinée au dépôt

Memory of Voice

Nicole REVEL Langues-Musiques-Sociktbs LMS (UMR 8099) http:/www.vi f. cnrs. fr/lms

Abstract:

In today's world, an outstanding poetic and musical heritage is present in living memories and practices among many Austronesian languages and cultures. In spite of the many studies conducted since World War Ill a great deal remains to be listened to, collected, safeguarded, transcribed, translated, analyzed and comprehended. Modern technologies are now available which can be explored and developed in our respective attempts to do SO.

This paper focuses on the Nusantara region, the Western branch of the Austronesian family, and more particularly on the Philippines, where, since 1992, we have begun to build up a "Philippine Oral Epics Archive", based at Ateneo de Manila University. This tridimensional audio, audio-video and computerized archive will be presented briefly.

A pluridisciplinary approach is employed in order to capture a "mental text", and its elaboration in performance. Specific aspects considered include:

the role of deixis related to action and text processing, particularly in repetitive segments and in the macrostructure of the plot; -the close relationship between words and music; -the projection of time in space in establishing the first manuscript of a sung poetic composition; the referential knowledge encapsulated in the narratives and the teachings they convey, will be successively refered to and illustrated. Finally, epic singing is analyzed as a vocal "skilled activity.

Linguistics, Ethnography, Ethnomusicology and Cognitive Linguistics-Anthropology will contribute to our understanding of this creative process.

Memory of Voice I. A Philippine Oral Epics Archive

A

  tri-dimensional audio, audio -video and computerized archive 11. ' Mental Text ' and its Projection in Performance -The singer oftales and his singing body -Two formulaic styles and their treatment : -Tala-andig Bukidnon parallelisms and their interplay -Lexicometric treatment of a Palawan composition 111. Deixis and the Macrostructure of the Plot an Ihgao hudhud a Palawan tultul a Sama kata-kata IV. ' Mim6sis of Action ' and Memory -Singing Epic as a ' Skilled Activity ' 10-ICAL conference, 2006


Florence Haudin

Xavier Noblin

Yann Bouret

Médéric Argentina

Christophe Raufaste

Bubble dynamics inside an

Keywords: numbers: 47, 55, D-87, 19, rh 82, 40, Ck

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

Bubble growth inside a non-Newtonian fluid mixes numerous fundamental and applied fields. Among them, failure in amorphous solids results from cavitation inside a viscoplastic zone at the end of a crack tip [START_REF] Bouchbinder | Dynamic failure in amorphous solids via a cavitation instability[END_REF]. Cavitation and bubble growth occur inside adhesives under tension [START_REF] Lakrout | Direct Observation of Cavitation and Fibrillation in a Probe Tack Experiment on Model Acrylic Pressure-Sensitive-Adhesives[END_REF][START_REF] Chikina | Cavitation in Adhesives[END_REF], or inside a drying gel [START_REF] Scherer | Cavitation during drying of a gel[END_REF]. Growing a single bubble at the tip of syringe is used as a local probe to characterize the mechanical properties of soft materials and biological tissues [START_REF] Zimberlin | Cavitation rheology for soft materials[END_REF][START_REF] Kundu | Cavitation and fracture behavior of polyacrylamide hydrogels[END_REF][START_REF] Zimberlin | Cavitation rheology of the vitreous : mechanical properties of biological tissue[END_REF]. Bubble nucleation and growth can also be provoked by pulsed lasers for cutting biological tissues [START_REF] Palanker | Dynamics of ArF excimer laser-induced cavitation bubbles in gel surrounded by a liquid medium[END_REF].

The case of an oversaturated fluid containing dissolved gases leads to diffusion-fed bubble growth mechanisms. After fermentation of wheat flour dough [START_REF] Babin | Fast X-ray tomography analysis of bubble growth and foam setting during breadmaking[END_REF][START_REF] Turbin-Orger | Growth and setting of gas bubbles in a viscoelastic matrix imaged by X-ray microtomography : the evolution of cellular structures in fermenting wheat flour dough[END_REF] or the processing of polymer foams [START_REF] Villamizar | Studies on structural foam processing II. Bubble dynamics in foam injection molding[END_REF], the morphologies of the bubbles and of their formed network have a direct impact on the properties of the final product. In geology, it is encountered in the outgassing of volcanic magmas [START_REF] Divoux | Intermittent outgassing through a non-Newtonian fluid[END_REF] and mantle melts [START_REF] Lensky | Bubble nucleation as a trigger for xenolith entrapment in mantle melts[END_REF].

In newtonian fluids, the bubble growth from dissolved gases also happens in multiple situations, and in particular in carbonated beverages. When a champagne bottle is opened, the initially pressurized wine undergoes a drastic pressure decrease : bubble growth is then observed (see * corresponding author : Christophe.Raufaste@unice.fr a review on the topic [START_REF] Liger-Belair | Recent advances in the science of champagne bubbles[END_REF]). The rheological fluid properties are then close to water and the spatial distribution of bubbles, which is directly correlated to nucleation sites, is rapidly influenced by gravity and finally leads to specific bubble columns. Theoretically, Epstein and Plesset [START_REF] Epstein | On the Stability of Gas Bubbles in Liquid-Gas Solutions[END_REF] studied the diffusion-fed growth of a spherical bubble in an oversaturated liquid. Accounting for the liquid-bubble interface advection and the effect of surface tension, they calculated the radial concentration of the dissolved species around the bubble by a competition between the dissolved species diffusion and their outgassing at the liquid-gas interface. Several studies extended the case of Newtonian liquids [START_REF] Barlow | Diffusion of Gas from a Liquid into an Expanding Bubble[END_REF] to more complex rheological behaviors [START_REF] Street | Dynamics of Phase Growth in Viscous, Non-Newtonian Liquids. Initial Stages of Growth[END_REF][START_REF] Patel | Bubble growth in a viscous Newtonian liquid[END_REF][START_REF] Han | Studies on structural foam processing. IV. Bubble growth during mold filling[END_REF][START_REF] Arefmanesh | A numerical study of bubble growth during low pressure structural foam molding process[END_REF].

The models of a single bubble growth deal with the spherical geometry which seems in agreement with ductile-like materials [START_REF] Han | Studies on structural foam processing. IV. Bubble growth during mold filling[END_REF]. This is the case as long as bubbles does not interact. At most, bubbles enter into contact or percolate and can lead to very complex and anisotropic structures [START_REF] Lakrout | Direct Observation of Cavitation and Fibrillation in a Probe Tack Experiment on Model Acrylic Pressure-Sensitive-Adhesives[END_REF][START_REF] Turbin-Orger | Growth and setting of gas bubbles in a viscoelastic matrix imaged by X-ray microtomography : the evolution of cellular structures in fermenting wheat flour dough[END_REF]. It can be noted that the geometry can evolve from the regular sphere to a fracture-like anisotropic shape inside brittle materials when the bubble growth time scale is faster than the stress relaxation time [START_REF] Kundu | Cavitation and fracture behavior of polyacrylamide hydrogels[END_REF][START_REF] Tabuteau | Microscopic Mechanisms of the Brittleness of Viscoelastic Fluids[END_REF].

In this context, we report an experimental study of the diffusion-fed bubble growth inside a hydrogel. Such a gel exhibits both a non-Newtonian behavior and contains a significant amount of dissolved gases at standard pressure-temperature conditions. The supersatura-tion is triggered by a pressure decrease and the hydrogel confinement between two glass plates in a Hele-Shaw geometry enables a direct imaging of the bubbles growth and morphology in a quasi 2D geometry. The originality of this work is related to the concomitant focus on both the growth rate and morphology of the bubbles, and to the study of the onset of the bubble interaction.

As the pressure decreases, bubbles nucleate and start to grow. The growth dynamics is twofold. In a first regime, the bubbles behave independently and the growth rate is only related to the imposed pressure regardless of the bubble size and history. The growth rate is driven by the gas concentration profile in the vicinity of the bubble. It depends on the diffusive properties in the hydrogel bulk and the saturation equilibrium at the hydrogel-bubble interface. This regime is denoted "diluted" since bubbles are too far away from each other to interact. Bubbles morphology is homogeneous and rather isotropic regardless of the rheological properties of the hydrogel in our parameters range. In a second regime, denoted "dense", bubbles interact. This corresponds to gas fractions around 5-10%. The growth rate decreases due to a competition for the dissolved gases and the bubble morphology becomes more and more anisotropic with a significant effect of the hydrogel rheological properties. Bubbles of the same size begin to interact when the center-to-center bubble distance becomes comparable to the bubble size.

The article is organized as follows : Section II is dedicated to the experimental setup. Section III deals with the qualitative observations that are quantitatively described in section IV for the diluted regime and in section VI for the dense regime. A model for the diluted regime is presented in section V. The setup consists in a Hele-Shaw cell filled with a hydrogel and connected to a vacuum pump to drive the pressure decrease (Fig. 1). The cell is built with two glass plates (200 × 200 × 10 mm) clamped on both sides of a 1 mm thick waterproof spacer (in a sake of simplicity, the clamping system is not shown in Fig. 1). The thickness of the spacer was varied between 0.5 and 2 mm, and this spacing did not affect qualitatively the bubble dynamics. In what follows, 1 mm is chosen as a good compromise to get a good contrast and prevent superposition of bubbles in the confining dimension. A hole is drilled through the upper plate to enable the cell filling and the vacuum pump connection.

II. MATERIALS AND METHODS

Our hydrogels are made of distilled water and Laponite R RD, a synthetic clay manufactured by Rockwood. Composition of the hydrogel ranges between 0 and 5% of Laponite in mass. Laponite powder is added in 20 minutes to water under strong stirring to ensure a homogeneous dispersion and prevent flocculation. After 20 minutes of extra-stirring the mixture is still relatively liquid and is poured inside the cell. The higher the Laponite content, the higher the initial gelation of the hydrogel and small bubbles are trapped for the 5% hydrogel. For all compositions, pH measurements give a value around 10, which is coherent with the fact that the Laponite suspension behaves as a buffer solution. The cell is then kept at rest for 24 hours to ensure the complete gelation of the mixture. Concentrations smaller than 2% did not exhibit any gelling behavior. For higher concentrations, the hydrogel sustains its shape and does not flow under gravity. The visco-elasto-plastic properties are measured by a cone plate rheometer (Tab. I). The rigidity of the hydrogel changes significantly : both the yield stress and the elastic modulus vary over a factor 500 between the 3% and 5% compositions.

The pressure decrease is driven by a vacuum pump. It is connected to the cell through a tubing connection system made of a three-way valve and two taps (Fig. 1). The valve enables to start and stop the pressure decrease process. The first tap connects the cell and the pump while the second tap connects the cell with a closed and large reservoir initially at room pressure. The second tap generates a leak and decreases the efficiency of the vacuum pump. By configuring both taps, both the pressure decrease rate and the final pressure value are prescribed. The latter is always chosen larger than 30 mbar to prevent the water vaporization. A pressure sensor and an I/O card record continuously the pressure evolution in the system during the experiment. This pressure corresponds to those imposed at the free interface of the hydrogel. Note that the cell is never completely filled (Fig. 1) so that the hydrogel does not enter the tubing connection system when it expands due to bubble growth.

Finally, the cell is placed on a transmitted light illumination system and the dynamics inside the cell is recorded with a digital camera (Fig. 1). Each bubble is characterized by its projected area A based on the outermost contour. The growth rate is given by dA/dt.

III. QUALITATIVE OBSERVATIONS

The main features of the hydrogel evolution are represented in Fig. 2 for a 4% hydrogel. These concern bubble nucleation, growth, interaction and finally percolation of gaseous domains. The vacuum pump reduces the pressure from 1000 mbar to 50 mbar in typically 10 minutes. Once bubbles appear, they drift toward the pumping direction to account for the gas generation and the whole expansion of the hydrogel. Bubbles do not move inside the hydrogel matrix itself when gelation is strong enough. This occurs for composition larger than 3%. We recorded a temperature decrease, at most 10 • C, inside the hydrogel during an experiment.

Around 500 mbar, nucleations start (Fig. 2a) and bubbles appear continuously leading to typical densities around a few bubbles/cm 2 . While they are significantly smaller than the millimeter scale, it is difficult to observe a deviation from circularity. When their sizes approach the millimeter scale (Fig. 2bc), they develop anisotropy. Fewer supplementary nucleation events are observed when the system enters the dense regime (Sec. VI).

When bubbles are far from each other, typically for bubbles separated from their neighbors within a few bubble radii (Sec. VI), they tend to elongate like ellipses in the direction perpendicular to the pumping direction. Their shape and growth rate are not influenced by their neighbors. This regime is called "diluted". Note that some isolated bubbles can be slightly oriented toward the pumping direction. In this regime, the smaller the pressure, the larger the growth rate of the bubbles. The higher the amount of the gelling agent, the more anisotropic the bubbles (Fig. 3).

When bubbles are close to each other (Fig. 2cd), they tend to grow in the direction of their neighbors. This characterizes the "dense" regime and the beginning of the interaction between bubbles. In this regime, the bubbles growth rate decreases in comparison with the diluted regime. The elliptical shapes develop very curved parts in the direction of the closest neighbors. Again, the higher the amount of the gelling agent, the more anisotropic the bubbles (Fig. 3).

Finally, they might enter into contact during their evolution and form thin films between them. If two bubbles in contact have a very different size, a coarsening-like behavior is observed and the small bubble decreases its size for the benefit of the largest one due to gas transfer through the thin film (Fig. 2d). Eventually, the breakage of this films result in the formation of a connected network inside the hydrogel (Fig. 2e). If the latter percolates and connects with the free interface, gas is released outside the hydrogel in a few tenths of a second. After that, the hydrogels heals and cavities close. Some smaller cavities are still present at the end where former large cavities were found (Fig. 2f)

Changing the pressure decrease rate reveals some specific behaviors of our 2D geometry. The pressure decrease rate is characterized by its average value between 250 and 150 mbar, since the pressure evolution is not linear with time. The effect of three different pressure rates are shown in Fig. 4. For the lowest pressure decrease (-0.7 mbar.s -1 ), the bubbles evolution is homogeneous whatever the position inside the cell. For the intermediate case (-1.4 mbar.s -1 ), a gradient of size appears through the cell. The closer to the free interface the bubbles, the larger their areas. This effect is even clearer and more dramatic for the higher pressure decrease rate (-30 mbar.s -1 ). In that case, bubbles close to the interface percolate very quickly while bubbles in bulk remain very small. This effect may be explained by the pressure drop inside the hydrogel when the material is expanding too quickly. In what follows, imposed pressure rates will always be relatively slow, ranging between -0.33 and -0.81 mbar.s -1 , such that the internal pressure could be considered as homogeneous inside the hydrogel. In that limit, we ensure that the hydrogel pressure equals the pressure at the free interface imposed by the pump and measured by our pressure sensor.

mm

Figure 2. Image sequence of bubbles growth inside a 4% hydrogel with 100 s between two images and a pressure rate -0.67 mbar.s -1 . Bubbles are advected toward the pumping direction, here on the right hand side, due to the global expansion of the hydrogel triggered by the nucleation and growth of bubbles.

mm mm

IV. DILUTED REGIME -EXPERIMENTS

The diluted regime is characterized by a bubble dynamics that is not influenced by the presence of the neighboring bubbles. Experiments are performed on several hydrogels and for various pressure evolutions (Tab. II). For each of them, we track the area of the bubbles. This area corresponds to the darker and outer part of the projected area of the bubble. Data are recorded as soon as the first bubble becomes measurable (area around 0.5 mm 2 ) and can be tracked by an automatized procedure. For each experiment, a few tenth of bubbles inside the diluted regime are selected and tracked until they reach the crossover between the diluted and dense regimes (Sec. VI). Three sets of experiments are presented. The first one consists of a 3% hydrogel subjected to either a constant pressure or a pressure decrease. The second one consists of the same 3% hydrogel subjected to three different pressure decreases. The third one consists of three different hydrogels (3, 4 and 5%) subjected to the same pressure decrease.

Set 1 deals with the comparison between a constant pressure experiment and a decreasing pressure experiment both performed with the same hydrogel composition 3% (Fig. 5). For the constant pressure, the area of each bubble increases almost linearly in time. Therefore the growth rate is constant, around 0.0008 mm 2 .s -1 . It seems to be the same for all bubbles even with different areas, due to different nucleation times. For the decreasing pressure experiment, the growth rate seems to increase with time leading to parabola-like curves for the areas. Given that bubbles have nucleated at different times, the later the bubble is nucleated, the quicker the growth rate. Plotting the area as a function of pressure gives a significant dispersion of the data. But we observed that the data of all bubbles collapse while plotting the growth rate as a function of the pressure. This means that the growth rate is homogeneous in space for a given pressure whatever the bubble size.

Set 2 deals with the comparison between three decreasing pressure experiments with the same hydrogel composition 3% (Fig. 6). Different pressure rate are used with a factor 2.5 between the highest and the lowest (Tab. II). For a given experiment, the data confirm that plotting the area rate dA/dt as a function of the pressure P allows to collapse all the experimental points on a single curve. The three curves are very similar and suggest that the bubble dynamics in the diluted regime is quasistatic in our experimental range. The small differences between them do not enable to discriminate the role or the in- For the constant pressure experiment, all bubbles grow with a quasi constant and homogeneous growth rate. Its value is centered around 0.8×10 -9 m 2 .s -1 . For the decreasing pressure experiment, the bubbles do not nucleate simultaneously and each of them follows its own area-time or area-pressure trajectory. For a given time, the growth rate is the same for all bubbles. It increases as the pressure decreases and when plotting the growth rate as a function of the imposed pressure, all the data points collapse onto the same trend.

fluence of the pressure rate (the data of the intermediate pressure rate are not found in between the ones of the two extreme pressure rates) and the effect of this parameter can not be resolved with the precision of the setup. The constant pressure experiment performed in set 1, which can be seen as an extremely low pressure rate experiment, gives a data range that is very close to the trends obtained with the decreasing pressure experiments (not plotted in Fig. 6 for simplicity), which is consistent with the quasistatic regime hypothesis.

Set 3 deals with the comparison between three different hydrogels (compositions 3, 4 and 5 %) subjected to the same pressure decrease (Fig. 6). It is worth noting that the 5% hydrogel contains initially small bubbles (Sec II). Therefore their initial areas are different from zero. Again, plotting dA/dt as a function of P seems to be the best choice to display a single trend for a given expe-riment. We notice that the three curves do not exactly collapse but are relatively similar while the rheological properties of the three hydrogels are rather different (up to a factor 500 for the elastic properties, Tab. I, while we have at most a factor 2 for the area rate at a given pressure). Again, the effect of this parameter can not be resolved with the precision of our setup since the data of the 4% hydrogel are not found in between the ones of the 3 and 5% hydrogels. To conclude, the composition and the rheological properties do not seem to play a significant role to account for the bubble growth rate in the diluted regime. As observed before they have a small influence on the morphology in the diluted regime, since bubbles are much more anisotropic for the highest concentrations. For a given experiment, all data points collapse onto a single trend when plotting the growth rate as a function of the imposed pressure. By varying the pressure rate or the hydrogel composition, trends are maintained. The small differences observed do not enable to discriminate the impact of these two parameters on the evolution of the growth rate (see text).

V. DILUTED REGIME -MODEL

The experiments described previously suggest that the bubble growth rate is a function only of the pressure and does not depend neither on the bubble area nor on the pressure rate nor on the rheological properties of the hydrogel : the smaller the pressure, the larger the growth rate. We propose a model to account for these observations. It assumes that the bubble growth is mainly driven by the diffusion of the dissolved gas through the hydrogel and its outgassing at the reactive-advected hydrogel-gas interface.

We first assume that the pressure is uniform through the whole hydrogel and inside the gas bubbles. As observed previously, the pressure loss through the hydrogel is small when the pressure rate is small enough. By taking a surface tension of order 10 -2 N.m -1 and a bubble radius around 0.1 -1 mm, the Laplace pressure at the hydrogel-bubble interface is found around 10-100 Pa and is negligible compared to the absolute pressure inside the hydrogel. Following these considerations the pressure P (t), which is the control parameter, is uniform and is the same inside the hydrogel and inside the bubbles. The temperature change is around 3% so that we will assume a constant temperature in the model.

Given the Hele-Shaw geometry, we expect a 2D-3D geometry crossover when the bubble radius reaches 0.5 mm or when the projected area reaches A c = 0.8 mm 2 equivalently. In most of the data, A < A c and the 3D problem has to be considered. Nevertheless, bubbles might nucleate close to one of the glass plates and trigger specific confining effects. As seen below, the 3D approach seems to capture the main features of the growth dynamics, even though some corrections could be introduced by considering the confinement, which is out of the scope of our study. In the diluted regime, all bubbles be-have independently and we consider only the case of a single bubble with the spherical geometry. The bubble is centered at r = 0 in the spherical coordinates reference frame. We note R its radius, V = 4πR 3 /3 its volume and A = πR 2 its projected area.

The evolution of the dissolved gas concentration c(r, t) (units of mol.m -3 ) at a distance r > R in the hydrogel is given by the advection-diffusion equation :

∂c ∂t + u • ∇c = D∆c ( 1
)
where D is the diffusion coefficient of the dissolved gas in the hydrogel and u is the velocity produced in the hydrogel by bubble growth. The initial gas concentration is noted c 0 . It corresponds to the saturation concentration in equilibrium with the initial pressure P 0 (here the room pressure for our experiments, approximated by 1000 mbar). Boundary conditions are c(r, 0) = c 0 for r > R(0) and c(r, t) = c 0 for r → ∞ and t > 0. Due to the outgassing at the hydrogel-bubble interface, the concentration at r = R is the saturation concentration in equilibrium with the pressure inside the bubble (note that it corresponds to the imposed pressure P that is uniform through the whole hydrogel). We note c(R(t), t) = c s (t) for t > 0. Solving the advection-diffusion equation is not an easy task. Following Epstein and Plesset [START_REF] Epstein | On the Stability of Gas Bubbles in Liquid-Gas Solutions[END_REF], we can get a reasonable physical approximation by neglecting first the advection term, solving the diffusion equation and make a coordinate change in a second step to account for the moving boundary condition. In that case, the concentration gradient at the moving boundary is written :

∂c ∂r r=R = c 0 -c s R 1 + R √ πDt (2)
As suggested in the same article, we have checked that R/ √ πDt << 1 (this factor equals 0.25 at most in our experiments -data not shown). This supports the physical approximation made above and shows that R is the relevant length scale of the spatial concentration variations around a bubble.

Gas D H D m 2 s -1 mol.Pa -1 .m -3 m 2 .s -1 N2 1.9 × 10 -9 6.1 × 10 -6 4.6 × 10 -12 O2 2.1 × 10 -9 1.3 × 10 -5 1.1 × 10 -11 Ar 2.0 × 10 -9 1.4 × 10 -5 1.1 × 10 -11 CO2 1.9 × 10 -9 3.4 × 10 -4 2.6 × 10 -10
Table III. Diffusion coefficient D, Henry's constant H and the constant D found in the model for nitrogen, oxygen, argon and carbon dioxide gases in water solution at 25 • C [START_REF] Cussler | Diffusion : Mass Transfer in Fluid Systems[END_REF][START_REF] Sander | Compilation of Henry's law constants (version 4.0) for water as solvent[END_REF]. For the calculations of D, RT is taken equal to 2500 Pa.m 3 .mol -1 .

The amount of substance rate entering into the bubble is given by

dn dt = 4πR 2 D ∂c ∂r r=R ≃ 4πR 2 D c 0 -c s R (3)
where n is the amount of substance (units of mol) inside the bubble. At the same time, the amount of substance balance equation leads to

dn dt = 4πR 2 n V dR dt = 4πR 2 P RT dR dt (4)
where we take the ideal gas equation P V = nRT inside the bubble and assume dn/dV ≃ n/V or equivalently that the pressure is a slowly decreasing function of time and the temperature is constant. From the former equations we deduce

R dR dt = DRT c 0 -c s P = DRT H P 0 -P P (5)
where c 0 = HP 0 and c(R(t), t) = HP (t) are given by the Henry's law and its solubility constant H. In terms of the projected area A = πR 2 , we write

dA dt = D P 0 -P P (6)
with D = DRT H 2π the relevant parameter. Note that D has the dimension of a diffusion coefficient as well. This relation is tested for every experiment with P 0 = 1000 mbar. For a given experiment, all data points are in agreement with Eq. 6 and an experimental measurement of the proportionality factor D is obtained by a linear fit (Fig. 7). Results are summarized in Tab. II. values are comparable and range between 1.9 and 3.8 × 10 -10 m 2 .s -1 . The values of the diffusion coefficients and Henry's constants of air gases dissolved in a water solution at room pressure and temperature are summarized in Tab. III. We observe that carbon dioxide has the largest value for D and should correspond to the fastest outgassing gas in a water solution. Its theoretical value 2.6 × 10 -10 m 2 .s -1 is in very good agreement with the values found experimentally with our hydrogels (Fig. 8) and shows that carbon dioxide is the gas at play in our experiments as well. Small differences in the value of D are observed for the same hydrogel and the three decreasing pressure rates. As expected the pressure rate is not a significant parameter. It is difficult to conclude if the slight differences are linked to the dispersion of the data or to a slight sensitivity to the experimental protocol . It might be that the hydrogel composition plays a little role on the coefficient D as well since 4 and 5% hydrogels have smaller values than 3% hydrogels (Tab II). Possibly this could come from the deviation from sphericity assumed in the model or slight changes in the value of the diffusion coefficient D due to a change in the chemical composition. At this stage, with the dispersion of the data, it is not possible to draw strong conclusions about this effect. Modeling the deviation from sphericity observed in 4 and 5% hydrogels inside the diluted regime would require to account for the hydrogel rheology and to calculate the stress, strain and strain rate tensors around the bubbles. The larger bubble growth in the direction perpendicular to the pulling direction is redolent of the behavior of a soft inclusion inside an elastic material under uniaxial tension [START_REF] Timoshenko | Theory of elasticity[END_REF] : the whole visco-elasto-plastic behavior of the material should be accounted for to accurately model this effect.

VI. DENSE REGIME

When bubbles are too close, they begin to interact and enter into what we define the dense regime. The interaction is two-fold and has characteristics both on the bubble growth dynamics and morphologies. First, bubbles compete for the same dissolved gas and the concentration field calculated previously for the case of isolated bubbles does not hold anymore. As a consequence, the growth rate decreases in comparison to what is expected in the diluted regime (data not shown). Second, the stress fields in the hydrogel around bubbles overlap, resulting in a deviation from the elliptical shape observed in the diluted regime. We expect that a change in the bubble morphology might also alter the bubble growth rate as it would modify the dissolved gas concentration field.

The effect of the interaction on the bubble morphology is characterized by the deformation and the growth of very curved parts toward the position of the neighboring bubble (Fig. 9). To perform quantitative measurements, we have considered pairs of bubbles only with the following criteria : 1) they are far enough from other bubbles so that we can assume a two bodies interaction and neglect the effect of surrounding bubbles -2) their areas are relatively similar at least inside the diluted regime. For each pair of bubbles and for each time step, we have measured the radii from the bubble centers, r 1 (θ) and r 2 (θ) respectively, as a function of the angular position θ. For each of them, 360 values were taken. < r 1 > and < r 2 > denote the angular average values. For each time step, the morphology correlation function C(α, t) is defined as the dimensionless function :

C(α, t) = 1 360 θ=360 o θ=1 o (r 1 (θ)-< r 1 >) < r 1 > × (r 2 (θ + α)-< r 2 >) < r 2 > . (7)
A typical angle-time diagram is given in Fig. 10a. It exhibits a band-like feature with maximal and minimal values. For times shorter than 185 s, the maxima are centered around 0 deg and 180 deg, and the minima are found in between. Both maxima and minima are more and more exacerbated as time proceeds and bubbles grow. This marks an increase of the bubble anisotropy. The values of the maxima, found at α = 0 deg and 180 deg, are similar (Fig. 10b). These features are characteristics of the diluted regime and the ellipse-like geometry of the bubbles. Above 185 s, the features change significantly and the difference between maxima and minima decreases. The trends of C for both 0 deg and 180 deg are not similar anymore : the bubbles correlate now at most when oriented in opposite directions, pointing toward each other. This marks the onset of the dense regime. The bands in the angle-time diagram are not horizontal anymore and reflect the evolution of interacting bubbles into more complex geometries.

In Fig. 11, the center-to-center distance ℓ cc between the two bubbles is plotted as a function of their radius R at the onset of the dense regime. Based on the morphological criterion, we have studied the effect of the composition. For a given hydrogel, the higher the area of the bubbles, the larger the distance ℓ cc for the crossover between the diluted and dense regimes. The correlation does not seem to depend significantly on the hydrogel composition and rheological properties. The data are rather dispersed but a linear interpolation gives ℓ cc ≃ 4.3R. It means that the bubbles begin to interact at a distance comparable to the bubble size regardless of the hydrogel composition. A model of the bubble interaction is still missing to account for the observations.

VII. CONCLUSION

The decreasing pressure experiment proposed in this study has provided insights about the bubble dynamics inside an outgassing hydrogel. Carbon dioxide is the gas at play and such a study should help to understand some aspects of carbon dioxide sequestration in non-Newtonian fluids. We have focused on both the growth rate and the morphology of the bubbles and have identified a crossover between a diluted regime of isolated bubbles and a dense regime where bubbles interact. Other rheological behaviors should be studied in the future and performing the study to smaller length scales should exhibit surface tension effects. Models should also be derived to account for the gas concentration and stress fields overlap in the dense regime.

Figure 1 .

 1 Figure 1. Representation of the experimental setup.

Figure 3 .

 3 Figure 3. Image sequence of bubbles growth : (a-d) inside a 3% hydrogel with 130 s between two images and a pressure rate -0.30 mbar.s -1 , (e-h) inside a 5% hydrogel with 300 s between two images and a pressure rate -0.45 mbar/s. Pumping from the right side.

Figure 4 .

 4 Figure 4. a) Imposed pressure for three different pressure decrease rates : -30, -1.4 and -0.7 mbar.s -1 respectively. b) Experimental snapshots obtained at the times t1, t2, ..., t6. The hydrogel concentration is 4%. The observation field is 2.6 cm ×3.3 cm large.

Figure 5 .

 5 Figure 5. Diluted regime -Set 1. Constant pressure ( ) and decreasing pressure (•) experiments with the same hydrogel composition 3%. For both experiments, several bubbles are tracked and their respective data points are connected by solid lines. Pressure as a function of time (a), area as a function of time (b), area as a function of pressure (c) and area growth rate as a function of pressure (d).For the constant pressure experiment, all bubbles grow with a quasi constant and homogeneous growth rate. Its value is centered around 0.8×10 -9 m 2 .s -1 . For the decreasing pressure experiment, the bubbles do not nucleate simultaneously and each of them follows its own area-time or area-pressure trajectory. For a given time, the growth rate is the same for all bubbles. It increases as the pressure decreases and when plotting the growth rate as a function of the imposed pressure, all the data points collapse onto the same trend.

Figure 6 .

 6 Figure 6. Diluted regime -Sets 2 and 3. (a-b) Three decreasing pressure experiments at different pressure rates (-0.33 (△), -0.45 (•) and -0.81 (⋄) mbar.s -1 ) with the same hydrogel composition 3% : pressure as a function of time (a) and area rate as a function of pressure (b).(c-d) Three hydrogel compositions (3% (•), 4% ( * ) and 5% (+)) with the same pressure rate -0.45 mbar.s -1 : pressure as a function of time (c) and area growth rate as a function of pressure (d). For a given experiment, all data points collapse onto a single trend when plotting the growth rate as a function of the imposed pressure. By varying the pressure rate or the hydrogel composition, trends are maintained. The small differences observed do not enable to discriminate the impact of these two parameters on the evolution of the growth rate (see text).

2 .s - 1 )Figure 7 .

 217 Figure 7. Area rate dA/dt vs (P0 -P )/P . For simplicity, only four sets of data are plotted (Tab. II). All time-steps for every bubble of each experiment are represented. Measurements are interpolated by straight lines modeled by Eq. 6 with P0 = 1000 mbar and the best fits for the parameter D.

Figure 8 .

 8 Figure 8. Model parameter D for gases in the air composition (black disks and Tab. III) and measured in the experiments (see Tab. II for symbols).

Figure 9 .

 9 Figure 9. Experimental snapshots of two close-neighbor growing bubbles : a) t=50 s, b) t=100 s, c) t=150 s, d) t=200 s, e) t=250 s, f) t=300 s, g) t=350 s and h) t=400 s. Outer contours, bubble radii for given angles and the center-to-center distance ℓcc are indicated. Very curved parts begin to grow on the contour of the bubbles between t=150 and 200 s ; this marks the onset of the bubble interaction. The hydrogel concentration is 4% and the pressure is constant, 100 mbar. The observation field is 2.5 mm×4.6 mm large.

Figure 10 .

 10 Figure 10. Morphology correlation measurements for the two bubbles of Fig. 9. a) Angle-time diagram of the correlation function C(α, t). b) C versus time for two specific values of the angle α : 0 deg (red circles) and 180 deg (black triangles) respectively. The crossover is observed around 185 s (vertical dashed line).

Figure 11 .

 11 Figure 11. Center-to-center distance ℓcc as a function of the bubble radius R for pairs of bubbles at the crossover between the diluted and dense regimes. The criterion is based either on the bubble morphology for hydrogels of composition 3% (•), 4% ( * ) and 5% (+). Error bars are plotted in light grey and are inferred from the area difference between the two interacting bubbles. The dashed line represents ℓcc = 4.3R.

Table I .

 I Rheological properties of the hydrogels measured by a cone plate rheometer (angle 1 deg). The Young modulus E is determined by oscillatory shear measurements at 5 Hz. The yield stress σY and the viscosity η are obtained by fitting the stress-shear rate curve of continuous shear experiments by a Bingham fluid model.

	hydrogel elastic modulus E yield stress σY viscosity η
	composition	(Pa)	(Pa)	(Pa.s)
	3%	2 ± 2	0.2 ± 0.2	0.025 ± 0.005
	4%	500 ± 50	50 ± 15	0.012 ± 0.002
	5%	1090 ± 40	115 ± 15	0.010 ± 0.003

Table II

 II

. Experiments parameters and measurements : hydrogel composition, set(s) of experiments, pressure characteristics, symbols, model parameter D (Eq. 6).

ACKNOWLEDGMENTS

This project was supported by the "Agence Nationale de la Recherche" (CAVISOFT : ANR-2010-JCJC-0407 01).