Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France
This work brings new insight to the question of heat transfer in **near**–**critical** **fluids** under Earth gravity conditions. The interplay between buoyant convection and ther- moacoustic heat transfer (piston effect) is investigated in a two-dimensional non- insulated cavity containing a local heat source, to reproduce the conditions used in recent experiments. The results were obtained by means of a finite-volume numerical code solving the Navier–Stokes equations written for a low-heat-diffusing **near**-**critical** van der Waals fluid. T hey s how t hat h ydrodynamics g reatly a ffects thermoacoustics in the vicinity of the upper thermostated wall, leading to a rather singular heat transfer mechanism. Heat losses through this wall govern a cooling piston effect. Thus, the thermal plume rising from the heat source triggers a strong enhancement of the cooling piston effect when it strikes the middle of the top boundary. During the spreading of the thermal plume, the cooling piston effect drives a rapid thermal quasi-equilibrium in the bulk fluid s ince i t i s f urther e nhanced s o a s t o b alance the heating piston effect generated by the heat source. Then, homogeneous fluid heating is cancelled and the bulk temperature stops increasing. Moreover, diffusive and con- vective heat transfers into the bulk are very weak in such a low-heat-diffusing fluid. Thus, even though a steady state is not obtained owing to the strong and seemingly continuous instabilities present in the flow, the bulk temperature is expected to remain quasi-constant. Comparisons performed with a supercritical fluid at initial conditions further from the **critical** point show that this thermalization process is peculiar to **near**-**critical** **fluids**. E ven e nhanced b y t he t hermal p lume, t he c ooling p iston effect does not balance the heating piston effect. Thus, overall piston-effect heating lasts much longer, while convection and diffusion progressively affect the thermal field much more significantly. Ultimately, a classical two-roll convective-diffusive structure is obtained in a perfect gas, without thermoacoustic heat transfer playing any role.

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Rapid Thermal Relaxation in **Near**-**Critical** **Fluids** and **Critical** Speeding Up: Discrepancies Caused by Boundary Effects
Arnaud Jounet, 1 Bernard Zappoli, 1,2 and Abdelkader Mojtabi 1
1 Institut de Mécanique des Fluides de Toulouse, UMR 5502 CNRS-INP 兾UPS, UFR MIG, Université Paul Sabatier,

T = 1 + l
.
Figure 1. The centre-heated two-dimensional square cavity.
et al. (1996) have shown that the PE was not destroyed by convection in another heating configuration. Thus, something had to be clarified concerning the effects of the thermal boundary conditions and of the flow on the PE thermalization. In the present study, our numerical code is adapted to approach as far as possible the experimental configuration. Beside the explanation of the specific mechanism involved in this experiment, this paper sheds new light on the general question of heat transfer in **near**-**critical** **fluids** under 1 g conditions. Comparisons with the structure of the flow and the thermal field obtained for a normally compressible fluid (perfect gas) emphasize the present results. Even though buoyant convection in the **near**-**critical** fluid does not contribute directly to bulk heating, which remains driven by the PE, its interplay with diffusion in the ‘cold’ boundary layer formed at the top isothermal boundary, in which a cooling PE originates, is found to significantly accelerate thermal equilibrium in the bulk fluid. This equilibrium is found to result from a balance between the two competing PEs. Additional calculations show that this mechanism is not witnessed further from the **critical** point.

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Abstract Within an averaging approach, the governing equations and effective boundary conditions describing both the average and pulsation motion of a near-critical fluid subjected to hi[r]

decreases (ﬁgure 2b).
3. Isothermal cavity under weightlessness
The most interesting result obtained in the preceding § 2 is a possibility of inducing the average temperature variations by vibration, which was shown to result in convective ﬂows even in an isothermal container. Here we examine problems where the average motion is induced by this **near**-**critical** mechanism in zero gravity conditions. Another aim of the section is to compare the results obtained using the system of averaged equations with the results by Jounet et al. (2000) obtained by direct numerical simulations using the Navier–Stokes equations for a compressible medium. We consider an isothermal container ﬁlled with a **near**-**critical** ﬂuid under microgravity conditions. The container undergoes translational vibrations with linear polarization. The pulsation velocity (the leading term of the pulstational velocity of the ﬂuid equals the velocity of the container, see Part 1) may be represented as

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5. Discussion and conclusions
In Part 1 and here we have examined the inﬂuence of vibration and gravity on the behaviour of a single-phase **near**-**critical** ﬂuid. The main results are two closed theoretical models of thermal vibrational convection for two diﬀerent types of imposed heating, and a model for one exceptional case of isothermal boundaries and weightless conditions. Anomalously high compressibility and heat conductivity of a ﬂuid result in qualitative diﬀerences of the equations from the standard theory (Gershuni & Lyubimov 1998).

In compressible ﬂuids, additional sources of motion under vibrations can originate in bulk phases via the bulk pressure gradient, thus provoking non-solenoidal motion in thermally homogeneous ﬂuids. In hyper-compressible **near**-**critical** ﬂuids, thermoacoustic coupling in the dissipative layers may change the bulk boundary conditions from those simply given by the presence of rigid walls, and also generate ﬂuid motion. Carles & Zappoli (1995) and Jounet et al. (2000a) showed that extremely strong ﬂuid expansion or contraction in thermal adaptation layers is able to generate signiﬁcant boundary-layer-like ﬂows. This changes the dynamic boundary conditions of the bulk momentum equation and may drastically modify the bulk ﬂow structure. These thermomechanical couplings are due to the diverging compressibility of **near**- **critical** ﬂuids and more generally to the **critical** anomalies of the transport parameters. These anomalies result in unexpected behaviour of **near**-**critical** ﬂuids such as the existence of a fourth heat transfer mechanism of a thermoacoustic nature, named the piston eﬀect, which allows heat to be transferred much faster than by diﬀusion or convection (see Boukari et al. 1990; Onuki, Hao & Ferrell 1990; Zappoli et al. 1990). The piston eﬀect was recently shown to strongly inﬂuence the hydrodynamics of **near**-**critical** ﬂuids (see Garrabos et al. 1998; Polezhaev, Emelianov & Gorbunov 1998).

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of a circular region with a well-defined border. In this region, fringes are distorted and the fringe spacing is smaller than in the bulk region. By decreasing t[r]

We should emphasize the diﬀerence between the problem examined here and the works by Zappoli et al. (1996) and Polezhaev & Soboleva (2000), where the behaviour of **near**-**critical** ﬂuid ﬁlling a square cavity was also studied. The main diﬀerence lies in the temperature boundary conditions. In the works mentioned, the temperature at the boundary is non-stationary. The authors are interested in transitional mechanisms taking place after heat impulse on a boundary. We, on the other hand, focus on the ﬂows developing on long time scales (much longer than the typical scale of the piston eﬀect) when all transitional processes have ended. The problem considered in this section is a classical problem of convective theory with typical boundary conditions (see Gershuni & Zhukhovitsky 1976), but with a non-classical ﬂuid model.

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Abstract Within an averaging approach, the governing equations and effective boundary conditions describing both the average and pulsation motion of a near-critical fluid subjected to hi[r]

The response of a **near**-**critical** fluid to low-frequency vibrations is investigated by means of numerical simulations. Its characteristics are first established by one-dimensional analysis. It is shown that the strong thermo-mechanical coupling occurring in the boundary layers tends to make the fluid oscillate homogeneously at low frequencies, and with a larger amplitude than in a normal gas. The numerical results obtained in this first part are found to confirm earlier predictions made in pioneering theoretical work. Then, the study is extended to a two-dimensional configuration. In a square cavity, the wall shear stresses developing along the longitudinal boundaries do not affect the one-dimensional regime, since the viscous layer present in these areas behaves like the Stokes boundary layer. By contrast, thermostatting these boundaries, like the others, generates local curvature of the stream lines. The fluid response to the homogeneous acceleration field then takes some more pronounced two-dimensional patterns, but remains driven by the strong alternating expansions and retractions of the fluid in the thermal boundary layers, which are specific to **near**-**critical** **fluids**. © 2000 American Institute of Physics. 关S1070-6631共00兲00901-6兴

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I. INTRODUCTION
For a decade, investigations concerning **critical** phenom- ena have shown that heat and mass transfer in a pure fluid **near** its liquid–vapor **critical** point is very specific. In par- ticular, experiments performed in microgravity 1–4 have led to the conclusion that the temperature of a **near**-**critical** fluid submitted to heating or cooling from the outside of its con- tainer responds quickly despite its very low thermal diffusiv- ity. This phenomenon has been theoretically discussed, using thermodynamical considerations 5 or from the asymptotic analysis of the Navier–Stokes equations. 6 It is based on the anomalies of transport properties **near** the **critical** point. The thermal diffusivity of **near**-**critical** **fluids** becomes very small on approaching the **critical** point, but their compressibility and their thermal-expansion coefficients diverge at the same time. As a consequence, when heat is injected into a **near**- **critical** fluid, the fluid located in the thin thermal boundary layer strongly expands, compressing the rest of the fluid 共the bulk 兲 adiabatically. The compression wave involves a rapid and homogeneous rise of the temperature, which is signifi- cant on a much shorter time scale than diffusion. This mechanism has been called the ‘‘piston effect,’’ and since effects linked to compressibility become increasingly intense compared to those of diffusion when approaching the **critical**

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at T exit in a neighborhood of Q ± of size δ arbitrarily small, and hence step 5 ensures
that the solution will either blow up type I or dissipate to zero forward in time after T exit .
The paper is organized as follows. In Section 2 we recall the standard results on the semilinear heat equation and on Q, leading to a preliminary study of solutions **near** Q. The local well-posedness is addressed in Proposition 2.1. In Proposition 2.2 we describe the spectral structure of the linearized operator. In particular, for functions orthogonal to its instable eigenfunction and to its kernel, it displays some coercivity properties stated in Lemma 2.3. The nonlinear decomposition Lemma 2.5 then allows to decompose in a suitable way solutions around Q. Under this decomposition, the adapted variables are defined in Definition 2.4, the variation of the energy is studied in Lemma 2.7, the modulation equations are established in Lemma 2.6 and the energy bounds for the remainder on the infinite dimensional subspace are stated in Lemma 2.9. A direct consequence is the non-degeneracy of the scale and the central point, subject of the Lemma 2.11 ending the section. In the following one, Section 3, we first construct the minimal solutions in Proposition 3.1 and then give the proof of the Liouville-type theorem for minimal solutions 1.2. Finally, in Section 4 we give the proof of the classification Theorem 1.1. In Lemma 4.1 we characterize the instability time. If this time never happens, the solution is proved to dissipate to a soliton in Lemma 4.2. If it happens, then it blows up with type I blow up or dissipate according to Lemma 4.4. Eventually, Section 5 contains the proof of the stability of type I blow up.

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The solution predicted by this theorem is a superposition of k bubbles with respective blow-up orders ε 1 2 −j for j = 1, . . . , k.
Bubbling solutions for semilinear equations **near** the **critical** exponent has been the object of various works in the literature. In particular we refer the reader to [1, 7, 6, 10, 12] and references therein for construction of bubbling solutions in relation to Green’s function of the domain. The results above do have analogues for dimension N ≥ 4, which we shall state in the last section. It should be remarked that when N ≥ 4 we have that λ ∗ = 0. The object

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In a recent work, D. Funfschilling and G. Ahlers (FA) [10] observe a new effect, that they interpret as spurious (non-Boussinesq), in their convection cell working with ethane, **near** its **critical** point. They remark that most of the working points of the works observing the Kraichnan like behaviour [3, 4, 5, 6] are nearer the **critical** point and/or the gas-liquid coexistence curve of helium than those not observing it [7, 8, 9]. They thus propose that this spurious effect explains the difference quoted above, which would mean that the Kraichnan regime has not yet been experimentally observed.

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[SS98] Jalal Shatah and Michael Struwe. Geometric wave equations, volume 2 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 1998. [Tao05] Terence Tao. Global well-posedness and scattering for the higher-dimensional energy-**critical** nonlinear
Schr¨ odinger equation for radial data. New York J. Math., 11:57–80 (electronic), 2005.

about acoustic cavitation)”, suggests that the lack of cavitation damage could be understandable.
5. Conclusions
A numerical application of the characteristic parameters of the thermal effects has been performed at the beginning of this study. It has been suggested that thermal effects have significant implica- tions on the bubble dynamics in lc-CO 2 (liquid CO 2 **near** the **critical** point) by inhibiting early the motion of the bubble interface. The collapse of a CO 2 bubble has therefore been predicted to feature a very slow contraction of the interface. A bubble dynamics model has then been proposed and successfully benchmarked with a pre- vious study made on the collapse of a sodium bubble. Simulation results of the bubble collapse in lc-CO 2 have confirmed the very slow contraction of the bubble interface and demonstrated the absence of noticeable pressure rise. These also showed that the thermal layer inside the bubble has important effects on the bub- ble dynamics since it transfers a complementary heat flux to the interface. This heat flux has to be evacuated by the one from the interface to the liquid region, thereby altering the rate of conden- sation. With a sufficient increase in the liquid pressure, vaporiza- tion could occur during the bubble collapse, which was traditionally experienced only during the bubble growth. In addi- tion, simulation results have suggested that if the increase in the liquid pressure is maintained, the vapor would enter the supercrit- ical phase before approaching the **critical** point. The liquid region **near** the bubble interface would also reach this state by closely fol- lowing the saturation line.

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equal to the solubility y 2 sat predicted from this EoS.
Figures 3 illustrate the effect of temperature at a given pressure (P=79 bars for naphthalene and P=75.4 bars for benzoic acid and β-carotene) close to the UCEP of the mixtures. For all solutes, the thermodynamic factor exhibits a sudden variation **near** the mixture **critical** point, exactly like the osmotic susceptibility as described by Levelt Sengers [15]. The more dilute the mixture is, the most sudden is the variation of the thermodynamic factor and the smallest is the extent of this phenomenon.

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2. Overview
The study of linear stability of compressible boundary layers on a flat plate is not new. Motivated by applications related to the civil or military aeronautical sector, different works have very quickly sought to understand the influence of real gas effects on the stability of a boundary layer, including vibrational excitation, dissociation and recombination of gas species, ionization, radiation and surface ablation (for example, Malik & Anderson 1991 ) and thermochemical non-equilibrium reactions (for example, Stuckert & Reed 1994 ). However, these studies do not deal with certain aspects of non-ideal gases, such as a strong thermodynamic stratification or the modification of transport properties (speed of sound, viscosity, conductivity, etc.), that occur when the fluid evolves close to the **critical** point. Non-ideal **fluids** represent a research field of great importance for a wide range of applications; the work of Ren et al. ( 2019b ) aims to fill this gap.

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roundness and sphericity, and density. Sand is commonly used, as are other materials like resin or polymer-based particles, ceramics or carbides, and hollow glass spheres may be added as proppants or as components in a mixture. As mentioned in Section 4, fibers are also added, not as proppant per se, but to modify fluid rheology and decrease settling rates ( 132 ) although fibers can also mechanically bridge the fracture. At the pump, the volume fraction (vol/vol) of the flowing suspension can range from 0% to 5% for waterfrac/slickwater applications ( 133 ), and upwards of 20% in treatments using more viscous **fluids** ( 134 ); these values increase along the fracture due to leak-off. The size of the proppant particles used is guided by the expected width of the fracture during pumping, and proppant concentration by the desired width during production which is directly linked to the final amount of proppant per fracture area after closure. Particle size is specified using the mesh size of a sieve, larger mesh values correspond to smaller particles. 40/70 mesh sand is commonly used, corresponding to particles with approximate diameters between 210 and 420 µm. Volume fraction: Typically, solids

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