A new form of governing equations of fluids arising from Hamilton's principle

Gavrilyuk, Sergey; Gouin, Henri

doi:10.1016/s0020-7225(98)00131-1

Cited by 49 publications

(53 citation statements)

References 16 publications

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“…The generalisation, from mono to multiphase theories, of the variational procedure is not straightforward and we followed methods available in the literature, see e.g. Bedford and Drumheller (1978;1979), Batra et al (1986), Batra and Bedford (1988), Gouin (1990), Gavrilyuk et al (1998), Gavrilyuk and Gouin (1999), Sciarra et al (2003), Gouin and Ruggeri (2003), Gavrilyuk (2005). Two distinct families of variations have been defined and the balance equations have been deduced.…”

Section: Historical Backgroundmentioning

confidence: 99%

“…The first one regards the placement map (χ s or χ f ) which has to be varied, the second is about the particle one should follow in the variation. Among other approaches, we use that belonging to Gouin (1990), Gavrilyuk et al (1998), Gouin and Gavrilyuk (1999), Gavrilyuk and Gouin (1999), Gouin and Ruggeri (2003) and to Gavrilyuk (2005). Apparently, specifying a variation scheme affects the form of the derived field equations.…”

Section: A Variational Approach For the Derivation Of Balance Equationsmentioning

confidence: 99%

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Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

Placidi

Dell’Isola

Ianiro

et al. 2008

European Journal of Mechanics - A/Solids

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Fluid saturated porous media are modelled by the theory of mixtures and the placement maps of the solid and of the fluid are considered. The momentum balance equations are derived in the framework of a variational approach: We take an action functional and two families of variations and assume that the sum of the virtual work of the external forces and the variation of such an action along each variation are zero. Constitutive equations for the two Cauchy stress tensors and for the interaction force are derived taking into account a general state of pre-stress for the solid and for the fluid species. Governing equations are therefore formulated, however, for the sake of simplicity, only the case of pure initial pressure is further investigated. The propagation of bulk (transversal and longitudinal) waves and the influence of pre-stress are studied: In particular, stability analyses are carried out starting from dispersion relations and the role of pre-stress is investigated. Finally, a numerical example is established for a given state of pre-stress, deriving the phase velocities and the attenuation coefficients of transversal and longitudinal waves.

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Section: Historical Backgroundmentioning

confidence: 99%

Section: A Variational Approach For the Derivation Of Balance Equationsmentioning

confidence: 99%

Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

Placidi

Dell’Isola

Ianiro

et al. 2008

European Journal of Mechanics - A/Solids

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“…Now we have to couple the equation (2) with the governing equations of non-equilibrium compressible mixtures. First, we will construct a dissipation-free equilibrium model by using Hamilton's principle of stationary action (Gavrilyuk & Gouin, 1999, Gavrilyuk & Saurel, 2002. Consider the Lagrangian which is the difference between the kinetic and the potential energy of the mixture :…”

Section: Variational Principlementioning

confidence: 99%

“…The variations of unknown variables in Eulerian coordinates in terms of virtual displacements δx are given by (Gavrilyuk & Gouin, 1999) :…”

Section: Variational Principlementioning

confidence: 99%

“…Then, dissipation terms which are compatible with the entropy inequality, are introduced (see examples of such a derivation in Gavrilyuk & Gouin (1999), and Gavrilyuk & Saurel (2002)). The model generalizes the capillary model of Perigaud & Saurel (2005) with the introduction of heat and mass transfer relaxation terms to describe phase transition.…”

Section: Introductionmentioning

confidence: 99%

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Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

Berry¹,

Saurel²,

Petitpas³

et al. 2008

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8 MotivationIn nuclear reactor safety and optimization there are key issues that rely on in-depth understanding of basic two-phase flow phenomena with heat and mass transfer. Within the context of multiphase flows, two bubble-dynamic phenomena -boiling (heterogeneous) and flashing or cavitation (homogeneous boiling), with bubble collapse, are technologically very important to nuclear reactor systems. The main difference between boiling and flashing is that bubble growth (and collapse) in boiling is inhibited by limitations on the heat transfer at the interface, whereas bubble growth (and collapse) in flashing is limited primarily by inertial effects in the surrounding liquid. The flashing process tends to be far more explosive (and implosive), and is more violent and damaging (at least in the near term) than the bubble dynamics of boiling. However, other problematic phenomena, such as crud deposition, appear to be intimately connecting with the boiling process. In reality, these two processes share many details.Flashing occurs in flowing liquid systems when the pressure falls sufficiently low in some region of the flow, reaching a metastable state where the temperature is higher than the saturated one at the reduced pressure of this expanded state. Then the superheated liquid releases its metastable energy (stored as internal energy) very quickly, even explosively, producing either pure vapor (bubble) or liquid-vapor mixture at high velocity, [2]. Expansion effects in nuclear reactor systems are often due to geometrical effects, as for example in nozzles where flashing appears at locations where the pressure is relatively low and the liquid superheated. In the case of twophase blowdown (from the superheated liquid state), bubble collapse is usually not important, but the flashing of superheated liquid strongly influences critical flow rates. In other cases, besides the performance limitations which this cavitation may cause in flow systems, subsequent bubble collapse may be responsible for damage to nearby solid surfaces.Many nuclear reactor applications rely on convective nucleate boiling to efficiently remove high heat fluxes from heated surfaces. Nucleate boiling is a very effective heat transfer mechanism, however it is well known that there exists a critical value of the heat flux at which nucleate boiling transitions to film boiling (departure from nucleate boiling (DNB) and boiling crisis), a very poor heat transfer mechanism. In most practical applications it is imperative to maintain the operating heat flux below such critical value, which is called the Critical Heat Flux (CHF). In this case, the presence of a nearby solid surface is necessary for the rapid supply of the latent heat inherent in the phase change. The presence of these surfaces is known to modify the flow patterns and other characteristics of these multiphase flows, and therefore must be interactively coupled with analyses of these phenomena. And again, as mentioned above, DNB is believed to play an integral role in performance degradatio...

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Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

Germain

LeFloch

2015

Comm Pure Appl Math

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This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a certain symmetry (for instance, plane symmetry); these solutions may have unbounded amplitude and contain cavitation regions in which the mass density vanishes. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. We encompass a broad class of nonlinear Navier-Stokes-Korteweg constitutive laws, which is determined by two main conditions relating the viscosity and capillarity coefficients, that is, on one hand the strong coercivity condition (as we call it) which provides a favorable sign for the integrated dissipation associated with an effective energy, and on the other hand the tame capillarity condition (as we call it), which restricts pointwise the strength of the capillarity relatively to the viscosity. Rather mild conditions on the growth of the constitutive functions are aso imposed, which are required in order to define finite energy weak solutions to the Navier-Stokes-Korteweg system, even in the presence of cavitation. Our method of proof relies on fine algebraic properties of the Euler system and combines together energy and effective energy estimates, dissipation and effective dissipation estimates, a nonlinear Sobolev inequality, highintegrability properties for the mass density and for the velocity, and compactness properties based on entropies.

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A new form of governing equations of fluids arising from Hamilton's principle

Cited by 49 publications

References 16 publications

Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

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