Korteweg Fluids Research Articles

Interpretation of Second Law of Thermodynamics in Extended Procedures for the Exploitation of the Entropy Inequality: Korteweg Fluids and Strain-Gradient Elasticity as Examples.

In continuum physics the dissipation principle, first proposed by Coleman and Noll in 1963, regards second law of thermodynamics as a unilateral differential constraint on the constitutive equations. In 1996, Muschik and Ehrentraut provided a rigorous proof of such an approach under the assumption that, at an arbitrary instant, t0, in an arbitrary point, P0, of a continuous system, the entropy production is zero if, and only if, P0 is in thermodynamic equilibrium. In 2022, Cimmelli and Rogolino incorporated such an assumption in a more general formulation of the second law of thermodynamics. In this paper, we prove that the same conclusions hold if both the fundamental balance laws and their gradients are substituted into the entropy inequality. Such a methodology is applied to analyze the strain-gradient elasticity.

Korteweg-Type Fluids and Thermodynamic Modelling via Higher-Order Gradients

This paper investigates the modelling of Korteweg-type fluids and hence the dependence of the stress tensor on gradients of mass density. This topic, originating from the need for describing capillarity effects, is mainly of interest in connection with nanosystems where the mean free path may be comparable with the geometric dimensions of the system. In addition to the Korteweg fluid model, the paper gives a review of the stress tensor function arising in quantum fluid hydrodynamics. Next, thermodynamic consistency is established for a fluid involving first- and second-order density gradients. The modelling investigated is a generalization of the classical Korteweg fluid and allows a better understanding of previous thermodynamic restrictions. The restrictions determined for the general scheme with second-order gradients are applied to the particular cases of the Korteweg fluid and the quantum fluid. Further, to allow for discontinuity wave solutions with finite speed of propagation, a model is established which involves higher-order derivatives and reduces to the Korteweg fluid in stationary conditions.

Holographic fluids: A thermodynamic road to quantum physics

Quantum mechanics, superfluids, and capillary fluids are closely related: It is thermodynamics that links them. In this paper, the Liu procedure is used to analyze the thermodynamic requirements. A comparison with the traditional method of divergence separation highlights the role of spacetime. It is shown that perfect Korteweg fluids are holographic. The conditions under which a complex field can represent the density and velocity fields of the fluid, and where the complex scalar field becomes a wave function of quantum mechanics, are explored. The bridge between the field and particle representations of a physical system is holography, and the key to holography is the Second Law of Thermodynamics.

Entropy Principle and Shock-Wave Propagation in Continuum Physics

According to second law of thermodynamics, the local entropy production must be nonnegative for arbitrary thermodynamic processes. In 1996, Muschik and Ehrentraut observed that such a constraint can be fulfilled in two different ways: either by postulating a suitable form of the constitutive equations, or by selecting among the solutions of the systems of balance laws those which represent physically realizable thermodynamic processes. Hence, they proposed an amendment to the second law which assumes that reversible process directions in state space exist only in correspondence with equilibrium states. Such an amendment allowed them to prove that the restriction of the constitutive equations is the sole possible consequence of non-negative entropy production. Recently, Cimmelli and Rogolino revisited the classical result by Muschik and Ehrentraut from a geometric perspective and included the amendment in a more general formulation of the second law. Herein, we extend this result to nonregular processes, i.e., to solutions of balance laws which admit jump discontinuities across a given surface. Two applications of these results are presented: the thermodynamics of an interface separating two different phases of a Korteweg fluid, and the derivation of the thermodynamic conditions necessary for shockwave formation. Commonly, for shockwaves the second law is regarded as a restriction on the thermodynamic processes rather than on the constitutive equations, as only perturbations for which the entropy continues to grow across the shock can propagate. We prove that this is indeed a consequence of the general property of the second law of thermodynamics that restricts the constitutive equations for nonregular processes. An analysis of shockwave propagation in different thermodynamic theories is developped as well.

Two-dimensional equilibrium configurations in Korteweg fluids

In this paper, after reviewing the form of the constitutive equations for a third grade Korteweg fluid, recently derived by means of an extended Liu procedure, an equilibrium problem is investigated. By considering a two-dimensional setting, a single nonlinear elliptic equation is derived such that the equilibrium conditions are identically satisfied. Such an equation is discussed both analytically and numerically. Moreover, by considering a particular boundary value problem of Dirichlet type, some preliminary numerical solutions are presented.

Thermodynamical analysis and constitutive equations for a mixture of viscous Korteweg fluids

A complete thermodynamical analysis for a binary mixture of viscous Korteweg fluids with two velocities and two temperatures is developed. The constitutive functions are allowed to depend on the diffusion velocity and the specific internal energies of both constituents, together with their first gradients, on the symmetric part of the gradient of barycentric velocity as well as on the mass density of the mixture and the concentration of one of the constituents, together with their first and second gradients. Compatibility with the entropy principle is analyzed by applying the extended Liu procedure, and a complete solution of the set of thermodynamical restrictions is recovered in three space dimensions. Finally, the equilibrium configurations are investigated, and it is proved that no restrictions arise on the admissible phase boundaries. The theoretical results here provided may serve as a basis for experimental and/or numerical investigations, in particular for determining the surface levels of phase boundaries at equilibrium and making a comparison with the experimental profiles.

On the characterization of constitutive equations for third-grade viscous Korteweg fluids

We consider a model of a third-grade viscous Korteweg-type fluid in three space dimensions and apply the extended Liu procedure in order to explicitly solve the constraints imposed by the entropy principle on the nonlocal constitutive relations. We detail the algorithm we use and are able to characterize the material functions involved in the constitutive equations. In a natural way, the application of the extended Liu procedure allows us to recover an extra-term in the entropy flux, preserving all the features of third-grade viscous Korteweg-type fluids. Moreover, a further constraint, in order to avoid that at equilibrium only very special phase boundaries are admissible, is investigated.

Continua with non-local constitutive laws: Exploitation of entropy inequality

In this paper, we consider a system of balance laws sufficiently general to contain the equations describing the thermomechanics of a one-dimensional continuum; this system involves some constitutive functions depending on the elements of the so called state space assumed to contain the spatial gradients of some of the unknown fields. The compatibility of the constitutive equations with an entropy-like principle is considered via an extended Liu procedure by using as constraints both the balance equations and some of their gradient extensions. This procedure is then applied to the equations of a fluid whose description involves an internal variable and first order non-local constitutive relations, and to a Korteweg fluid with second order non-localities. In both cases, the restrictions placed by an entropy inequality are solved, and an explicit solution for the constitutive equations is provided.

On a thermodynamic framework for developing boundary conditions for Korteweg-type fluids

We provide a derivation of several classes of boundary conditions for fluids of Korteweg-type using a simple and transparent thermodynamic approach that automatically guarantees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier’s slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of a Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.

Weakly nonlocal thermodynamics of binary mixtures of Korteweg fluids with two velocities and two temperatures

We provide a thermodynamic framework for binary mixtures of Korteweg fluids with two velocities and two temperatures. The constitutive functions are allowed to depend on the diffusion velocity and the specific internal energy of both constituents, together with their first gradients, as well as on the mass density of the mixture and the concentration of one of the constituents, the latters together with their first and second gradients. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure. In the one-dimensional case, a complete solution of the set of thermodynamic restrictions is obtained by postulating a possible form of the constitutive equations for the partial heat fluxes and stress tensors. Taking a first order expansion in the gradients of the specific entropy, the expression of the entropy flux is determined. This contains the classical terms (namely, the sum of the ratios between the heat fluxes and the temperatures of the constituents) and some additional contributions accounting for nonlocal effects.

Nonlinear wave-mechanical effects in Korteweg fluid magma transport

Statistical mechanics arguments and Madelung hydrodynamical presentation are applied to the transport of magma in volcanic conduits. An effective wave equation with logarithmic nonlinearity becomes apparent in systems of this kind, which describes the flow of a two-phase barotropic Korteweg fluid with capillarity, and allows multiple eigensolutions thus leading to wave-mechanical effects. We study spontaneous symmetry breaking in the erupting lava which flows up the conduit, so that fluid fragmentation and nucleation of density inhomogeneities occur; therefore, changing temperature can trigger a transition between the “magma-dissolved gas” fluid and magmatic foam phases. This phase structure is studied by both analytical and numerical methods. In the fluid phase, cell-like inhomogeneities occur which are described by solitary wave solutions with a Gaussian density profile; we derive the many-body interaction potential for these inhomogeneities. For the foam phase, we demonstrate the existence of topological kink solitons which describe bubble-type inhomogeneities; their stability is ensured by the conservation of a topological charge.

Phase-field evolution in Cahn–Hilliard–Korteweg fluids

In this paper, the diffusion of different phases in a third-grade Korteweg fluid is modeled by introducing a phase-field as a new independent thermodynamic variable. The constitutive equations are supposed to depend on the mass density and its spatial derivatives up to the second order, as well as on specific internal energy, barycentric velocity and phase-field, together with their first-order spatial derivatives. The compatibility of the model with the second law of thermodynamics is exploited by applying a generalized Liu procedure. For isothermal and isochoric phases, a general evolution equation for the phase-field, which generalizes the classical Cahn–Hilliard equation, is derived. Specific entropy and free energy are proved to depend on the basic unknown fields as well as on their gradients. A general constitutive equation for the Cauchy stress, which encompasses the classical one postulated by Korteweg in 1901, is obtained.

Fréedericksz transition in the director-density coupling theory.

We show that the director-density coupling theory gives rise to a singular behavior for the mass density. To overcome this drawback, we propose to supplement the theory with a term that can be derived by regarding liquid crystals as anisotropic Korteweg fluids. We thus show that the static bevahior of the resulting theory predicts a Fréedericksz transition accompanied by a modulation in the mass density.

A nonlocal phase-field model of Ginzburg–Landau–Korteweg fluids

A thermodynamic model of Korteweg fluids undergoing phase transition and/or phase separation is developed within the framework of weakly nonlocal thermodynamics. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure recently introduced in the literature. Possible forms of the free energy and of the stress tensor, which generalize some earlier ones proposed by several authors in the last decades, are carried out. Owing to the new procedure applied for exploiting the entropy principle, the thermodynamic potentials are allowed to depend on the whole set of variables spanning the state space, including the gradients of the unknown fields, without postulating neither the presence of an energy or entropy extra-flux, nor an additional balance law for microforce.

A free boundary model for Korteweg fluids as a limit of barotropic compressible Navier-Stokes equations

We consider the limit of some barotropic compressible fluid model with Korteweg forcing term, studied in (Bresch, Desjardin and Lin, 2002), as the exponent of the barotropic law goes to infinity. This provides a free boundary problem model, with capillary effects, and therefore generalizes the free boundary model obtained by (Lions and Masmoudi, 1999). Our interest for such free boundary problem stems from a study of the Leidenfrost effect.

On the Thermodynamics of Korteweg Fluids with Heat Conduction and Viscosity

A model of third-grade Korteweg fluid with heat conduction and viscosity is developed. The restrictions placed by the Dissipation Principle are investigated by applying two different methods which generalize the classical Coleman-Noll and Liu procedures. Compatibility with thermodynamics is achieved for arbitrary form of the energy and entropy fluxes. In the one-dimensional case a particular solution of the system of thermodynamic restrictions is provided.

Exploitation of the entropy principle: Proof of Liu theorem if the gradients of the governing equations are considered as constraints

The exploitation of the entropy principle in thermodynamics with Lagrange multipliers (the so called Liu procedure) is based on the celebrated Liu theorem. In some recent papers, where either rigid heat conductors or Korteweg fluids have been considered, the Liu procedure has been generalized by using the gradients of the governing equations as additional constraints. Here, a rigorous proof of the validity of the extended procedure to arbitrary first-order nonlocal continua is provided, thus explicitly generalizing the Liu theorem. As an application, heat conducting first-order Korteweg-type fluids with a scalar internal variable are analyzed. The thermodynamic restrictions are determined by the extended Liu procedure. Moreover, a comparison with the thermodynamic analysis obtained by the application of an extended Coleman–Noll procedure is performed.

A new perspective on the form of the first and second laws in rational thermodynamics: Korteweg fluids as an example

A generalized Coleman–Noll procedure for the exploitation of the entropy principle is presented. It is shown that it is able to dismantle some physical properties of nonlocal materials, which do not emerge if one uses the classical procedure. As an application, Korteweg fluids with a scalar internal variable are studied.

Spectral transverse instability of solitary waves in Korteweg fluids

The motion of Korteweg fluids is governed by the Euler–Korteweg model, which admits planar solitary waves for nonmonotone pressure laws such as the van der Waals law below critical temperature. In an earlier work with Danchin, Descombes and Jamet, it was shown by variational arguments and numerical computations that some of these solitary waves are stable in one space dimension. The purpose here is to study their stability with respect to transverse perturbations in several space dimensions. By Evans functions techniques and Rouché's theorem, it is shown that transverse perturbations of large wave length always destabilize solitary waves in the Euler–Korteweg model, whereas energy estimates show that perturbations of short wave length tend to stabilize them.

A note on a Korteweg fluid and the hydrodynamic form of the logarithmic Schrödinger equation

It is shown that the hydrodynamic form of nonlinear logarithmic Schrödinger equation (NLSE), in 3 + 1 dimensions, describes a solitonic flow of an isothermal, inviscid, compressible, capillary (Korteweg) fluid. This formulation corresponds to the so-called diffuse interface models used in hydrodynamics for two-phase fluids. The negative thermodynamic pressure furnished by the model indicates a possible application of the model to the description of a gas-saturated magma melt, rising in a volcanic conduit, in presence of cavitation processes.