arXiv subjects

Matteo Gorgone

Publications and source records attributed to Matteo Gorgone.

16 recordsLinked to original sources

arXiv2021

A thermodynamical description of third grade fluid mixtures

A complete thermodynamical analysis for a non-reacting binary mixture exhibiting the features of a third grade fluid is analyzed. The constitutive functions are allowed to depend on the mass density of the mixture and the concentration of one of the constituents, together with their first and second order gradients, on the specific internal energy of the mixture with its first order gradient, as well as on the symmetric part of the gradient of barycentric velocity. Compatibility with second law of thermodynamics is investigated by applying the extended Liu procedure. An explicit solution of the set of thermodynamic restrictions is obtained by postulating a suitable form of the constitutive relations for the diffusional mass flux, the heat flux and the Cauchy stress tensor. Taking a first order expansion in the gradients of the specific entropy, the expression of the entropy flux is determined. It includes an additional contribution due to non-local effects.

math-ph↗

arXiv2021

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 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 experimental profiles.

physics.flu-dyn↗

arXiv2021

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 non--local 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.

physics.flu-dyn↗

arXiv2021

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.

physics.flu-dyn↗

arXiv2021

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.

math-ph↗

arXiv2021

Lie remarkable partial differential equations characterized by Lie algebras of point symmetries

Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular, we determine hierarchies of second order partial differential equations uniquely characterized by affine transformations of $\mathbb{R}^{n+m}$, and a system of two third order partial differential equations in two independent variables uniquely determined by the Lie algebra of projective transformations of $\mathbb{R}^4$.

math-ph↗

arXiv2021

Generalized Hamiltonian for a two-mode fermionic model and asymptotic equilibria

In some recent papers, the so called $(H,\rho)$-induced dynamics of a system $\mathcal{S}$ whose time evolution is deduced adopting an operatorial approach, has been introduced. According to the formal mathematical apparatus of quantum mechanics, $H$ denotes the Hamiltonian for $\mathcal{S}$, while $\rho$ is a certain rule applied periodically on $\mathcal{S}$. In this approach the rule acts at specific times $k\tau$, with $k$ integer and $\tau$ fixed, by modifying some of the parameters entering $H$ according to the state variation of the system. As a result, a dynamics admitting an asymptotic equilibrium state can be obtained. Here, we consider the limit for $\tau\rightarrow 0$, so that we introduce a generalized model leading to asymptotic equilibria. Moreover, in the case of a two-mode fermionic model, we are able to derive a relation linking the parameters involved in the Hamiltonian to the asymptotic equilibrium states.

math-ph↗

arXiv2021

Approximate Q-conditional symmetries of partial differential equations

Following a recently introduced approach to approximate Lie symmetries of differential equations which is consistent with the principles of perturbative analysis of differential equations containing small terms, we analyze the case of approximate $Q$--conditional symmetries. An application of the method to a hyperbolic variant of a reaction--diffusion--convection equation is presented.

math-ph↗

arXiv2021

$(H,\rho)$--induced political dynamics: facets of the disloyal attitudes into the public opinion

A simple model, suitable to describe the dynamics of a political system consisting of three macro--groups affected by turncoat--like behaviors and the influence of the opportunistic attitudes of politicians on voters' opinion, is presented. The model is based on raising and lowering fermionic operators whose dynamics is ruled by a suitable quadratic Hamiltonian operator with the addition of specific rules (depending on the variations of the mean values of the observables) able to adjust periodically the model to the political environment, \emph{i.e.}, we move in the framework of the so called $(H,\rho)$--induced dynamics approach.

physics.soc-ph↗

arXiv2021

Political dynamics affected by turncoats

An operatorial theoretical model based on raising and lowering fermionic operators for the description of the dynamics of a political system consisting of macro--groups affected by turncoat--like behaviors is presented. The analysis of the party system dynamics is carried on by combining the action of a suitable quadratic Hamiltonian operator with specific rules (depending on the variations of the mean values of the observables) able to adjust periodically the conservative model to the political environment.

physics.soc-ph↗

arXiv2021

A Consistent Approach to Approximate Lie Symmetries of Differential Equations

Lie theory of continuous transformations provides a unified and powerful approach for handling differential equations. Unfortunately, any small perturbation of an equation usually destroys some important symmetries, and this reduces the applicability of Lie group methods to differential equations arising in concrete applications. On the other hand, differential equations containing \emph{small terms} are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, \emph{i.e.}, to establish an approximate symmetry theory. There are two widely used approaches to approximate symmetries: the one proposed in 1988 by Baikov, Gazizov and Ibragimov, and the one introduced in 1989 by Fushchich and Shtelen. Moreover, some variations of the Fushchich--Shtelen method have been proposed with the aim of reducing the length of computations. Here, we propose a new approach that is consistent with perturbation theory and allows to extend all the relevant features of Lie group analysis to an approximate context. Some applications are also presented.

math-ph↗

arXiv2021

Approximately invariant solutions of creeping flow equations

In this paper, the steady creeping flow equations of a second grade fluid in cartesian coordinates are considered; the equations involve a small parameter related to the dimensionless non--Newtonian coefficient. According to a recently introduced approach, the first order approximate Lie symmetries of the equations are computed, some classes of approximately invariant solutions are explicitly determined, and a boundary value problem is analyzed. The main aim of the paper is methodological, and the considered mechanical model is used to test the reliability of the procedure in a physically important application.

math-ph↗

arXiv2021

On the decoupling problem of general quasilinear first order systems in two independent variables

The paper deals with the decoupling problem of general quasilinear first order systems in two independent variables. We consider either the case of homogeneous and autonomous systems or the one of nonhomogeneous and/or nonautonomous systems. Necessary and sufficient conditions for the partial or full decoupling of the systems at hand are provided. The conditions involve the properties of eigenvalues and eigenvectors of the coefficient matrix, and provide the differential constraints whose integration leads to the decoupling transformation. Some applications of physical interest are also given.

math-ph↗

arXiv2021

Nonlinear first order partial differential equations reducible to first order homogeneous and autonomous quasilinear ones

A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system. Some applications to relevant partial differential equations are given.

math-ph↗

arXiv2021

Nonlinear first order PDEs reducible to autonomous form polynomially homogeneous in the derivatives

It is proved a theorem providing necessary and sufficient conditions enabling one to map a nonlinear system of first order partial differential equations, polynomial in the derivatives, to an equivalent autonomous first order system polynomially homogeneous in the derivatives. The result is intimately related to the symmetry properties of the source system, and the proof, involving the use of the canonical variables associated to the admitted Lie point symmetries, is constructive. First order Monge-Amp\`ere systems, either with constant coefficients or with coefficients depending on the field variables, where the theorem can be successfully applied, are considered.

math-ph↗

arXiv2021

Reduction of balance laws in (3+1)--dimensions to autonomous conservation laws by means of equivalence transformations

A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.

math-ph↗