Cattaneo Model Research Articles

Growth and decay of shock and acceleration waves in a traffic flow model with relaxation

The usual Ficken-based constitutive relation for traffic flux is replaced with one based on the Maxwell–Cattaneo model. The resulting flux law, which now takes into account the reaction time of driver and vehicle, results in a second-order, hyperbolic generalization of Burgers’ equation as the PDE governing the traffic density. An analytical study of this equation is presented with an emphasis on shock and related kinematic wave phenomena. Specifically, the exact traveling wave solution (TWS) is derived and it is shown that shock formation is possible only if the diffusivity is non-vanishing. Using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained and their temporal evolution determined. The exact upper bound of the reaction time parameter is established and connections between singular surface and TWS results are noted.

An overview of advances in heat conduction models and approaches for prediction of thermal conductivity in thin dielectric films

We first provide an overview of some predominant theoretical methods currently used for predicting thermal conductivity of thin dielectric films: the equation of radiative transfer, the temperature‐dependent thermal conductivity theories based on the Callaway model, and the molecular dynamics simulation. This overview also highlights temporal and spatial scale issues by looking at a unified theory that bridges physical issues presented in the Fourier and Cattaneo models. This newly developed unified theory is the so‐called C‐ and F‐processes constitutive model. This model introduces the notion of a new dimensionless heat conduction model number, which is the ratio of the thermal conductivity of the fast heat carrier F‐processes to the total thermal conductivity comprised of both the fast heat carriers F‐processes and the slow heat carriers C‐processes. Illustrative numerical examples for prediction of thermal conductivity in thin films are primarily presented.

THERMOELASTIC WAVES IN A HELIX WITH PARABOLIC OR HYPERBOLIC HEAT CONDUCTION

Of concern is the steady-state thermoelastodynamics of a helix, whose material is governed by a coupled thermoelastic model. Heat transfer follows either a parabolic (Fourier) or a hyperbolic (Maxwell–Cattaneo) rule. As a result, in the one-dimensional setting, there are three coupled differential equations that lead to a fast (primarily axial) and a slow (primarily torsional) wave. Particularly for harmonic motions, we find that the thermal diffusivity and the relaxation time (of the Maxwell–Cattaneo model) have minor effects, whereas the thermoelastic coupling constant is dominant—it speeds up these waves—and overall there is damping. Additionally, we review and note differences from solutions of equations governing plane harmonic waves in a non-centrosymmetric, thermoelastic micropolar continuum in three dimensions.

Cattaneo models for chemosensitive movement

We derive models for chemosensitive movement based on Cattaneo's law of heat propagation with finite speed. We apply the model to pattern formation as observed in experiments with Dictyostelium discoideum, with Salmonella typhimurium and with Escherichia coli. For Salmonella typhimurium we make predictions on pattern formation which can be tested in experiments. We discuss the relations of the Cattaneo models to classical models and we develop an effective numerical scheme.

ON A NEW C- AND F-PROCESSES HEAT CONDUCTION CONSTITUTIVE MODEL AND THE ASSOCIATED GENERALIZED THEORY OF DYNAMIC THERMOELASTICITY

Emanating from the Boltzmann transport equation, a new C- and F-processes heat conduction constitutive model is derived. The model acknowledges the notion of the simultaneous coexistence of both the slow Cattaneo-type C-processes and fast Fourier-type F-processes in the mechanisms of heat conduction. The C- and F-processes heat conduction constitutive model and the corresponding temperature equation that results from coupling the constitutive model with the energy equation naturally lead to a generalization of the macroscale in space one-temperature theory for heat conduction in solids of the Jeffreys'-type model, Cattaneo model, and the Fourier model for heat conduction in solids. This is unlike the Jeffreys'-type phenomenological model, which cannot reduce to the classical Fourier model (but only to a Fourier-like representation with relaxation) and it cannot explain the underlying physics associated with the C- and F-processes model. Additionally, the microscale in space two-temperature theory for pulse heating of metals is also highlighted via the C- and F-processes heat conduction constitutive model. Emphasis is placed on the development of a new C- and F-processes heat conduction model based on generalized thermoelastic theory to study the dynamic thermoelastic behavior of solids with special features that can lead to and explain the classical and nonclassical dynamic thermoelastic theories. Finally some conceptual pitfalls that appear in the literature are addressed.

Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems

In this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type. In particular we obtain a complete theory for the 2×2 systems of genuinely nonlinear hyperbolic balance laws in 1≳D with a strong dissipative term. A different method, which combines the div–curl lemma with accretive operators, is then applied to study the limiting profiles in the case of nonhomogeneous isentropic gas dynamics. We also investigate relaxation results for some 2≳D cases, which include the Cattaneo model for nonlinear heat conduction and the compressible Euler flow. Moreover, convergence result is also obtained for general semilinear systems in 1≳D.

Second sound and characteristic temperature in solids.

A temperature pulse propagating in a solid is studied by simple wave theory. The aim is to determine the shape change of the pulse, which initially is given by a Gaussian function, using a generalized nonlinear Cattaneo model proposed, in the framework of extended thermodynamics, by Ruggeri and co-workers. We prove that the characteristic temperature \ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{}, already pointed out in previous papers, plays an essential role in the shape change of the propagating second sound wave. In fact, three different shape changes can occur: (i) The back branch of the wave becomes steeper than the front one, (ii) vice versa, the steepening is forwards, or (iii) a double shock effect arises. These possibilities depend on the relation of the unperturbed temperature of the body to the characteristic temperature and, in some cases, on the wave amplitude. We discuss both the usual experimental case of a hot wave and the case of a cold wave, proving that this last process is not symmetric with respect to the previous one. Nevertheless, our calculations show that the effects described here are not so evident in the usual experimental conditions. Finally, for NaF and Bi crystals we discuss the best choice of some parameters in order to see clearly the changes of the wave profile and, in particular, the double shock effect. \textcopyright{} 1996 The American Physical Society.

Relaxation model for heat conduction and generation and the second law

This paper presents a phenomenological analysis of the relaxation model for heat conduction and generation from the point of view of its consistency with the second law of thermodynamics. The formulation of the second law based on classical irreversible thermodynamics as well as that based on extended irreversible thermodynamics are considered. It is shown that introduction of the transient capacity of the heat source into the energy conservation equation does not lead to any inconsistency with the second law. An analogy between the heat transport and the motion of a particle in the resistive medium is presented. The analysis shows that an inherent property of Cattaneo's model for heat conduction is that heat may flow spontaneously from cold to hot regions.

Hyperbolic heat conduction and the second law

It is well known that Cattaneo's modification of the constitutive equation for heat flux yields a telegraph equation for determining temperature that models hyperbolic heat conduction with a finite thermal wave speed. By analyzing the solution of an example of thermal equilibration of an initially inhomogeneous state it is shown that Cattaneo's model violates a notion of the second law of thermodynamics because it predicts that heat may flow from cold to hot regions during finite time periods. In this paper we discuss modified restrictions associated with the second law and we propose an alternative formulation of the equations for heat conduction which retains the usual Fourier equation for heat flux but modifies the constitutive equations for internal energy, Helmholtz free energy, and entropy to include dependence on temperature rate. This alternative formulation satisfies the restrictions associated with the second law and produces the same telegraph equation for hyperbolic heat conduction as the Cattaneo model. However, since Fourier heat conduction is retained the new model predicts that heat always flows from hot to cold regions. An additional example of uniform heating is considered to show that the solution for temperature of the model proposed here differs from the solution associated with the parabolic and Cattaneo models.