Cattaneo Law Research Articles

On stability of hyperbolic thermoelastic Reissner–Mindlin–Timoshenko plates

In the present article, we consider a thermoelastic plate of Reissner–Mindlin–Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absence of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, and so on. We present a well‐posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Restricting the problem to the rotationally symmetric case, we further prove that a single frictional damping merely for the bending component is sufficient for exponential stability. To this end, we construct a Lyapunov functional incorporating the Bogovskiĭ operator for irrotational vector fields, which we discuss in the appendix. Copyright © 2014 John Wiley & Sons, Ltd.

Second-sound phenomena in inviscid, thermally relaxing gases

We consider the propagation of acoustic and thermal waves in a class of inviscid, thermally relaxing gases wherein the flow of heat is described by the Maxwell--Cattaneo law, i.e., in Cattaneo--Christov gases. After first considering the start-up piston problem under the linear theory, we then investigate traveling wave phenomena under the weakly-nonlinear approximation. In particular, a shock analysis is carried out, comparisons with predictions from classical gases dynamics theory are performed, and critical values of the parameters are derived. Special case results are also presented and connections to other fields are noted.

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

AbstractIn this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo‐propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping. Copyright © 2013 John Wiley & Sons, Ltd.

The Timoshenko system with history and Cattaneo law

In this paper we study a fully hyperbolic thermoelastic Timoshenko system with past history where the thermal effects are given by Cattaneo’s law. We obtain exponential stability of solutions if and only if a new condition on the wave speed of propagation is verified. Otherwise, when that condition fails, we obtain polynomial stability of solutions. In this case optimal rates of decay are established.

Stability of a Damped Hyperbolic Timoshenko System Coupled with a Heat Equation

AbstractWe consider a one‐dimensional damped hyperbolic Timoshenko beam that is coupled with a heat equation. When its wave speeds are different, it is known that the Timoshenko beam that is coupled with a heat equation, under Cattaneo's law, does not have exponential stability. With two internal dampings being introduced, in this paper, we show that the system under Cattaneo's law is exponentially stable. We also show the exponential stability when the system is considered under Fourier's law.

Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same

We consider the Cauchy problem for the one-dimensional Timoshenko system coupled with heat conduction, wherein the latter is described by either the Cattaneo law or the Fourier law. We prove that heat dissipation alone is sufficient to stabilize the system in both cases, so that additional mechanical damping is unnecessary. However, the decay of solutions without the mechanical damping is found to be slower than that with mechanical damping. Furthermore, in contrast to earlier results of Said-Houari and Kasimov (2012) [10] and Fernández Sare and Racke (2009) [12], we find that the Timoshenko–Fourier and the Timoshenko–Cattaneo systems have the same decay rate. The rate depends on a certain number α (first identified by Santos et al., 2012 [11] in a related study in a bounded domain), which is a function of the parameters of the system.

A conserved phase-field system based on the Maxwell–Cattaneo law

Our aim in this paper is to study a generalization of the conserved Caginalp phase-field system based on the Maxwell–Cattaneo law for heat conduction and endowed with Neumann boundary conditions. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators.

Effect of Non-Uniform Temperature Gradient on the Onset of Rayleigh–Bénard–Magnetoconvection in Micropolar Fluid with Maxwell–Cattaneo Law

The effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard magnetoconvection in a Micropolar fluid with Maxwell-Cattaneo law is studied using the Galerkin technique. The eigenvalue is obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal condition on the spin-vanishing boundaries. A linear stability analysis is performed. The influence of various parameters on the onset of convection has been analyzed. One linear and five non-linear temperature profiles are considered and their comparative influence on onset of convection is discussed. The classical approach predicts an infinite speed for the propagation of heat. The present non-classical theory involves a wave type heat transport (Second Sound) and does not suffer from the physically unacceptable drawback of infinite heat propagation speed.

WAVE-RELAXATION DUALITY OF HEAT PROPAGATION IN FERMI–PASTA–ULAM CHAINS

In our recent work [Gusev and Lurie, Int. J. Mod. Phys. B26 (2012) 125003], we have presented the theory of spacetime elasticity. The theory advocates its own law of heat conduction in solids, involving both wave and relaxation (diffusion) modes of heat propagation. Here we use that heat conduction law — together with the classical Fourier's and Maxwell–Cattaneo laws — to analyze the temperature dynamics occurring in Fermi–Pasta–Ulam (FPU-β) chains. We focus on the acoustic limit and use a combination of the Langevin and molecular dynamics to study the corresponding temperature decay functions. We have found that all the three laws suit well for describing the high-temperature, exponential-form temperature decay functions. However, at low temperatures the wave dynamics becomes prevailing and it is only the spacetime-elasticity heat conduction law that provides the appropriate functional forms. At all the temperatures and wavelengths studied, the observed temperature dynamics is anomalous, and we discuss both theoretical and practical implications of such anomalous behavior

THEORY OF SPACETIME ELASTICITY

We present the theory of spacetime elasticity and demonstrate that it involves traditional thermoelasticity. Assuming linear-elastic constitutive behavior and using spacetime transversely-isotropic elastic constants, we derive all principal thermodynamic relations of classical thermoelasticity. We introduce the spacetime principle of virtual work, and use it to derive the equations of motion for both reversible and dissipative thermoelastic dynamics. We show that spacetime elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, spacetime elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity, complemented by the spectrum of boundary and interface conditions. We argue that the presented framework of spacetime elasticity should prove adequate for describing the thermoelastic phenomena occurring at low temperatures, for interpreting the results of molecular simulations of heat conduction in solids, and also for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.

Exponential stability of a general network of 1-d thermoelastic rods

We consider a finite planar network of 1-$d$ thermoelastic rods using Fourier's law or Cattaneo's law for heat conduction, we show that the system is exponentially stable in the two cases.

Decay property of Timoshenko system in thermoelasticity

We investigate the decay property of a Timoshenko system of thermoelasticity in the whole space for both Fourier and Cattaneo laws of heat conduction. We point out that although the paradox of infinite propagation speed inherent in the Fourier law is removed by changing to the Cattaneo law, the latter always leads to a solution with the decay property of the regularity‐loss type. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We derive L2 decay estimates of solutions and observe that for the Fourier law the decay structure of solutions is of the regularity‐loss type if the wave speeds of the first and the second equations in the system are different. For the Cattaneo law, decay property of the regularity‐loss type occurs no matter what the wave speeds are. In addition, by restricting the initial data to with a suitably large s and γ ∈ [0,1], we can derive faster decay estimates with the decay rate improvement by a factor of t−γ/2. Copyright © 2011 John Wiley & Sons, Ltd.

On a phase-field system based on the Cattaneo law

Our aim in this paper is to study a generalization of the Caginalp phase-field system based on the Maxwell–Cattaneo law for heat conduction and endowed with Neumann boundary conditions. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators. We also prove, when the enthalpy is conserved, the existence of the global attractor. We finally study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist and have a proper (spatial) decay at infinity.

Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U 0 ∈ H s (ℝ) ∩ L 1(ℝ). In addition, by restricting the initial data to U 0 ∈ H s (ℝ) ∩ L 1,γ(ℝ) and γ ∈ [0, 1], we can derive faster decay estimates with the decay rate improvement by a factor of t −γ/2.

Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound

In this article we are concerned with the strong stabilization of models for the Reissner–Mindlin plate equations with second sound, that is, models that include thermal effects described according to Cattaneo's law of heat conduction instead of Fourier's law in classical thermoelasticity. Two models will be considered which are distinct with respect to the property of compactness or non-compactness of the resolvent of the generator of the underlying semigroup. In accordance with the compactness or non-compactness of the resolvent operator, a different criterion for strong stability is implemented to achieve the strong stabilization of each model. In the compact resolvent case we avail ourselves of a result given by Benchimol [C.D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), pp. 373–379] and in the non-compact case we resort to a stability criterion Tomilov [Y. Tomilov, A resolvent approach to stability of operator semigroups, J. Operator Theory 46 (2001), pp. 63–98].

Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Second Sound

We consider a thermoelastic diffusion problem in one space dimension with second sound. The thermal and diffusion disturbances are modeled by Cattaneo′s law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier′s law. The system of equations in this case is a coupling of three hyperbolic equations. It poses some new analytical and mathematical difficulties. The exponential stability of the slightly damped and totally hyperbolic system is proved. Comparison with classical theory is given.

The influence of thermal relaxation on the oscillatory properties of two-gradient convection in a vertical slot

We study the effects of the Maxwell–Cattaneo (MC) law of heat conduction on the flow of a Newtonian fluid in a vertical slot subject to both vertical and horizontal temperature gradients. Working in one spatial dimension (1D), we employ a spectral expansion involving Rayleigh’s beam functions as the basis set, which are especially well-suited to the fourth order boundary value problem (b.v.p.) considered here, and the stability of the resulting dynamical system for the Galerkin coefficients is investigated. It is shown that the absolute value of the (negative) real parts of the eigenvalues are reduced, while the absolute values of the imaginary parts are somewhat increased, under the MC law. This means that the presence of the time derivative of the heat flux increases the order of the system, thus leading to more oscillatory regimes in comparison with the usual Fourier case. Moreover, no eigenvalues with positive real parts were found, which means that in this particular situation, the inclusion of thermal relaxation does not lead to destabilization of the motion.

Heat conduction: a telegraph-type model with self-similar behavior of solutions

For the heat flux q and temperature T, we introduce a modified Fourier–Cattaneo law qt + lq/(t + τ) = −kTx. The consequence of it is a non-autonomous telegraph-type equation. This model already has a typical self-similar solution which may be written as a product of two traveling waves modulo a time-dependent factor and might play a role of intermediate asymptotics.

Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis

We consider a model of chemotaxis with finite speed of propagation based on Cattaneo's law in two space dimensions. For this system we present a global existence theorem for the smooth solutions to the Cauchy problem. This result is obtained using estimates of the Green functions of the linearized operators. We illustrate the behavior of solutions by some numerical simulations.

Oscillatory convection and the Cattaneo law of heat conduction

The problem of thermal convection is investigated for a layer of fluid when the heat flux law of Cattaneo is adopted. The boundary conditions are those appropriate to two fixed surfaces. It is shown that for small Cattaneo number the critical Rayleigh number initially increases from its classical value of 1707.765 until a critical value of the Cattaneo number is reached. For Cattaneo numbers greater than this critical value a notable Hopf bifurcation is observed with convection occurring at lower Rayleigh numbers and by oscillatory rather than stationary convection. The aspect ratio of the convection cells likewise changes.