Cattaneo Law Research Articles

Blow‐up of smooth solutions to the relaxed compressible Navier‐Stokes equations

In this paper, the full compressible Navier‐Stokes equations with Cattaneo's heat transfer law or revised Maxwell's law are considered. We present a loss of initial smoothness of smooth solutions within a finite time and give an upper bound of such time. It is not required that the initial data has compact support or contains vacuum in any finite regions. Moreover, the blow‐up result has no restriction about the specific heat ratio. The main approach is motivated by previous studies and constructing a differential inequality.

Convective heat transfer in Brinkman–Darcy–Kelvin–Voigt fluid with couple stress and generalized Maxwell–Cattaneo law

This article investigates thermal convection in Kelvin–Voigt fluids saturating a Brinkman–Darcy-type porous medium. We examine the linear (stationary and oscillatory), nonlinear, and unconditional nonlinear stability of this fluid under the generalized Maxwell–Cattaneo law with couple stress effects. Using the normal mode technique, we calculate the critical Rayleigh number for the linear stability under stress-free boundary conditions for both stationary and oscillatory convection. Additionally, we employ the energy method to determine the critical Rayleigh number for nonlinear and unconditional nonlinear stabilities under the same boundary conditions. All critical values were determined numerically, and various graphs were plotted to illustrate the results. Our findings reveal that a higher couple stress parameter leads to increased critical Rayleigh numbers for stationary, oscillatory, and nonlinear stability, indicating greater fluid stability and reduced susceptibility to convection. Additionally, the Kelvin–Voigt parameter significantly affects oscillatory convection, though it remains crucial within the nonlinear stability framework. These findings provide a detailed understanding of the stability behavior in this complex fluid system.

Retrieving the time-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeIn order to include the non-negligible lag relaxation time feature that is characteristic of heat transfer in biological bodies, the classical Fourier's law of heat conduction has to be generalized as the Maxwell–Cattaneo law resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the unknown time-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from (non-intrusive) measurements of the temperature on an accessible sub-portion of the boundary that may be taken with an infrared scanner.Design/methodology/approachThe nonlinear and ill-posed problem is reformulated as a nonlinear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite-difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Numerical results for a benchmark test example are presented and thoroughly discussed, shedding light on the performance and effectiveness of the proposed methodology.FindingsThe inverse problem of obtaining the time-dependent blood perfusion coefficient and the temperature in the thermal-wave model of bio-heat transfer from extra boundary temperature measurement has been solved. In particular, the uniqueness of the solution to this inverse problem has been established. Furthermore, our proposed computational method demonstrated successful attainment of the perfusion coefficient and temperature, even when dealing with noisy data.Originality/valueThe originalities of the present paper are to account for such a more representative thermal-wave model of heat transfer in biological bodies and to investigate the possibility of determining its time-dependent blood perfusion coefficient from non-intrusive boundary temperature measurements.

Generalized approach for stability numbers of shear-damped Bresse systems with Cattaneo couplings

In this paper, by means of two general lemmas applicable to a wide variety of cases, we present a general formulation for the stability numbers associated with the thermoelastic Bresse system with heat flux given by Cattaneo's law and use it to show exponential stability of shear-damped versions of this system.

Exponential stability of piezoelectric beams with delay and second sound

AbstractIn this paper, we consider a fully‐dynamic piezoelectric beam model subjected to a magnetic effect, where the heat flux is given by Cattaneo's law. It is well known that, in the absence of delay, the dissipation produced by the heat conduction is strong enough to make the piezoelectric beams exponentially stable. However, time delay effects may destroy this behavior. Here, we show the existence and uniqueness of solutions through the semigroup theory. Furthermore, under a smallness condition on the delay, we prove an exponential stability result via establishing the appropriate Lyapunov functional. Finally, we numerically illustrate the asymptotic behavior of the solution.

Decay of the compressible Navier-Stokes equations with hyperbolic heat conduction

This paper is concerned with the global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction, where the heat flux is assumed to satisfy Cattaneo's law. When the initial perturbation is small and regular, we establish the global existence of the classical solution without any additional assumption on the relaxation time τ. Furthermore, assuming the initial perturbation belongs to a negative Sobolev space, we obtain the optimal time-decay rates of the high-order spatial derivatives of solutions by using a pure energy method instead of complicated spectral analysis. It is also observed that the heat flux, due to its damping structure, decays to the motionless state at a faster time-decay rate compared to density, velocity, and temperature.

Well-posedness and stability result for Timoshenko system with thermodiffusion effects and time-varying delay term

The main aim of the present paper is to investigate a new Timoshenko beam model with thermal and mass diffusion effects combined with a time-varying delay. Heat and mass exchange with the environment during a thermodiffusion in the Timoshenko beam, where the heat conduction is given by the classical Fourier law and acts on both the rotation angle and the transverse displacements. The heat conduction is given by the Cattaneo law. Under an appropriate assumption on the weights of the delay and the damping, we prove a well-posedness result, more precisely, we prove the existence of the weak solution. Then we proceed to the strong solution using the classical elliptic regularity and we get the result by applying the Lax-Milgram theorem, the Lumer-Phillips corollary and the Hille-Yosida theorem. We show the exponential stability result of the system in the case of nonequal speeds of wave propagation by using a multiplier technique combined with an appropriate Lyapunov functions. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory. To achieve our goals, we employ of the semigroup method and the energy method.

Stability results of a laminated beam with Cattaneo's thermal law and structural damping

Stability results of a laminated beam with Cattaneo's thermal law and structural damping

Three-dimensional rotatory micropolar fluid flow between two stretchable disks with Maxwell–Cattaneo law

Three-dimensional rotatory micropolar fluid flow between two stretchable disks with Maxwell–Cattaneo law

A generalized thermal memory effect of Caputo–Fabrizio fractional integral on natural convection flow of hybrid nanofluid

In this paper, the natural convective flow of hybrid nanofluid over the vertical plate has been examined. Cattaneo law of thermal flux is used to characterize thermal transport. The novel model for fractional constitutive equation is expressed by the time fractional Caputo–Fabrizio integral. The analytical solution of the generalized convective flow of viscous fluid over the vertical plate along generalized conditions is obtained by using the Laplace transform. Mathcad is used to determine numerically the influence of the physical and fractional parameters on the temperature and velocity field to depict the results graphically.

EXISTENCE AND UNIQUENESS OF THE CAGINALP PHASE-FIELD SYSTEM BASED ON THE CATTANEO LAW

Our aim in this paper is to study the existence and the uniqueness of the Caginalp phase-field system based on the Cattaneo Law,with initial conditions,Dirichlet Boundary Conditions and Regular Potentiels.

Thermal Timoshenko beam system with suspenders and Kelvin–Voigt damping

In the present study, we consider a thermal-Timoshenko-beam system with suspenders and Kelvin–Voigt damping type, where the heat is given by Cattaneo's law. Using the existing semi-group theory and energy method, we establish the existence and uniqueness of weak global solution, and an exponential stability result. The results are obtained without imposing the equal-wave speed of propagation condition.2010 MSC:35D30, 35D35, 35B35.

Upper semicontinuity of the global attractor for Bresse system with second sound

In this paper, we study the long-time dynamics of Bresse system under mixed homogeneous Dirichlet–Neumann boundary conditions. The heat conduction is given by Cattaneo's law. Only the shear angle displacement is damped via the dissipation from the Cattaneo's law, and the vertical displacement and the longitudinal displacement are free. Under quite general assumptions on the source term and based on the semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy. Finally, we prove the upper semicontinuous with respect to the relaxation time τ as it converges to zero.

Numerical Analysis of a Swelling Poro-Thermoelastic Problem with Second Sound

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flux. The numerical analysis of this problem is performed applying the classical finite element method with linear elements for the spatial approximation and the backward Euler scheme for the discretization of the time derivatives. Then, we prove the stability of the discrete solutions and we provide an a priori error analysis. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximations, the exponential decay of the discrete energy and the dependence on a coupling parameter.

Stability analysis for abstract theomoelastic systems with Cattaneo's law and inertial terms

We study the stability of an abstract thermoelastic system, which consists of an abstract elastic equation with or without inertial term and a heat equation of Cattaneo type. A complete analysis is carried out on how the decay rates of the system depend on the parameter region and the so-called 'wave speeds'. Specifically, the parameter region for exponential stability of such a system is characterized. Furthermore, optimality of the rate of polynomial decay is obtained under certain conditions. The proofs rely on interpolation inequalities, frequency domain condition for stability, and detailed spectral analysis.

A numerical study of swelling porous thermoelastic media with second sound

In this work, we numerically consider a swelling porous thermoelastic system with a heat flux given by the Maxwell–Cattaneo law. We study the numerical energy and the exponential decay of the thermoelastic problem. First, we give a variational formulation written in terms of the transformed derivatives corresponding to a coupled linear system composed of four first-order variational equations. A fully discrete algorithm is introduced and a discrete stability property is proven. A priori error estimates are also provided. Finally, some numerical results are given to demonstrate the behavior of the solution.

Decay rate of the solutions to the Bresse-Cattaneo system with distributed delay

<abstract><p>This study examines the pace at which solutions to a Bresse system in combination with the Cattaneo law of heat conduction and the dispersed delay term degradation. We establish our major finding utilizing the energy approach in the Fourier space.</p></abstract>

Timoshenko systems with Cattaneo law and partial Kelvin–Voigt damping: well-posedness and stability

This work considers thermoelastic Timoshenko beam systems with full and partial Kelvin–Voigt damping. The heat conduction is governed by the Cattaneo law. By applying the semi-group method, we establish the existence and uniqueness of weak global solution, then with the multiplier method, we as well prove exponential stability results. In most cases, stabilizing a Timoshenko system with Cattaneo law requires obtaining stability number or some equal wave speed propagation condition. However, interestingly in this work our stability result does not require any stability number nor equal wave speed propagation condition.

Analytical solutions for time-fractional diffusion equation with heat absorption in spherical domains

Under the time-variable Dirichlet condition, the time-fractional diffusion equation with heat absorption in a sphere is taken into consideration. The time-fractional derivative with the power-law kernel is used in the generalized Cattaneo constitutive equation of the thermal flux. The Laplace transform and a suitable transformation of the independent variable and function are used to determine the analytical solution of the problem in the Laplace domain. To obtain the temperature distribution in the real domain, the inverse Laplace transforms of two functions of exponential type are obtained. These formulae are new in the literature. The particular cases of the classical Cattaneo law of heat conduction and of the classical Fourier's law are obtained from the solutions corresponding to the time-fractional generalized Cattaneo law.

Thermal convection in a Brinkman–Darcy–Kelvin–Voigt fluid with a generalized Maxwell–Cattaneo law

We investigate thoroughly a model for thermal convection of a class of viscoelastic fluids in a porous medium of Brinkman–Darcy type. The saturating fluids are of Kelvin–Voigt nature. The equations governing the temperature field arise from Maxwell–Cattaneo theory, although we include Guyer–Krumhansl terms, and we investigate the possibility of employing an objective derivative for the heat flux. The critical Rayleigh number for linear instability is calculated for both stationary and oscillatory convection. In addition a nonlinear stability analysis is carried out exactly.