Maxwell Cattaneo Law Research Articles

Dynamic model of fractional thermoelasticity due to ramp-type heating with two relaxation times

This is an attempt to design a fractional heat conduction model in a bounded cylindrical region exposed to axisymmetric ramp-type heat flux and discuss its thermal behaviour. This model is drafted using the classical theory of fractional thermoelasticity due to the involvement of two relaxation times to the heat conduction equation and the equations of motion. In order to achieve finite thermal wave speed, the heat conduction equation is derived from the viewpoint of Maxwell–Cattaneo law in the context of fractional derivative. Analytical results for the distribution of temperature, displacement and thermal stresses are obtained using integral transforms in the Laplace domain. The Gaver–Stehfest method has been used to invert the results of Laplace domain. The convergence of infinite series solutions has been discussed. As a specific case this model has been applied to a thick circular plate subjected to the axisymmetric ramp-type heat flux. The results obtained for the thermal variations are validated by comparing to coupled and generalized theory of thermoelasticity. These comparisons are shown graphically.

Finite difference approach and successive over relaxation (SOR) method for MHD micropolar fluid with Maxwell–Cattaneo law and porous medium

This analysis considered the flow of micropolar fluid to describe the micro-rotational influence of fluid particles. The oscillatory motion of stretchable disk is responsible for the fluid flow. The impacts of magnetic field and porous medium are incorporated in a momentum equation. The Maxwell–Cattaneo law of material-invariant version having damped-hyperbolic equation is proposed in order to overcome the limitations of Fourier’s law. The model of three-dimensional flow problem is then normalized by adopting suitable similarity variables. The resulting normalized system of partial differential equations is solved in a numerical way using first and second order central and backward finite difference approximations by converting the semi-infinite domain into the finite one. The resulting expressions are solved iteratively using the successive over relaxation parameter method. The impacts of various dimensionless parameters are elaborated through graphs and tables. Higher values of porosity and magnetic parameters reduced the radial velocity curves. Micropolar parameters, namely vortex and spin gradient viscosity parameters, enhanced the microrotational tangential velocity profiles. Radial shear stresses and frictional torque are decreased due to larger values of the vortex viscosity parameter. Couple stresses along radial and tangential directions are enhanced for increasing values of porosity parameter. The three-dimensional and two-dimensional flow phenomenon is also sketched. The numerical results for the limiting scenario are compared for various physical parameters to validate the numerical scheme.

Dispersive mixed-order systems in $L^p$-Sobolev spaces and application to the thermoelastic plate equation

We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation with and without inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.

Hyperbolic thermoelasticity in gas medium

A numerical approach is used to investigate waves propagation due to laser radiation. A thin layer of gas medium is considered, and gas is modeled based on ideal gas concept. The aim of the research is to examine time relaxation constant’s influence on the coupled thermoelastic system. Heat flux relaxation according to Maxwell–Cattaneo law is studied. The spatial description is used in order to describe continuum fields. Density, temperature, heat flux and velocity are evaluated through the system of balance equations: mass, energy and momentum balances are taken in integral form. The high-speed thermal impact is modeled by defining the distribution of heat sources in the volume for the semitransparent medium with help of Bouguer law. The power of the laser pulse depends on time as the Dirac delta function or as the Heaviside function do. A number of problems with different time relaxation constant are considered, and the interaction between thermal and acoustic waves is examined. For numerical calculation, an explicit technique is applied.

Utilization of Maxwell-Cattaneo law for MHD swirling flow through oscillatory disk subject to porous medium

The present study aims to investigate the salient features of incompressible, hydromagnetic, three-dimensional flow of viscous fluid subject to the oscillatory motion of a disk. The rotating disk is contained in a porous medium. Furthermore, a time-invariant version of the Maxwell-Cattaneo law is implemented in the energy equation. The flow problem is normalized by obtaining similarity variables. The resulting nonlinear system is solved numerically using the successive over-relaxation method. The main results are discussed through graphical representations and tables. It is perceived that the thermal relaxation time parameter decreases the temperature curves and increases the heat transfer rate. The oscillatory curves for the velocity field demonstrate a decreasing tendency with the increasing porosity parameter values. Two- and three-dimensional flow phenomena are also shown through graphical results.

Model Interpretation of BSPM Based SCD Risk Markers

This paper deals with the numerical (finite volume) approximation of reaction–diffusion systems with relaxation, among which the hyperbolic extension of the Allen–Cahn equation – given by where τ,μ>0 and f is a cubic-like function with three zeros – represents a notable prototype.Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process, given by the substitution of the Fourier's law with the Maxwell–Cattaneo's law. A corresponding pseudo-kinetic scheme is also proposed for the Guyer–Krumhansl's law.For the Maxwell–Cattaneo case, numerical experiments1 are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves under a sign condition on the damping term g:=1−τf′. Moreover, important evidence of the validity of those results beyond the formal hypotheses g(u)>0 for any u is numerically established.

On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials

Our aim in this article is to study generalizations of the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Furthermore, we give some numerical simulations, obtained with the $\mathtt{FreeFem++}$ software [ 24 ], comparing the nonconserved Caginalp phase-field model with regular and logarithmic nonlinear terms.

Kinetic schemes for assessing stability of traveling fronts for the Allen–Cahn equation with relaxation

This paper deals with the numerical (finite volume) approximation of reaction–diffusion systems with relaxation, among which the hyperbolic extension of the Allen–Cahn equation – given byτ∂ttu+(1−τf′(u))∂tu−μ∂xxu=f(u), where τ,μ>0 and f is a cubic-like function with three zeros – represents a notable prototype.Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process, given by the substitution of the Fourier's law with the Maxwell–Cattaneo's law. A corresponding pseudo-kinetic scheme is also proposed for the Guyer–Krumhansl's law.For the Maxwell–Cattaneo case, numerical experiments1 are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves under a sign condition on the damping term g:=1−τf′. Moreover, important evidence of the validity of those results beyond the formal hypotheses g(u)>0 for any u is numerically established.

On the nature of the relaxation time, the Maxwell–Cattaneo and Fourier law in the thermodynamics of a continuous medium, and the scale effects in thermal conductivity

We present the theory of space–time elasticity and demonstrate that it is the extended reversible thermodynamics and gives the coupled model of thermoelasticity and heat conductivity and involves traditional thermoelasticity. We formulate the generally covariant variational model’s dynamic thermoelasticity and heat conductivity in which the basic kinematic and static variables are unified tensor objects (subject, matter). Variation statement defines the whole set of the initial-boundary problems for the 4D vector governing equation (Euler equation), the spatial projections of which define motion equations and the time projection gives the heat conductivity equation. We show that space–time elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, space–time elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity. Moreover, we establish that the Maxwell–Cattaneo law and Fourier law can be defined for the reversible processes as compatibility equations without introducing dissipation. We argue that the present framework of space–time elasticity should prove adequate to describe the thermoelastic phenomena at low temperatures for interpreting the results of molecular simulations of heat conduction in solids and for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.

Effect of Gravity Modulation on Single Component Convection in a Couple Stress Fluid with Maxwell-Cattaneo Law

The outset of convection in a thin layer of couple stress fluid is analyzed using the linear stability analysis when the fluid is heated from below. In order to assimilate the inertial effects Maxwell-Cattaneo law is used in lieu of the classical Fourier's heat conduction law. The normal mode analysis is used to arrive at the eigenvalues of the perturbed state and a regular perturbation method to find the analytical solutions. The effect of Cattaneo number, couple stress parameter and Prandtl number is discussed and it is concluded that gravity modulation can delay or advance the onset of convection.

Nonlinear heat conduction equations with memory: Physical meaning and analytical results

We study nonlinear heat conduction equations with memory effects within the framework of the fractional calculus approach to the generalized Maxwell–Cattaneo law. Our main aim is to derive the governing equations of heat propagation, considering both the empirical temperature-dependence of the thermal conductivity coefficient (which introduces nonlinearity) and memory effects, according to the general theory of Gurtin and Pipkin of finite velocity thermal propagation with memory. In this framework, we consider in detail two different approaches to the generalized Maxwell–Cattaneo law, based on the application of long-tail Mittag–Leffler memory function and power law relaxation functions, leading to nonlinear time-fractional telegraph and wave-type equations. We also discuss some explicit analytical results to the model equations based on the generalized separating variable method and discuss their meaning in relation to some well-known results of the ordinary case.

Global weak solutions to a model of micropolar fluids with Maxwell–Cattaneo heat transfer law

This work is devoted to the study of a system describing the dynamics of a heated micropolar fluid under the action of an applied magnetic field. The heat transfer is subjected to Maxwell–Cattaneo law instead of the usual Fourier law. Thanks to regularization and compactness methods the global existence in time of weak solutions with finite energy has been proved.

On the Caginalp phase‐field systems with two temperatures and the Maxwell–Cattaneo law

Our aim in this paper is to study generalizations of the nonconserved and conserved Caginalp phase‐field systems based on the Maxwell–Cattaneo law with two temperatures for heat conduction. In particular, we obtain well‐posedness results and study the dissipativity of the associated solution operators. Copyright © 2016 John Wiley & Sons, Ltd.

Deviation from the Maxwell-Cattaneo law: Role of asymmetric interparticle interactions.

Nonstationary heat conduction in a few one-dimensional nonlinear lattices is studied numerically based on the Maxwell-Cattaneo (MC) law. We simulate the relaxation process and calculate the magnitudes of the temperature oscillation A(T)(t) and the local heat current oscillation A(j)(t). A phase difference between A(T)(t) and A(j)(t) is observed, which not only verifies the existence of the time lag τ in the MC law but also provides a better way of determining the critical wavelength L(*) that separates between oscillatory and diffusive relaxation modes. However, clear deviations from the MC law are observed. Not only do the decay exponents differ from the theoretical expectations, but, more importantly, suboscillation in the diffusive regime, which is not expected by the MC law, is found in the lattices with asymmetric interactions as well. These findings imply that higher-order effects must be considered in order to well describe the nonstationary heat conduction process in these systems.

Gradient on Rayleigh–Bénard – Marangoni – Magnetoconvection in a Micropolar Fluid with Maxwell – Cattaneo Law

The effect of non-uniform temperature gradient on the onset of Rayleigh-Bénard-Marangoni- Magneto-convection in a Micropolar fluid with Maxwell-Cattaneo law is studied using the Galerkin technique. The eigen value is obtained for rigid-free velocity boundary combination with isothermal and adiabatic 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 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.

Global Weak Solutions to Magnetic Fluid Flows with Nonlinear Maxwell–Cattaneo Heat Transfer Law

We discuss the equations describing the dynamic of the heat transfer in a magnetic fluid flow under the action of an applied magnetic field. Instead of the usual heat transfer equation we use a generalization given by the Maxwell–Cattaneo law which is a system satisfied by the temperature and the heat flux. We prove a global existence of weak solutions to the system having a finite energy.

A reformulation of the Caginalp phase-field system based on the Maxwell–Cattaneo law

Our aim in this paper is to study the well-posedness and the dissipativity for a reformulation of the Caginalp phase-field system based on the Maxwell–Cattaneo law, instead of the usual Fourier law, for heat conduction. In particular, instead of the equation for the relative temperature (or the thermal displacement variable), we consider here the equation for the enthalpy.

Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law

Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, for the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law, instead of the usual Fourier law, for heat conduction. The system consists of the equation for the order parameter and that for the enthalpy, instead of the relative temperature or the thermal displacement variable.

Gravitational Instability in a Ferromagnetic Fluid Saturated Porous Medium with Non-Classical Heat Conduction

The problem of Rayleigh-Benard convection in a ferromagnetic fluid saturated porous medium with the Maxwell-Cattaneo law is studied by the method of small perturbation. Modified Darcy-Brinkman model is used to describe the fluid motion. The horizontal porous layer is cooled from the upper boundary, while an isothermal boundary condition is imposed at the lower boundary. The non-classical Maxwell-Cattaneo heat flux law involves a wave type heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. The resulting eigenvalue problem is solved exactly for simplified boundary conditions and the thresholds for the marginal stability are determined. Some of the known cases are derived as special cases. The influence of porous, magnetic, and non-magnetic parameters on the onset of ferroconvection has been analyzed. It is found that the Benard problem for a Maxwell-Cattaneo ferromagnetic fluid is always less stable than the classical ferroconvection problem. It is shown that the destabilizing influence of the Cattaneo number is not attenuated by that of magnetic forces and vice versa, and that the aspect ratio of the convection cells changes when the parameters involved in the study vary with the porous structure bringing out considerable influence.

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.