Heat Flux Law Research Articles

On a model of phase relaxation for the hyperbolic Stefan problem

In this paper we study a model of phase relaxation for the Stefan problem with the Cattaneo–Maxwell heat flux law. We prove an existence and uniqueness result for the resulting problem and we show that its solution converges to the solution of the Stefan problem as the two relaxation parameters go to zero, provided a relation between these parameters holds.

Modulation of mantle plumes and heat flow at the core mantle boundary by plate-scale flow: results from laboratory experiments

We report results from analog laboratory experiments, in which a large-scale flow is imposed upon natural convection from a hot boundary layer at the base of a large tank of corn syrup. The experiments show that the subdivision of the convective flow into four regions provides a reasonable conceptual framework for interpreting the effects of large-scale flow on plumes. Region I includes the area of the hot thermal boundary layer (TBL) that is thinned by the large-scale flow, thereby suppressing plumes. Region II encompasses the critically unstable boundary layer where plumes form. Region III is the area above the boundary layer that is devoid of plumes. Region IV comprises the area of hot upwelling and plume conduits. Quantitative analysis of our experiments results in a scaling law for heat flux from the hot boundary and for the spatial extent of plume suppression. When applied to the Earth's core–mantle boundary (CMB), our results suggest that large-scale mantle flow, due to sinking lithospheric plates, can locally thin the TBL and suppress plume formation over large fractions of the CMB. Approximately 30% of heat flow from the core may be due to increased heat flux from plate-scale flow. Furthermore, CMB heat flux is non-uniformly distributed along the CMB, with large areas where heat flux is increased on average by a factor of 2. As a consequence, the convective flow pattern in the outer core may be affected by CMB heat-flux heterogeneity and sensitive to changes in plate-scale mantle flow. Because of plume suppression and ‘focusing’ of hot mantle from the CMB into zones of upwelling flow, plume conduits (hotspots) are expected to be spatially associated with lower-mantle regions of low seismic velocities, inferred as hot upwelling mantle flow.

Universal Attractor for a Penrose-Fife System with Special Heat Flux Law

A Penrose-Fife system for non isothermal phase transitions with non conserved order parameter is introduced. A linear growth of the latent heat density with respect to the phase field is allowed. Continous dependence on data and the existence of the universal attractor for the associated nonlinear semigroup are shown. These properties hold with respect to a strong metric accounting for the nonlinear and even singular terms characterizing the system. The present analysis extends a former result by the same authors, holding in the case of a constant latent heat.

On a Penrose-Fife phase-field model with nonhomogeneous Neumann boundary conditions for the temperature

This work is concerned with the study of an initial and boundary value problem for a nonconserved system of phase field equations arising from the Penrose-Fife approach to phase transitions problems. Several works deal with variations of the same problem coupled with third type boundary conditions for the heat flux. On the contrary, our aim is to consider the case of the nonhomogeneous Neumann boundary condition for the heat flux, to find well-posedness for a weak formulation of this problem, and to prove a regularity result in case of smoother data and a slightly less general heat flux law.

Three scale thermomechanical theory for swelling biopolymeric systems

A three-scale theory for the swelling polymeric/biopolymeric media is developed via the hybrid mixture theory. At the microscale, the solid polymeric matrix interacts with the solvent through surface contact. At the mesoscale, the homogeneous mixture of vicinal fluid and solid polymers exchanges thermodynamic properties with two bulk fluids, one of which is of the same type as the vicinal fluid. The relaxation processes within the polymeric matrix are incorporated by modeling the solid phase as viscoelastic and the solvent phases as viscous at the macroscale. We obtain novel equations for the total stress tensor, chemical potential of the solid phase, heat flux and Darcy's law all at the macroscale. Viscoelastic stress components in Darcy's law make it applicable for both Fickian and non-Fickian fluid transport. The form of the generalized Fick's law is similar to that obtained in earlier works involving colloids. Thermoviscoelastic and thermoviscous effects are incorporated by coupling thermal gradients with strain-rate tensors for the solid phase and the deformation-rate tensors for the liquid phases.

Temperature inversion in granular fluids under gravity

We study, via hydrodynamic equations, the granular temperature profile of a granular fluid under gravity and subjected to energy injection from a base. It is found that there exists a turn-up in the granular temperature and that, far from the base, it increases linearly with height. We show that this phenomenon, observed previously in experiments and computer simulations, is a direct consequence of the heat flux law, different form Fourier's, in granular fluids. The positive granular temperature gradient is proportional to gravity and a transport coefficient μ 0, relating the heat flux to the density gradients, that is characteristic of granular systems. Our results provide a method to compute the value μ 0 for different restitution coefficients. The theoretical predictions are verified by means of molecular dynamics simulations, and the value of μ 0 is computed for the two-dimensional inelastic hard sphere model. We provide, also, a boundary condition for the temperature field that is consistent with the modified Fourier's law.

The conserved Penrose–Fife system with Fourier heat flux law

A system of phase field equations of Penrose–Fife-type governing the dynamics of phase transitions with a conserved order parameter is considered. As in the original model, the heat flux is assumed to be given by the Fourier law. Existence of weak solutions is proved for the related initial–Neumann boundary value problem.

Visco-elastic MHD flow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work

The present paper deals with the study of momentum, heat and mass transfer characteristics in a viso-elastic fluid flow over a porous sheet, where the flow is generated due to linear stretching of the sheet and influenced by a uniform magnetic field applied vertically and a continuous injection of the fluid through porous boundary. In the flow region, heat balance is maintained with a temperature dependent heat source/sink, viscous dissipation, dissipation due to elastic deformation and stress work produced as the result of magnetic field on the non-Newtonian fluid. In mass transfer analysis we have taken into account the loss of mass of the chemically reactive diffusive species by means of first order chemical conversion rate. Using suitable similarity transformations on the highly non-linear partial differential equations we derive several closed form analytical solutions for non-dimensional temperature, concentration, heat flux, mass flux profiles in the form of confluent hyper geometric (Kummer's) functions and some other elementary functions as its special form, for two different cases of the boundary conditions, namely, (i) wall with prescribed second order power law temperature (PST) and prescribed second order power law concentration (ii) wall with prescribed second order power law heat flux (PHF) and prescribed second order power law mass flux. The effect of the non-dimensional magnetic parameter on momentum, heat and mass transfer characteristics for non-isothermal boundary condition and different physical situations of the fluid, having various degrees of visco-elasticity, Prandtl number, heat source/sink strength and Schmidt number, are discussed in detail. Some of the several important findings reported in this paper are: (i) The combined effect of magnetic field, visco-elasticity and impermeability of the wall is to increase skin-friction largely at the wall; (ii) maximum enhancement of wall-temperature profile due to the application of transverse magnetic field occurs when the boundary heating is maintained with prescribed heat flux, boundary wall is porous and Prandtl number of the fluid is low; (iii) the effect of transverse uniform magnetic field is to increase concentration in the flow region, however, enhancement of concentration is higher when the stretching wall is porous and subjected to injection and (iv) the reduction of concentration due to chemical conversion is of significant order near the wall in PHF case when the wall is maintained with prescribed power low mass flux, in comparison with the PST case when the wall is maintained with prescribed power law surface concentration.

The Conserved Penrose-Fife Phase Field Model with Special Heat Flux Laws and Memory Effects

In this paper a phase-field model of PenroseFife type is considered for diffusive phase transitions with conserved order parameter. Different motivations lead to investigate the case when the heat flux is the superposition of two different contributions; one part is the gradient of a function of the absolute temperature θ, behaving like 1/θ as θ approaches to 0 and like −θ as θ ↗ +∞, while the other is given by the Gurtin-Pipkin law introduced in the theory of materials with thermal memory. An existence result for a related initial-boundary value problem is proven. Strengthening some assumptions on the data, the uniqueness of the solution is also achieved.

Convergence to the Stefan problem of the phase relaxation problem with Cattaneo heat flux law

In this paper we show that the solutions of the phase change problem with the Cattaneo-Fourier heat flux law and phase relaxation, converge to the solution of the Stefan problem as the two relaxation parameters go independently to zero.

Heat and mass transfer in a visco-elastic fluid flow over an accelerating surface with heat source/sink and viscous dissipation

In this paper we present a mathematical analysis of heat and mass transfer phenomena in a visco–elastic fluid flow over an accelerating stretching sheet in the presence of heat source/sink, viscous dissipation and suction/blowing. Similarity transformations are used to convert highly non-linear partial differential equations into ordinary differential equations. Several closed form analytical solutions for non-dimensional temperature, concentration, heat flux, mass flux profiles are obtained in the form of confluent hypergeometric (Kummer's) functions for two different cases of the boundary conditions, namely, (i) wall with prescribed second order power law temperature and second order power law concentration (PST), and (ii) wall with prescribed second order power law heat flux and second order power law mass flux (PHF). The effect of various physical parameters like visco–elasticity, Eckert number, Prandtl number, heat source/sink, Schmidt number and suction/blowing parameter on temperature and concentration profiles are analysed. The effects of all these parameters on wall temperature gradient and wall concentration gradient are also discussed.

Asymptotic analysis of a diffusive model for shape memory alloys with Cattaneo-Maxwell heat flux law

This work is concerned with a diffusive model for some austenite-martensite phase transition processes ruled by the Cattaneo-Maxwell heat flux law, namely assuming for the heat flux a relaxed version of the classical Fourier law. A rigorous asymptotic analysis of the macroscopic model is performed and it is shown that such model is nothing but a singular perturbation of the standard Frémond model for shape memory alloys. Convergence results are proved along with error estimates.

Hyperbolic Phase-Field Dynamics with Memory

We consider a non-conserved phase-field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations. This model is derived from two basic assumptions: the heat flux law is of Gurtin–Pipkin type and the response of the order parameter to the variation of the free energy functional is delayed. These hypotheses might be a reasonable attempt to describe, for instance, the melt of He4 crystals. A suitable initial and boundary value problem is then associated with the system and its well-posedness is analyzed in detail.

Asymptotic behavior of a nonlinear hyperbolic heat equation with memory

In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an integro-differential equation describing the heat flow in a rigid heat conductor with memory. This model arises matching the energy balance, in presence of a nonlinear time-dependent heat source, with a linearized heat flux law of the Gurtin-Pipkin type. Existence and uniqueness of solutions for the corresponding semilinear system (subject to initial history and Dirichlet boundary conditions) is provided. Moreover, under proper assumptions on the heat flux memory kernel and the magnitude of nonlinearity, the existence of a uniform absorbing set is achieved.

Non-Fourier Heat-Flux Law for Diatomic Gases

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Statistical Mechanical Derivation of Cattaneo's Heat Flux Law

Statistical Mechanical Derivation of Cattaneo's Heat Flux Law

Asymptotic behavior of a semilinear problem in heat conduction with memory

This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved.

Regularity and convergence results for a phase-field model with memory

A phase–field model based on the Coleman–Gurtin heat flux law is considered. The resulting system of non-linear parabolic equations, associated with a set of initial and Neumann boundary conditions, is studied. Existence, uniqueness, and regularity results are proved. An asymptotic analysis is also carried out, in the case where the coefficient of the interfacial energy term tends to 0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.

Exact solution of heat transfer from a stretching surface with variable heat flux

An exact expression of the temperature distribution is constructed for the heat transfer from a stretching surface with prescribed power law heat flux. The stretching velocity is inversely proportional to the one third power of the distance measured along the surface from a thin slit. The final result is expressed in terms of hypergeometric functions. Although the exact solution is accomplished, some physically unrealistic phenomena are encounters for specific conditions. The temperature parameter which prescribe the surface heat flux, strongly affects those situations. Two types of temperature distribution are discussed: dimensionless temperatures with and without scaling to the dimensionless surface temperature. The expression of the temperature distribution without scaling is lucid to understand the heat transfer characteristics.

Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws

This paper is concerned with a thermodynamically consistent model for diffusive phase transitions proposed by Penrose and Fife. The model has recently received a good deal of attention, and the related initial-boundary value problems have been investigated from the mathematical point of view. In the case where the order parameter is not conserved, the common working assumption for the heat flow was that such a flux is proportional to the gradient of the inverse absolute temperature. This position seems to be helpful for the analysis, due to the coupling term in the phase-field equation which depends right on the inverse temperature. Here we prove the existence of weak solutions in a wide setting of constitutive laws for the heat flux, different from the previous ones and considerably more significant owing to their behaviour for high temperatures. We also discuss the limiting situation of zero interfacial energy for the order parameter.