Hyperbolic Stefan Problem Research Articles

Heat transfer in rapidly solidifying supercooled pure melt during final transient

The heat transfer model for a one-dimensional supercooled melt during the final stage of solidification is considered. The Stefan problem for the determination of the temperature distribution is solved under the condition that (i) the interface approaches the specimen surface with a constant velocity V; (ii) the latent heat of solidification linearly depends on the interface temperature; (iii) all the physical quantities given at the phase boundary are presented by linear combinations of the exponential functions of the interface position. First we find the solution of the corresponding hyperbolic Stefan problem within the framework of which the heat transfer is described by the telegraph equation. The solution of the initial parabolic Stefan problem is then found as a result of the limiting transition V/VH→0(VH→∞), where VH is the velocity of the propagation of the heat disturbances, in which the hyperbolic heat model tends to the parabolic one.

Quasilinear hyperbolic stefan problem with nonlocal boundary conditions

Using the method of contracting mappings, we prove, for small values of time, the existence and uniqueness of a generalized Lipschitz solution of a mixed problem with unknown boundaries for a hyperbolic quasilinear system of first-order equations represented in terms of Riemann invariants with nonlocal (nonseparated and integral) boundary conditions.

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.

Automatic Control via Thermostats of a Hyperbolic Stefan Problem with Memory

A hyperbolic Stefan problem based on the linearized Gurtin—Pipkin heat conduction law is considered. The temperature and free boundary are controlled by a thermostat acting on the boundary. This feedback control is based on temperature measurements performed by real thermal sensors located within the domain containing the two-phase system and/or at its boundary. Three different types of thermostats are analyzed: simple switch, relay switch, and a Preisach hysteresis operator. The resulting models lead to integrodifferential hyperbolic Stefan problems with nonlinear and nonlocal boundary conditions. Existence results are proved in all the cases. Uniqueness is also shown, except in the situation corresponding to the ideal switch.

Hyperbolic stefan problem in a curvilinear sector

The problem with unknown boundaries for a first-order semilinear hyperbolic system is studied in the case where the curve of definition of the initial conditions degenerates to a point. An existence and uniqueness theorem for a classical solution of the problem is proved for small t.

Existence and stability results for the hyperbolic Stefan problem with relaxation

The aim of this paper is to study a phase transition model, based on the Cattaneo-Fourier constitutive law for the heat flux and on a relaxed constitutive law for the phase variable. In turn, the model describes fast processes of melting and crystallization with supercooling and superheating effects. We give existence and stability results for the former phase transition problem. Uniqueness is deduced from the stability result

Hyperbolic phase change problems in heat conduction with memory

SynopsisThe aim of this paper is to formulate and study phase transition problems in materials with memory, based on the Gurtin–Pipkin constitutive assumption on the heat flux. As different phases are involved, the internal energy is allowed to depend on the phase variable (besides the temperature) and to take its past history into account. By considering the standard equilibrium condition at the interface between two phases, we deal with a hyperbolic Stefan problem reckoning with memory effects. Then, substituting this equilibrium condition with a relaxation dynamics, we represent some dissipation phenomena including supercooling or superheating. The application of a fixed point argument helps us to show the existence and uniqueness of the solution to the latter problem (still of hyperbolic type). Hence, by introducing a suitable regularisation and taking the limit as a kinetic parameter goes to zero, we prove an existence result for the former Stefan problem. Moreover, its uniqueness is deduced by contradiction.

Formulation and solution of hyperbolic Stefan problem

The interface condition for hyperbolic phase change problems, which includes sensible heat at the interface, is derived as an extension of the interface condition for standard parabolic phase change problems. The enthalpy formulation of the hyperbolic Stefan problem is presented and is used to numerically solve for the temperature distributions and the interface position. MacCormack’s predictor–corrector method is applied to solve the hyperbolic phase change problem and is validated by comparing the limiting case where the thermal relaxation parameter approaches zero to the parabolic phase change problem, in which the relaxation parameter is zero. Solutions with an applied surface temperature greater than the melt temperature are presented for two different Stefan numbers. It is noted that a discontinuity occurs at the phase change interface as well as at the thermal front.

HYPERBOLIC STEFAN PROBLEM WITH APPLIED SURFACE HEAT FLUX AND TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

Abstract The hyperbolic Stefan problem with an applied surface heat flux and temperature-dependent thermal conductivity is solved numerically for a semi-infinite slab using Mac-Cormack's predictor-corrector method. Solutions are presented for cases where the melt temperature is both below and above the instantaneous jump in surface temperature at time t = O+. The interface condition, surface temperature, and internal temperatures are presented for different Stefan numbers and melt temperatures, as well as thermal conductivity both increasing and decreasing with temperature. The results obtained from the hyperbolic solution are compared with those obtained from the parabolic solution.

A 3-dimensional hyperbolic Stefan problem with discontinuous temperature

The hyperbolic heat transfer model is obtained by replacing the classical Fourier’s law with the relaxation relation τ q t + q = − k ∇ T \tau {q_t} + q = - k\nabla T . The conditions are derived for the local existence and uniqueness of classical solutions for a 3-dimensional Stefan problem of hyperbolic heat transfer model where the temperature may sustain a jump across the phase change interface.

A one-phase hyperbolic Stefan problem in multi-dimensional space

The hyperbolic heat transfer model is obtained by replacing the classical Fourier’s law with the relaxation relation τ q → t + q → = − k ∇ T \tau \vec qt + \vec q = - k\nabla T . The sufficient and necessary conditions are derived for the local existence and uniqueness of classical solutions for multi- D {\text {D}} Stefan problem of hyperbolic heat transfer model where phase change is accompanied with delay of latent heat storage.

The well-posedness of a hyperbolic Stefan problem

On etudie le probleme de Stefan pour le modele de chaleur hyperbolique dans la formulation classique. Pour les problemes a une phase et a 2 phases on obtient les solutions locales et globales

Approximation of multidimensional Stefan-like problems via hyperbolic relaxation

An effective time delay in the governing equations, as proposed in [8], is further examined here. The resulting hyperbolic Stefan problem is mainly viewed as a perturbation of the traditional one. Asymptotic energy error estimates are derived for the three physical unknowns (heat flux, temperature and enthalpy) in accordance with various regularity assumptions on the initial data.

A hyperbolic Stefan problem

A free-boundary problem of Stefan type is presented under constitutive assumptions on flux and energy which contain an effective time delay. This contains the hyperbolic telegraphers equation and, hence, has the feature that propagation speed of disturbances is bounded. With the appropriate physically consistent condition on the interface this is shown to lead to a well-posed weak formulation of the problem.

A hyperbolic Stefan problem

Heat conduction is considered in a semi-infinite solid subjected to a high step change in surface heat flux, such that melting occurs. A time-dependent relaxation model for the energy flux is assumed, leading to a non-Fourier, non-linear equation for the thermal field, which is solved under suitable conditions on the interface displacement.