Exact solution for a two-phase Stefan problem with power-type latent heat

Zhou, Yang; Shi, Xiang-you; Zhou, Guoqing

doi:10.1007/s10665-017-9921-y

Cited by 13 publications

(16 citation statements)

References 26 publications

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“…As the right hand side of equation (3.16) is an increasing function that increases from 0 to +∞, we can assure that (3.16) has a unique positive solution. We remark here that the solution of the problem (3.2)-(3.8) was obtained in [23] by using results for a heat flux condition from an argument not so clear for us, and for this reason we have proved it with details.…”

Section: Case When α Is a Non-negative Integermentioning

confidence: 77%

“…The main contribution of this paper is to generalize the work that has been done in [18], [25] and [3], by obtaining the explicit solution of a one-dimensional two-phase Stefan problem for a semiinfinite material where a variable latent heat and a convective boundary condition at the fixed face is considered; as well as to obtain the results given in [23] when the coefficient that characterizes the convective boundary condition goes to infinity.…”

Section: Introductionmentioning

confidence: 99%

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Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

Bollati

Tarzia

2018

Z. Angew. Math. Phys.

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Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 1-11] for the classical twophase Lamé-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in

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Section: Case When α Is a Non-negative Integermentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

Bollati

Tarzia

2018

Z. Angew. Math. Phys.

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“…It is well known that a typical time-dependence of the function Γ(t, x, y) is power-law. In particular, the time-dependence Γ = x − t 1/2 was established by J. Stefan in the 1D space case for the ice melting problem mentioned above, and such dependence occurs in many other situations (see, e.g., the recent papers [24,25,27,28,35,36]). Another typical profile is Γ = x−vt (v is an unknown velocity of the moving boundary), which occurs, for example, in the model describing the metal melting and evaporation under power energy fluxes [37].…”

Section: The Model and Its Lie Symmetriesmentioning

confidence: 91%

“…In fact, the structure of such boundaries may depend on invariant variable(s) and this gives a possibility to reduce the given BVP to that of lover dimensionality. This is the reason why different authors applied the Lie symmetry method for solving BVPs with free boundaries [8,[21][22][23][24][25][26][27][28][29][30]. It should be stressed that a majority of these papers are devoted to solving of two-dimensional problems while only a few of them are dealing with multidimensional BVPs [21,23,26].…”

Section: Introductionmentioning

confidence: 99%

Lie symmetries, reduction and exact solutions of the (1+2)-dimensional nonlinear problem

Cherniha,

Davydovych

2019

Preprint

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The well known nonlinear model for describing the solid tumour growth [Byrne HM., et al. Appl Math Letters 2003;16:567-74] is under study using an approach based on Lie symmetries. It is shown that the model in the two-dimensional (in space) approximation forms a (1+2)-dimensional boundary value problem, which admits a highly nontrivial Lie symmetry. The special case involving the power-law nonlinearities is examined in details. The symmetries derived are applied for the reduction of the nonlinear boundary value problem in question to problems of lower dimensionality. Finally, the reduced problems with correctly-specified coefficients were exactly solved and the exact solutions derived were analysed, in particular, some plots were build in order to understand the time-space behaviour of these solutions and to discuss their biological interpretation.

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“…In [7], it was developed a mathematical model for the shoreline movement in a sedimentary basin using an analogy with the one-phase melting Stefan problem with a variable latent heat. Besides, in [8], it was introduced a two-phase Stefan problem with a general type of space-dependent latent heat from the background of the artificial ground-freezing technique.…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

Bollati¹,

Tarzia²

2020

Eur. J. Appl. Math

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The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

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Exact solution for a two-phase Stefan problem with power-type latent heat

Cited by 13 publications

References 26 publications

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

Lie symmetries, reduction and exact solutions of the (1+2)-dimensional nonlinear problem

Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

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