Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

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.

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification. Here we focus on Stefan problems, and their quasi-static variants, with applications to crystal growth. New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves interactions between gas, water and ice. The dynamics are driven by melting of the ice layer, while the pressurized gas also dissolves within the meltwater. The model incorporates the Stefan condition at the water–ice interface along with Henry's law for dissolution of gas at the gas–water interface. We employ a quasi-steady approximation for the phase temperatures and then derive a series solution for the interface positions. A non-standard feature of the model is an integral free boundary condition that arises from mass conservation owing to changes in gas density at the gas–water interface, which makes the problem non-self-adjoint. We derive a two-scale asymptotic series solution for the dissolved gas concentration, which because of the non-self-adjointness gives rise to a Fourier series expansion in eigenfunctions that do not satisfy the usual orthogonality conditions. Numerical simulations of the original governing equations are used to validate series approximations.

The numerical modeling of a binary solidification with a mushy layer mechanism is considered in this manuscript. The nonlinear coupled system of equations describes the heat and mass diffusions of a one-dimensional spatial variable in the semi-infinite interval. Also formulated is a transformed system in a finite interval. We propose numerical methods for solving the nonlinear system using a threshold strategy based on fixed computation-domain approach. Our calculated results and those from the LeadEx field experiment are well-matched in their tendencies.

The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs–Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law in a weak generalised BV-formulation.

The freezing of water to ice is a classic problem in applied mathematics, involving the solution of a diffusion equation with a moving boundary. However, when the water is salty, the transport of salt rejected by ice introduces some interesting twists to the tale. A number of analytic models for the freezing of water are briefly reviewed, ranging from the famous work by Neumann and Stefan in the 1800s, to the mushy zone models coming out of Cambridge and Oxford since the 1980s. The successes and limitations of these models, and remaining modelling issues, are considered in the case of freezing sea-water in the Arctic and Antarctic Oceans. A new, simple model which includes turbulent transport of heat and salt between ice and ocean is introduced and solved analytically, in two different cases—one where turbulence is given by a constant friction velocity, and the other where turbulence is buoyancy-driven and hence depends on ice thickness. Salt is found to play an important role, lowering interface temperatures, increasing oceanic heat flux, and slowing ice growth.

We consider a fully practical finite element approximation of the following degenerate system $$ {\frac{\partial }{\partial t}} \rho(u)- \nabla . ( \,\alpha(u) \,\nabla u ) \ni \sigma(u)\,|\nabla\phi|^2 ,\quad \nabla . (\, \sigma(u) \,\nabla \phi ) = 0$$ subject to an initial condition on the temperature, u,and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0is the electricalconductivity. The latter is zero in the frozen zone, u ≤ 0,which gives rise to the degeneracy in this Stefan system.In addition to showing stability bounds, we prove (subsequence) convergence of our finite element approximation intwo and three space dimensions. The latter is non-trivial due to the degeneracy in σ(u)and the quadratic nature of the Joule heating term forcing the Stefan problem.Finally, some numerical experiments are presented in two space dimensions.