The spreading of fluid under gravity occurs in both nature and man-made situations and has been the subject of many previous studies. Considerably less attention has been paid to cases in which there is a strong thermal coupling influencing the flow. In this paper, simple models of the spreading of materials with temperature-dependent viscosity are presented and features that are commonly seen in experiments, such as plateauing and fingering, are shown to result. A model which includes latent heat effects is also briefly outlined.

D. Kessler, O. Krüger, J. Rappaz and J.F. Scheid. A phase-field model for the isothermal solidification process of a binary alloy. CAMES 2000 (7)

We study a phase-field model for the isothermal solidification of a binary alloy which involves the relative concentration and the order parameter. We prove the existence of weak solutions as well as regularity and uniqueness results under Lipschitz and boundeness assumptions for the nonlinearities. A maximum principle holds that justifies these assumptions. A numerical approximation and some numerical results are also presented.

D. Gobin and P. Le Quéré. Melting from an isothermal vertical wall. Synthesis of a numerical comparison exercise. CAMES 2000 (7)

This paper provides an analysis of the results of a comparison exercise on the numerical 2D solution of melting from a vertical wall, dominated by natural convection in the liquid phase. The thirteen contributions to this exercise cover the great variety of mathematical models and numerical procedures most commonly used in this field. The main conclusions presented at the AMIF Workshop (PCC99) held in Warsaw in June 1999 and at the Moving Boundaries Seminar in Ljubljana are summarized in the paper. They emphasize the need for the definition of such reference validation tests.

M. Medale, M. Jaeger and A. Kaiss. Finite element analysis of the action of buoyancy-induced and thermocapillary flow on the melting of tin in a 2D square cavity. CAMES 2000 (7)

A finite element model has been developed for the computation of melting/solidifying process under the combined action of buoyancy and surface tension forces. Validated on the square cavity benchmark of Gobin and Le Quéré (Bertrand *et al.* [1], Gobin and Le Quéré [2]), the numerical model is used to extend this previous analysis to the free surface case where surface tension can drive the flow (capillary flow). A comparison of the results obtained for three types of boundary conditions applied at the top of the melting pool is performed. It shows that in the studied case of tin where the thermal Bond number is moderated (*Bo*=200), the flow is still mainly dominated by buoyancy effect as long as the melted pool is deep enough like in the square cavity case of the above mentioned benchmark.
[1] O. Bertrand *et al.*: Melting driven by natural convection. A comparison exercise. Int. J. of Therm. Sci., 38: 5-26, 1999
[2] D. Gobin, P. Le Quéré: Melting in enclosures: Coupled heat transfer and natural convection. In: T.A. Kowalewski *et al.*, eds., ESF-AMIF Workshop on Phase Change with Convection, Modeling and Validation, 13-18, 1999.
*Keywords:* phase change (melting, freezing), natural convection (buoyancy, thermocapillary flows), incompressible Navier-Stokes equations, finite element method.

M. Giangi, T.A. Kowalewski, F. Stella and E. Leonardi Natural convection during ice formation: numerical simulation vs. experimental results. CAMES 2000 (7)

A numerical and experimental study is presented of unsteady natural convection during freezing of water in a differentially heated cube shaped cavity. A boundary fitted grid as well as the enthalpy-porosity fixed grid numerical models are used in this study. Both numerical models show very good agreement with the experimental data only for pure convection and initial time of freezing process. As time passes the discrepancies between numerical predictions and the experiment became more significant. To elucidate these differences several numerical tests are performed, verifying assumptions made in the models.

J. Banaszek and P. Furmanski. FEM analysis of binary dilute system solidification using the anisotropic porous medium model of a mushy zone. CAMES 2000 (7)

Finite Element Method (FEM) calculations have been performed to address the problem of the influence of anisotropy of permeability and of thermal conductivity of a mushy region on a temporary flow pattern and temperature during solidification of binary mixtures. Computationally effective FEM algorithm is based on the combination of the projection method, the semi-implicit time marching scheme and the enthalpy-porosity model of the two-phase region. Example calculations are given for two different dilute solutions of ammonium chloride and water. The effect of permeability anisotropy considerably changes the shape of the mushy zone. Three different models of thermal conductivity, the first - based on a mixture theory, the second - fully anisotropic one and the third - the model of isotropic effective conductivity, have been analyzed and mutually compared. It has been found that the impact of the thermal conductivity anisotropy is visible only in the case when this property differs significantly in both phases.

T. Dumont. Numerical simulation of frontal polymerization. CAMES 2000 (7)

Frontal polymerization has been studied for many years experimentally and theoretically. This technique can exhibit a phase change between a liquid monomer and a solid polymer and many studies, both theoretical and experimental, have been devoted to the stability of the front which separates the two phases in the presence of thermal convection. We present here a new technique for the numerical simulation of this process which takes account of the chemical reaction, the phase transition and the hydrodynamics; it is based on the method of characteristics and a fictitious domain method. These two methods are known, but the coupling of them and the application to this problem is new. We also present and discuss some results of simulations.

J.S. Szmyd, K. Gajek and K. Suzuki. Yttrium distribution in Ba-Cu-O melt during growth of superconducting material Y123 by Czochralski method. CAMES 2000 (7)

Numerical computations of the yttrium distribution in the BaO-CuO melt were performed for the single crystal growth of yttrium barium copper oxide superconductor (YBa_2Cu_3O_(7-x)) with the Czochralski method. The finite volume method was used to calculate the fluid flow, heat transfer and yttrium distribution in the melt with staggered numerical grid. The flow in the melt was assumed to be axisymmetric and was modelled as an incompressible Newtonian, Boussinesque fluid. Mass transfer was due to both convection and diffusion. Calculations were presented for a buoyancy/crystal-rotation driven combined convective flow.

P. Furmanski. Modeling of transport phenomena during solidification of binary systems. CAMES 2000 (7)

The macroscopic equations describing the process of solidification in binary systems are usually introduced via the volume averaging technique. A different approach to obtain these equations, based on the ensemble averaging technique, is proposed in the paper. This technique was used to derive energy and solute conservation equations and the basic constitutive relations appearing in the macroscopic description of the solidification phenomena occurring in the mushy region. In general these relations are non-local and account for non-equilibrium processes. Problem of thermodynamic equilibrium (thermal and chemical) is also discussed. Formulae for enthalpy and porosity of the mushy zone, in the latter case, are given.

T. Blesgen. A fluid mechanical model for two-phase flows. CAMES 2000 (7)

A modified Allen-Cahn equation is combined with the compressible Navier-Stokes system. After a physically motivated modification of the stress tensor, for the resulting equations the second law of thermodynamics is valid. The model can be used to describe the forming of gas phases in a flowing liquid.

J. Kaupuzs and J. Rimshans. Non-steady state 2D numerical simulation of gas-liquid phase transition in metal vapour. CAMES 2000 (7)

A mathematical model of diffusion of vaporized interacting metal molecules in a fireproof material is considered. The model is based on microscopic kinetic equations describing the process under condition of a strongly non-homogeneous temperature field. A two-dimensional structure is examined, where the inner hot surface acts as the source of metal vapour and the outer surface - as a cooler. Due to interaction between metal molecules, a phase transition (condensation) proceeds near the outer surface. A conservative, monotonous, and absolutely stable difference scheme is developed on the basis of a special exponential substitution for the concentration of molecules. Results of 2D numerical experiments in non-steady state are presented.

J. Pieskä and E. Laitinen. Comparison of upwind and characteristic schemes for solving multiphase diffusion-convection equation. CAMES 2000 (7)

In this paper the multiphase diffusion-convection problem is solved numerically by using upwind and characteristic schemes. Discretization for the schemes are performed by finite difference method. For solving the algebraic equations on every time level the modified S.O.R. method is used. In the numerical results computing time, number of iterations and accuracy of the schemes are analysed.