# Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

## Abstract

A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201–211, 1990).

- Authors:

- CONICET-Universidad Austral, Depto. Matemática, Facultad de Ciencias Empresariales (Argentina)

- Publication Date:

- OSTI Identifier:
- 22769318

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; BOUNDARY CONDITIONS; HEAT TRANSFER; MATHEMATICAL SOLUTIONS; SOLIDIFICATION

### Citation Formats

```
Ceretani, Andrea N., E-mail: aceretani@austral.edu.ar, and Tarzia, Domingo A., E-mail: dtarzia@austral.edu.ar.
```*Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0442-0.

```
Ceretani, Andrea N., E-mail: aceretani@austral.edu.ar, & Tarzia, Domingo A., E-mail: dtarzia@austral.edu.ar.
```*Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model*. United States. doi:10.1007/S40314-017-0442-0.

```
Ceretani, Andrea N., E-mail: aceretani@austral.edu.ar, and Tarzia, Domingo A., E-mail: dtarzia@austral.edu.ar. Tue .
"Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model". United States. doi:10.1007/S40314-017-0442-0.
```

```
@article{osti_22769318,
```

title = {Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model},

author = {Ceretani, Andrea N., E-mail: aceretani@austral.edu.ar and Tarzia, Domingo A., E-mail: dtarzia@austral.edu.ar},

abstractNote = {A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201–211, 1990).},

doi = {10.1007/S40314-017-0442-0},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}