Analysis of a mathematical model related to Czochralski crystal growth
This paper is devoted to the study of a stationary problem consisting of the Boussinesq approximation of the Navier–Stokes equations and two convection–diffusion equations for the temperature and concentration, respectively. The equations are considered in 3D and a velocity–pressure formulation of t...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1998-01-01
|
| Series: | Abstract and Applied Analysis |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S108533759800058X |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850237353443983360 |
|---|---|
| author | Petr Knobloch Lutz Tobiska |
| author_facet | Petr Knobloch Lutz Tobiska |
| author_sort | Petr Knobloch |
| collection | DOAJ |
| description | This paper is devoted to the study of a stationary problem consisting of the Boussinesq approximation of the Navier–Stokes equations and two convection–diffusion equations for the temperature and concentration, respectively. The equations are considered in 3D and a velocity–pressure formulation of the Navier–Stokes equations is used. The problem is complicated by nonstandard boundary conditions for velocity on the liquid–gas interface where tangential surface forces proportional to surface gradients of temperature and concentration (Marangoni effect) and zero normal component of the velocity are assumed. The velocity field is coupled through this boundary condition and through the buoyancy term in the Navier–Stokes equations with both the temperature and concentration fields. In this paper a weak formulation of the problem is stated and the existence of a weak solution is proved. For small data, the uniqueness of the solution is established. |
| format | Article |
| id | doaj-art-020dff4d6b3e40d1943af276a957d42d |
| institution | OA Journals |
| issn | 1085-3375 |
| language | English |
| publishDate | 1998-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-020dff4d6b3e40d1943af276a957d42d2025-08-20T02:01:46ZengWileyAbstract and Applied Analysis1085-33751998-01-0133-431934210.1155/S108533759800058XAnalysis of a mathematical model related to Czochralski crystal growthPetr Knobloch0Lutz Tobiska1Institute of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské námĕstí 25, Praha 1 11800, Czech RepublicInstitute of Analysis and Numerics, Otto von Guericke University Magdeburg, Postfach 4120, Magdeburg 39016, GermanyThis paper is devoted to the study of a stationary problem consisting of the Boussinesq approximation of the Navier–Stokes equations and two convection–diffusion equations for the temperature and concentration, respectively. The equations are considered in 3D and a velocity–pressure formulation of the Navier–Stokes equations is used. The problem is complicated by nonstandard boundary conditions for velocity on the liquid–gas interface where tangential surface forces proportional to surface gradients of temperature and concentration (Marangoni effect) and zero normal component of the velocity are assumed. The velocity field is coupled through this boundary condition and through the buoyancy term in the Navier–Stokes equations with both the temperature and concentration fields. In this paper a weak formulation of the problem is stated and the existence of a weak solution is proved. For small data, the uniqueness of the solution is established.http://dx.doi.org/10.1155/S108533759800058XNavier–Stokes equationsBoussinesq aproximation nonstandard boundary conditionsweak solvabilityCzochralski method. |
| spellingShingle | Petr Knobloch Lutz Tobiska Analysis of a mathematical model related to Czochralski crystal growth Abstract and Applied Analysis Navier–Stokes equations Boussinesq aproximation nonstandard boundary conditions weak solvability Czochralski method. |
| title | Analysis of a mathematical model related to Czochralski crystal growth |
| title_full | Analysis of a mathematical model related to Czochralski crystal growth |
| title_fullStr | Analysis of a mathematical model related to Czochralski crystal growth |
| title_full_unstemmed | Analysis of a mathematical model related to Czochralski crystal growth |
| title_short | Analysis of a mathematical model related to Czochralski crystal growth |
| title_sort | analysis of a mathematical model related to czochralski crystal growth |
| topic | Navier–Stokes equations Boussinesq aproximation nonstandard boundary conditions weak solvability Czochralski method. |
| url | http://dx.doi.org/10.1155/S108533759800058X |
| work_keys_str_mv | AT petrknobloch analysisofamathematicalmodelrelatedtoczochralskicrystalgrowth AT lutztobiska analysisofamathematicalmodelrelatedtoczochralskicrystalgrowth |