Dynamic boundary conditions in the interface modeling of binary alloys

Igor B. Krasnyuk; Roman M. Taranets; Marina Chugunova; Igor B. Krasnyuk; Roman M. Taranets; Marina Chugunova

doi:10.3934/Math.2018.3.409

AIMS Mathematics

2018, Volume 3, Issue 3: 409-425. doi: 10.3934/Math.2018.3.409

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Research article Topical Sections

Dynamic boundary conditions in the interface modeling of binary alloys

1.
Donetsk Institute for Physics and Engineering, Donetsk, 83114, Ukraine
2.
Institute of Applied Mathematics and Mechanics of the NASU, Sloviansk, 84100, Ukraine
3.
Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA, 91711, USA

Received: 25 May 2018 Accepted: 13 September 2018 Published: 11 October 2018
MSC : 35B10, 34B60

We study the initial boundary value problem with dynamic boundary conditions to the Penrose-Fife equations with a 'memory e ect' for the order parameter and temperature time evolutions. The dynamic boundary conditions describe the process of production and degradation of surface crystallite near the walls, which confine the disordered binary alloy at a nearly melt temperature during the fast cooling process. The solid-liquid periodic distributions, which were obtained in 1D case, represent asymptotically periodic piecewise constant spatial-temporal impulses in a long time dynamics. It is confirmed that, depending on parameter values, the total number of discontinuity points of such periodic impulses can be finite or infinite. We refer to such wave solution types as relaxation or pre-turbulent, respectively. These results are compared with experimental data.
- phase-field equations,
- di erence equations,
- pre-turbulent,
- turbulent,
- global attractor
Citation: Igor B. Krasnyuk, Roman M. Taranets, Marina Chugunova. Dynamic boundary conditions in the interface modeling of binary alloys[J]. AIMS Mathematics, 2018, 3(3): 409-425. doi: 10.3934/Math.2018.3.409

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Abstract

We study the initial boundary value problem with dynamic boundary conditions to the Penrose-Fife equations with a 'memory e ect' for the order parameter and temperature time evolutions. The dynamic boundary conditions describe the process of production and degradation of surface crystallite near the walls, which confine the disordered binary alloy at a nearly melt temperature during the fast cooling process. The solid-liquid periodic distributions, which were obtained in 1D case, represent asymptotically periodic piecewise constant spatial-temporal impulses in a long time dynamics. It is confirmed that, depending on parameter values, the total number of discontinuity points of such periodic impulses can be finite or infinite. We refer to such wave solution types as relaxation or pre-turbulent, respectively. These results are compared with experimental data.

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