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Volume 15, Issue 3
Approximation of the Long-Term Dynamics of the Dynamical System Generated by a 3D NS-$α$ System with Phase Transition

T. Tachim Medjo & F. Tone

Int. J. Numer. Anal. Mod., 15 (2018), pp. 307-339.

Published online: 2018-03

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  • Abstract

In this article we study an approximate model for a binary fluid flow in a three-dimensional bounded domain. The governing equations consist of the Allen–Cahn equation for the order (phase) parameter $\phi$ coupled with the Navier–Stokes-$α$ (NS-$α$) system for the velocity $u$. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

  • Keywords

Navier–Stokes-$α$, phase transition, attractors, implicit Euler scheme, Gronwall Lemma.

  • AMS Subject Headings

35Q30, 35Q35, 35Q72

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

tachimt@fiu.edu (T. Tachim Medjo)

ftone@uwf.edu (F. Tone)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-307, author = {Medjo , T. Tachim and Tone , F.}, title = {Approximation of the Long-Term Dynamics of the Dynamical System Generated by a 3D NS-$α$ System with Phase Transition}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {3}, pages = {307--339}, abstract = {

In this article we study an approximate model for a binary fluid flow in a three-dimensional bounded domain. The governing equations consist of the Allen–Cahn equation for the order (phase) parameter $\phi$ coupled with the Navier–Stokes-$α$ (NS-$α$) system for the velocity $u$. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12518.html} }
TY - JOUR T1 - Approximation of the Long-Term Dynamics of the Dynamical System Generated by a 3D NS-$α$ System with Phase Transition AU - Medjo , T. Tachim AU - Tone , F. JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 307 EP - 339 PY - 2018 DA - 2018/03 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12518.html KW - Navier–Stokes-$α$, phase transition, attractors, implicit Euler scheme, Gronwall Lemma. AB -

In this article we study an approximate model for a binary fluid flow in a three-dimensional bounded domain. The governing equations consist of the Allen–Cahn equation for the order (phase) parameter $\phi$ coupled with the Navier–Stokes-$α$ (NS-$α$) system for the velocity $u$. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

T. Tachim Medjo & F. Tone. (2019). Approximation of the Long-Term Dynamics of the Dynamical System Generated by a 3D NS-$α$ System with Phase Transition. International Journal of Numerical Analysis and Modeling. 15 (3). 307-339. doi:
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