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Volume 22, Issue 2
The Navier-Stokes-$\omega$/Navier-Stokes-$\omega$ Model for Fluid-Fluid Interaction Using an Unconditionally Stable Finite Element Scheme

Qinghui Wang, Pengzhan Huang & Yinnian He

DOI: 10.4208/ijnam2025-1009

Int. J. Numer. Anal. Mod., 22 (2025), pp. 178-201.

Published online: 2025-02

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  • Abstract

In this article, for solving fluid-fluid interaction problem, we consider a Navier-Stokes-$ω$/Navier-Stokes-$ω$ model, which includes two Navier-Stokes-$ω$ equations coupled by some nonlinear interface conditions. Based on an auxiliary variable, we propose a fully discrete, decouple finite element scheme. We adopt the backward Euler scheme and mixed finite element approximation for temporal-spatial discretization, and explicit treatment for the interface terms and nonlinear terms. Moreover, the proposed scheme is shown to be unconditionally stable. Then, we establish error estimate of the numerical solution. Finally, with a series of numerical experiments we illustrate the stability and effectiveness of the proposed scheme and its ability to capture basic phenomenological features of the fluid-fluid interaction.

  • Keywords

Navier-Stokes-$ω$ model, fluid-fluid interaction, auxiliary variable, unconditional stability.

  • AMS Subject Headings

65M60, 76W05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-178, author = {Wang , QinghuiHuang , Pengzhan and He , Yinnian}, title = {The Navier-Stokes-$\omega$/Navier-Stokes-$\omega$ Model for Fluid-Fluid Interaction Using an Unconditionally Stable Finite Element Scheme}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {2}, pages = {178--201}, abstract = {

In this article, for solving fluid-fluid interaction problem, we consider a Navier-Stokes-$ω$/Navier-Stokes-$ω$ model, which includes two Navier-Stokes-$ω$ equations coupled by some nonlinear interface conditions. Based on an auxiliary variable, we propose a fully discrete, decouple finite element scheme. We adopt the backward Euler scheme and mixed finite element approximation for temporal-spatial discretization, and explicit treatment for the interface terms and nonlinear terms. Moreover, the proposed scheme is shown to be unconditionally stable. Then, we establish error estimate of the numerical solution. Finally, with a series of numerical experiments we illustrate the stability and effectiveness of the proposed scheme and its ability to capture basic phenomenological features of the fluid-fluid interaction.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1009}, url = {http://global-sci.org/intro/article_detail/ijnam/23820.html} }
TY - JOUR T1 - The Navier-Stokes-$\omega$/Navier-Stokes-$\omega$ Model for Fluid-Fluid Interaction Using an Unconditionally Stable Finite Element Scheme AU - Wang , Qinghui AU - Huang , Pengzhan AU - He , Yinnian JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 178 EP - 201 PY - 2025 DA - 2025/02 SN - 22 DO - http://doi.org/10.4208/ijnam2025-1009 UR - https://global-sci.org/intro/article_detail/ijnam/23820.html KW - Navier-Stokes-$ω$ model, fluid-fluid interaction, auxiliary variable, unconditional stability. AB -

In this article, for solving fluid-fluid interaction problem, we consider a Navier-Stokes-$ω$/Navier-Stokes-$ω$ model, which includes two Navier-Stokes-$ω$ equations coupled by some nonlinear interface conditions. Based on an auxiliary variable, we propose a fully discrete, decouple finite element scheme. We adopt the backward Euler scheme and mixed finite element approximation for temporal-spatial discretization, and explicit treatment for the interface terms and nonlinear terms. Moreover, the proposed scheme is shown to be unconditionally stable. Then, we establish error estimate of the numerical solution. Finally, with a series of numerical experiments we illustrate the stability and effectiveness of the proposed scheme and its ability to capture basic phenomenological features of the fluid-fluid interaction.

Wang , QinghuiHuang , Pengzhan and He , Yinnian. (2025). The Navier-Stokes-$\omega$/Navier-Stokes-$\omega$ Model for Fluid-Fluid Interaction Using an Unconditionally Stable Finite Element Scheme. International Journal of Numerical Analysis and Modeling. 22 (2). 178-201. doi:10.4208/ijnam2025-1009
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