Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions
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| Format: | Preprint |
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2025
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| author | Anguiano, María Suárez-Grau, Francisco J. |
| author_facet | Anguiano, María Suárez-Grau, Francisco J. |
| contents | This theoretical study deals with the Navier-Stokes equations posed in a 3D thin domain with thickness $0<\varepsilon\ll 1$, assuming power law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced in (Djoko et al., Comput. Math. Appl., 128 (2022) 198-213), represents a generalization of the Navier slip boundary condition. The goal is to study the influence of the power law slip boundary conditions with an anisotropic tensor of order $\varepsilon^{γ\over s}$, with $γ\in \mathbb{R}$ and flow index $1<s<2$, on the behavior of the fluid with thickness $\varepsilon$ by using asymptotic analysis when $\varepsilon\to 0$, depending on the values of $γ$. As a result, we deduce the existence of a critical value of $γ$ given by $γ_s^*=3-2s$ and so, three different limit boundary conditions are derived. The critical case $γ=γ_s^*$ corresponds to a limit condition of type power law slip. The supercritical case $γ>γ_s^*$ corresponds to a limit boundary condition of type perfect slip. The subcritical case $γ<γ_s^*$ corresponds to a limit boundary condition of type no-slip. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_14303 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions Anguiano, María Suárez-Grau, Francisco J. Analysis of PDEs This theoretical study deals with the Navier-Stokes equations posed in a 3D thin domain with thickness $0<\varepsilon\ll 1$, assuming power law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced in (Djoko et al., Comput. Math. Appl., 128 (2022) 198-213), represents a generalization of the Navier slip boundary condition. The goal is to study the influence of the power law slip boundary conditions with an anisotropic tensor of order $\varepsilon^{γ\over s}$, with $γ\in \mathbb{R}$ and flow index $1<s<2$, on the behavior of the fluid with thickness $\varepsilon$ by using asymptotic analysis when $\varepsilon\to 0$, depending on the values of $γ$. As a result, we deduce the existence of a critical value of $γ$ given by $γ_s^*=3-2s$ and so, three different limit boundary conditions are derived. The critical case $γ=γ_s^*$ corresponds to a limit condition of type power law slip. The supercritical case $γ>γ_s^*$ corresponds to a limit boundary condition of type perfect slip. The subcritical case $γ<γ_s^*$ corresponds to a limit boundary condition of type no-slip. |
| title | Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.14303 |