Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier-Stokes/Allen-Cahn System
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| Format: | Preprint |
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2023
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| _version_ | 1866914824072462336 |
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| author | Abels, Helmut Fischer, Julian Moser, Maximilian |
| author_facet | Abels, Helmut Fischer, Julian Moser, Maximilian |
| contents | We show convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility $m_\varepsilon>0$ in the Allen-Cahn equation tends to zero in a subcritical way, i.e., $m_\varepsilon= m_0 \varepsilon^β$ for some $β\in (0,2)$ and $m_0>0$. The proof proceeds by showing via a relative entropy argument that the solution to the Navier-Stokes/Allen-Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term $m_\varepsilon H_{Γ_t}$ in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_02997 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier-Stokes/Allen-Cahn System Abels, Helmut Fischer, Julian Moser, Maximilian Analysis of PDEs 76T06, 35Q30, 35Q35, 35R35, 76D05, 76D45 We show convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility $m_\varepsilon>0$ in the Allen-Cahn equation tends to zero in a subcritical way, i.e., $m_\varepsilon= m_0 \varepsilon^β$ for some $β\in (0,2)$ and $m_0>0$. The proof proceeds by showing via a relative entropy argument that the solution to the Navier-Stokes/Allen-Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term $m_\varepsilon H_{Γ_t}$ in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow. |
| title | Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier-Stokes/Allen-Cahn System |
| topic | Analysis of PDEs 76T06, 35Q30, 35Q35, 35R35, 76D05, 76D45 |
| url | https://arxiv.org/abs/2311.02997 |