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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2301.04181 |
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| _version_ | 1866929565379592192 |
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| author | Ghosh, Amrita Velázquez, Juan J. L. |
| author_facet | Ghosh, Amrita Velázquez, Juan J. L. |
| contents | In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of fluid, i.e. a fluid contained between two solid surfaces and part of the fluid surface is in contact with the air. The fluid is governed by Navier-Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the fluid thickness (lubrication approximation) and the contact angle between the fluid-solid and the fluid-gas interfaces is close to $π$. This resulting model is a free boundary problem for the equation $h_t + (h^3h_{xxx})_x = 0$, for which we have $h>0$ at the contact point (different from the usual thin film equation with $h=0$ at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Also the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from no-slip condition. Furthermore, we show the global stability of steady state solutions in a periodic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_04181 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A thin film model for meniscus evolution Ghosh, Amrita Velázquez, Juan J. L. Analysis of PDEs In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of fluid, i.e. a fluid contained between two solid surfaces and part of the fluid surface is in contact with the air. The fluid is governed by Navier-Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the fluid thickness (lubrication approximation) and the contact angle between the fluid-solid and the fluid-gas interfaces is close to $π$. This resulting model is a free boundary problem for the equation $h_t + (h^3h_{xxx})_x = 0$, for which we have $h>0$ at the contact point (different from the usual thin film equation with $h=0$ at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Also the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from no-slip condition. Furthermore, we show the global stability of steady state solutions in a periodic setting. |
| title | A thin film model for meniscus evolution |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2301.04181 |