Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation

[2502.14312] Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2502.14312 (math) [Submitted on 20 Feb 2025] Title: Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation Authors: Isidora Rapajić (1), Srboljub Simić (2), Endre Süli (3) ((1) Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia, (2) Department of Mathematics and Informatics, Faculty of Sciences, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia, (3) Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, United Kingdom) View a PDF of the paper titled Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation, by Isidora Rapaji\'c (1) and 14 other authors View PDF HTML (experimental) Abstract: The aim of this paper is to extend Washburn's capillary rise equation by incorporating a slip condition at the pipe wall. The governing equation is derived using fundamental principles from continuum mechanics. A new scaling is introduced, allowing for a systematic analysis of different flow regimes. We prove the global-in-time existence and uniqueness of a bounded positive solution to Washburn's equation that includes the slip parameter, as well as the continuous dependence of the solution in the maximum norm on the initial data. Thus, the initial-value problem for Washburn's equation is shown to be well-posed in the sense of Hadamard. Additionally, we show that the unique equilibrium solution may be reached either monotonically or in an oscillatory fashion, similarly to the no-slip case. Finally, we determine the basin of attraction for the system, ensuring that the equilibrium state will be reached from the initial data we impose. These results hold for any positive value of the nondimensional slip parameter in the model, and for all values of the ratio $h_0/h_e$ in the range $[0,3/2]$, where $h_0$ is the initial height of the fluid column and $h_e$ is its equilibrium height. Subjects: Analysis of PDEs (math.AP) ; Mathematical Physics (math-ph) Cite as: arXiv:2502.14312 [math.AP] (or arXiv:2502.14312v1 [math.AP] for this version) https://doi.org/10.48550/arXiv.2502.14312 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.physd.2025 Focus to learn more DOI(s) linking to related resources Submission history From: Srboljub Simić [ view email ] [v1] Thu, 20 Feb 2025 06:55:47 UTC (129 KB) Full-text links: Access Paper: View a PDF of the paper titled Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation, by Isidora Rapaji\'c (1) and 14 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2025-02 Change to browse by: math math-ph math.MP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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