Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface

Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface Jian Dong, Bilong Liu, Dezhao Li, Da-peng Tan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8817038/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The rolling angle calculation of a water droplet on an inclined surface is not only an everyday phenomenon but also forms an essential part of many industrial processes. Previous researchers always used Newtonian mechanics (the vector mechanics) to set up the differential equation by analyzing the forces affecting on the droplet on an inclined surface and get the rolling angle. Here, by constructing the single-valued mapping from the droplet mass center motion parameters to the moving droplet body, we use the Hamilton's principle (the energy method) and the Karush-Kuhn-Tucher (KKT) theory to set up Euler equation sets of mass center motions. In initial conditions of “sphere shape-zero velocity”, we solve Euler equation sets and gain the droplet mass center motion equations. By judging whether the singularities (velocity zero points) exist or not in the phase space of mass center motions, we judge the droplet “is pinned” or “continuously rolls” on the inclined surface. Finally, we calculate out the critical tilt angle of the inclined surface, which just makes the droplet “continuously roll”, as the droplet rolling angle. Our results from the energy method can also improve analyzing the kinetic behaviors of droplets in multi-physical fields, such as phase-change droplets, triboelectrically induced electrostatic droplets and polar droplets in fixed electrostatic fields. Physical sciences/Physics/Fluid dynamics Physical sciences/Materials science/Soft materials/Fluids the rolling angle calculation the single-valued mapping the Hamilton's principle the Karush-Kuhn-Tucher (KKT) theory the singularities the phase space Full Text Additional Declarations There is NO Competing Interest. Tables 2 and 3 are available in the Supplementary Files section. Supplementary Files NPSI5final.docx Supplementary Information for Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface Table2PDMSfinal.xlsx Table2 Table3.xlsx Table3 Partialexperimentalvideos.mp4 The motion of liquid droplets on the surface under high-speed cameras code.docx Supplementary code for Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface NPSIfinal.docx Supplementary Information for Droplet Hamilton Dynamics 1: the rolling angle of a water droplet on an inclined surface Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8817038","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":588339932,"identity":"868eb166-b837-4e8d-8c6d-da7141d40287","order_by":0,"name":"Jian 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