Waves beneath a drop levitating over a moving wall
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911999530631168 |
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| author | McKee, Kyle I. Primkulov, Bauyrzhan K. Hashimoto, Kotaro Tagawa, Yoshiyuki Bush, John W. M. |
| author_facet | McKee, Kyle I. Primkulov, Bauyrzhan K. Hashimoto, Kotaro Tagawa, Yoshiyuki Bush, John W. M. |
| contents | In recent experiments, Sawaguchi et al. directly probed the lubrication layer of air beneath a droplet levitating inside a rotating cylindrical drum. For small rotation rates of the drum, the lubrication film beneath the drop adopted a steady shape, while at higher rotation rates, travelling waves propagated along the drop's lower surface with roughly half the wall velocity. We here rationalize the physical origin of these waves. We begin with a simplified model of the lubrication flow beneath the droplet, and examine the linear stability of this base state to perturbations of the Tollmien--Schlichting type. Our developments lead to the Orr-Sommerfeld equation (OSE), whose eigenvalues give the growth rates and phase speeds of the perturbations. By considering wavelengths long relative to the lubrication film thickness, we solve the OSE perturbatively and so deduce the wavelength and phase velocity of the most unstable mode. We find satisfactory agreement between experiment and theory over the parameter regime considered in the laboratory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Waves beneath a drop levitating over a moving wall McKee, Kyle I. Primkulov, Bauyrzhan K. Hashimoto, Kotaro Tagawa, Yoshiyuki Bush, John W. M. Fluid Dynamics Mathematical Physics In recent experiments, Sawaguchi et al. directly probed the lubrication layer of air beneath a droplet levitating inside a rotating cylindrical drum. For small rotation rates of the drum, the lubrication film beneath the drop adopted a steady shape, while at higher rotation rates, travelling waves propagated along the drop's lower surface with roughly half the wall velocity. We here rationalize the physical origin of these waves. We begin with a simplified model of the lubrication flow beneath the droplet, and examine the linear stability of this base state to perturbations of the Tollmien--Schlichting type. Our developments lead to the Orr-Sommerfeld equation (OSE), whose eigenvalues give the growth rates and phase speeds of the perturbations. By considering wavelengths long relative to the lubrication film thickness, we solve the OSE perturbatively and so deduce the wavelength and phase velocity of the most unstable mode. We find satisfactory agreement between experiment and theory over the parameter regime considered in the laboratory. |
| title | Waves beneath a drop levitating over a moving wall |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2408.12357 |