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Klopp C, Trittel T, Harth K, Stannarius R. Coalescence of biphasic droplets embedded in free standing smectic A films. SOFT MATTER 2024; 20:1036-1046. [PMID: 38205564 DOI: 10.1039/d3sm01549a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/12/2024]
Abstract
We investigate micrometer-sized flat droplets consisting of an isotropic core surrounded by a nematic rim in freely suspended smectic A liquid-crystal films. In contrast to purely isotropic droplets which are characterized by a sharp edge and no long-range interactions, the nematic fringe introduces a continuous film thickness change resulting in long-range mutual attraction of droplets. The coalescence scenario is divided in two phases. The first one consists in the fusion of the nematic regions. The second phase involves the dissolution of a thin nematic film between the two isotropic cores. The latter has many similarities with the rupture of thin liquid films between droplets coalescing in an immiscible viscous liquid.
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Affiliation(s)
- Christoph Klopp
- Institute of Physics, Otto von Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany.
- MARS, Otto von Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany
| | - Torsten Trittel
- MARS, Otto von Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany
- Department of Engineering, Brandenburg University of Applied Sciences, Magdeburger Straße 50, D-14770 Brandenburg an der Havel, Germany
| | - Kirsten Harth
- MARS, Otto von Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany
- Department of Engineering, Brandenburg University of Applied Sciences, Magdeburger Straße 50, D-14770 Brandenburg an der Havel, Germany
| | - Ralf Stannarius
- Institute of Physics, Otto von Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany.
- MARS, Otto von Guericke University Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany
- Department of Engineering, Brandenburg University of Applied Sciences, Magdeburger Straße 50, D-14770 Brandenburg an der Havel, Germany
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Dolganov PV, Spiridenko NA, Zverev AS. Dynamics of viscous droplet coalescence in the confined geometry of optical cells. Phys Rev E 2024; 109:014702. [PMID: 38366422 DOI: 10.1103/physreve.109.014702] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2023] [Accepted: 12/28/2023] [Indexed: 02/18/2024]
Abstract
The dynamics of quasi-two-dimensional coalescence of isotropic droplets in nematic liquid crystal environment was studied. Investigations were made in confined geometry of a Hele-Shaw optical cell with different transverse droplet sizes. The existence of three distinct dynamic regimes was found for coalescence, namely, short-, middle-, and long-time regimes. The fast dynamics of bridge transformation was visualized. At short time the dynamics of droplet transformation is similar to the transformation of free (three-dimensional) droplets. At later stages, two regimes of the coalescence at different timescales are determined by Poiseuille flow. Experimental data are discussed on the basis of existing theories.
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Affiliation(s)
- P V Dolganov
- Osipyan Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow Region, Russia
| | - N A Spiridenko
- Osipyan Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow Region, Russia
| | - A S Zverev
- Osipyan Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow Region, Russia
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Ryu S, Zhang H, Anuta UJ. A Review on the Coalescence of Confined Drops with a Focus on Scaling Laws for the Growth of the Liquid Bridge. MICROMACHINES 2023; 14:2046. [PMID: 38004903 PMCID: PMC10673007 DOI: 10.3390/mi14112046] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2023] [Revised: 10/26/2023] [Accepted: 10/28/2023] [Indexed: 11/26/2023]
Abstract
The surface-tension-driven coalescence of drops has been extensively studied because of the omnipresence of the phenomenon and its significance in various natural and engineering systems. When two drops come into contact, a liquid bridge is formed between them and then grows in its lateral dimensions. As a result, the two drops merge to become a bigger drop. The growth dynamics of the bridge are governed by a balance between the driving force and the viscous and inertial resistances of involved liquids, and it is usually represented by power-law scaling relations on the temporal evolution of the bridge dimension. Such scaling laws have been well-characterized for the coalescence of unconfined or freely suspended drops. However, drops are often confined by solid or liquid surfaces and thus are a different shape from spheres, which affects their coalescence dynamics. As such, the coalescence of confined drops poses more complicated interfacial fluid dynamics challenges compared to that of unconfined drops. Although there have been several studies on the coalescence of confined drops, they have not been systematically reviewed in terms of the properties and geometry of the confining surface. Thus, we aim to review the current literature on the coalescence of confined drops in three categories: drop coalescence on a solid surface, drop coalescence on a deformable surface, and drop coalescence between two parallel surfaces with a small gap (i.e., Hele-Shaw cell), with a focus on power-law scaling relations, and to suggest challenges and outlooks for future research on the phenomena.
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Affiliation(s)
- Sangjin Ryu
- Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA; (H.Z.)
- Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, USA
| | - Haipeng Zhang
- Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA; (H.Z.)
| | - Udochukwu John Anuta
- Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588, USA; (H.Z.)
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Ohashi M, Toramaru A, Namiki A. Coalescence of two growing bubbles in a Hele-Shaw cell. Sci Rep 2022; 12:1270. [PMID: 35075182 PMCID: PMC8786952 DOI: 10.1038/s41598-022-05252-5] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2021] [Accepted: 01/10/2022] [Indexed: 12/03/2022] Open
Abstract
An understanding of the dynamics of growth-driven coalescence is important in diverse fields across natural science and engineering. Motivated by the bubble coalescence in magma during volcanic eruptions, we study both experimentally and theoretically the coalescence of two growing bubbles in a Hele–Shaw cell. In our system, bubbles grow by gas expansion due to decompression and the diffusional influx of dissolved gas in the liquid. Our experiments show that the evolution of film thickness and bubble shape depends on viscosity, decompression rate, and cell gap. Through a scaling analysis and a perturbation approximation, we find that the hydrodynamic interaction between two bubbles is characterized by a film capillary number \documentclass[12pt]{minimal}
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\begin{document}$$Ca_f=(\eta {\dot{R}}/\sigma )(R/D)^2$$\end{document}Caf=(ηR˙/σ)(R/D)2 depending on viscosity \documentclass[12pt]{minimal}
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\begin{document}$$\eta$$\end{document}η, bubble radius R, growth rate \documentclass[12pt]{minimal}
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\begin{document}$${\dot{R}}$$\end{document}R˙, interfacial tension \documentclass[12pt]{minimal}
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\begin{document}$$\sigma$$\end{document}σ, and cell gap D. The experimental results demonstrate that the film capillary number solely determines the bubble distortion just before coalescence. Under our experimental conditions, bubble coalescence occurs below a critical value of a nominal film capillary number defined as a film capillary number evaluated when two undeformed circular bubbles come into contact.
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Affiliation(s)
- Masatoshi Ohashi
- Department of Earth and Planetary Sciences, Kyushu University, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan.
| | - Atsushi Toramaru
- Department of Earth and Planetary Sciences, Kyushu University, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
| | - Atsuko Namiki
- Graduate School of Environmental Studies, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
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Dolganov PV, Zverev AS, Baklanova KD, Dolganov VK. Dynamics of capillary coalescence and breakup: Quasi-two-dimensional nematic and isotropic droplets. Phys Rev E 2021; 104:014702. [PMID: 34412240 DOI: 10.1103/physreve.104.014702] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2021] [Accepted: 06/04/2021] [Indexed: 11/07/2022]
Abstract
We observed the formation of small satellite droplets from the bridge at droplet coalescence. Investigations were made using a Hele-Shaw cell in the two-phase region at the nematic-isotropic phase transition. In previous works on coalescence it was considered that before the start of coalescence there exists a bridge between the outer fluid connecting regions on the two sides of the droplets (outer bridge). After the start of coalescence, a bridge connecting the two droplets appears (droplet bridge) and the outer bridge is broken. We have shown that there are coalescence processes where after the start of coalescence both the droplet bridge and the outer bridge can exist. This cardinally changes the coalescence process. During the first coalescence stage the size of the outer bridge decreases and the size of the droplet bridge increases. During the second stage the outer bridge becomes unstable which leads to pinch-off, formation of pointed end domains, secondary instability, splitting of pointed end domains, and formation of satellite droplets. We found the linear dependence of the minimum bridge radius on time near bridge breakup. This behavior confirms the capillary viscous regime of bridge breakup. Our work connects two areas of fluid dynamics: coalescence and breakup with formation of satellite droplets.
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Affiliation(s)
- P V Dolganov
- Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow district, Russia
| | - A S Zverev
- Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow district, Russia
| | - K D Baklanova
- Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow district, Russia.,National Research University Higher School of Economics, Moscow 101000, Russia
| | - V K Dolganov
- Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow district, Russia
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Shukla I, Kofman N, Balestra G, Zhu L, Gallaire F. Film thickness distribution in gravity-driven pancake-shaped droplets rising in a Hele-Shaw cell. JOURNAL OF FLUID MECHANICS 2019; 874:1021-1040. [PMID: 32981965 PMCID: PMC7116120 DOI: 10.1017/jfm.2019.453] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We study here experimentally, numerically and using a lubrication approach, the shape, velocity and lubrication film thickness distribution of a droplet rising in a vertical Hele-Shaw cell. The droplet is surrounded by a stationary immiscible fluid and moves purely due to buoyancy. A low density difference between the two media helps to operate in a regime with capillary number Ca lying between 0.03 and 0.35, where Ca =μo Ud /γ is built with the surrounding oil viscosity γ0 , the droplet velocity Ud and surface tension γ. The experimental data show that in this regime the droplet velocity is not influenced by the thickness of the thin lubricating film and the dynamic meniscus. For iso-viscous cases, experimental and three-dimensional numerical results of the film thickness distribution agree well with each other. The mean film thickness is well captured by the Aussillous & Quéré (Phys. Fluids, vol. 12 (10), 2000, pp. 2367-2371) model with fitting parameters. The droplet also exhibits the 'catamaran' shape that has been identified experimentally for a pressure-driven counterpart (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501). This pattern has been rationalized using a two-dimensional lubrication equation. In particular, we show that this peculiar film thickness distribution is intrinsically related to the anisotropy of the fluxes induced by the droplet's motion.
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Affiliation(s)
- Isha Shukla
- Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Nicolas Kofman
- Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Gioele Balestra
- Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Lailai Zhu
- Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
- Linné Flow Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, SE-100 44 Stockholm, Sweden
| | - François Gallaire
- Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
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