{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T06:42:43Z","timestamp":1783579363011,"version":"3.55.0"},"reference-count":105,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,13]],"date-time":"2023-04-13T00:00:00Z","timestamp":1681344000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Russian Science Foundation","award":["19-19-00598-P"],"award-info":[{"award-number":["19-19-00598-P"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The evolution of the falling drop substance transfer in a target fluid at rest was traced by high-speed video techniques. Two flow modes were studied: slow intrusive flow, when the KE of the drop was comparable or less than the available potential energy (APSE), and a fast impact flow, at a relatively high drop contact velocity. For the substance transfer visualization, a drop of alizarin ink solution at various concentrations was used. The use of transparent partially colored fluid allows tracing the drop matter motion in the bulk and on the fluid free surface. The traditional side and frontal view of flow patterns were registered and analyzed. In both flow modes, the substance of the drop partially remained on the free surface and partially went into the target fluid bulk, where it was distributed non-uniformly. In the intrusive mode, the drop substance partially remained on the surface, while the main mass of the drop flowed into the thickness of the target fluid, forming the lenticular colored domain. The intrusion was gradually transformed into an annular vortex. In the impact mode, the drop broke up into individual fibers during the coalescence, creating linear and reticular structures on the surface of the cavity and the crown. The flow patterns composed of individual fibers were rapidly rebuilt as the flow evolved and the splash emerged and decayed. The sizes of cavities and colored fluid domains were compared in different flow regimes as well. The total energy transfer and transformation impact on the flow structure formation and dynamics was revealed.<\/jats:p>","DOI":"10.3390\/axioms12040374","type":"journal-article","created":{"date-parts":[[2023,4,14]],"date-time":"2023-04-14T02:03:38Z","timestamp":1681437818000},"page":"374","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Intrusive and Impact Modes of a Falling Drop Coalescence with a Target Fluid at Rest"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7435-4344","authenticated-orcid":false,"given":"Yuli D.","family":"Chashechkin","sequence":"first","affiliation":[{"name":"Laboratory of Fluid Mechanics, Ishlinsky Institute for Problems in Mechanics RAS, Moscow 119526, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1405-1760","authenticated-orcid":false,"given":"Andrey Yu.","family":"Ilinykh","sequence":"additional","affiliation":[{"name":"Laboratory of Fluid Mechanics, Ishlinsky Institute for Problems in Mechanics RAS, Moscow 119526, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,13]]},"reference":[{"key":"ref_1","first-page":"246","article-title":"On the formation of rotating rings by air and liquids under certain conditions of discharge","volume":"26","author":"Rogers","year":"1858","journal-title":"Am. J. Sci. Second Ser."},{"key":"ref_2","first-page":"444","article-title":"On drops","volume":"13","author":"Guthrie","year":"1863","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_3","first-page":"457","article-title":"On drops\u2014Part II","volume":"13","author":"Guthrie","year":"1863","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_4","first-page":"417","article-title":"On the formation of vortex rings by drops falling into liquids, and some allied phenomena","volume":"29","author":"Thomson","year":"1885","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Thompson, D.W. (1992). On Growth and Form, Dover. [2nd ed.].","DOI":"10.1017\/CBO9781107325852"},{"key":"ref_6","unstructured":"Isaac Newton Institute for Mathematical Sciences (2022, October 08). The Programme \u201cGrowth Form and Self-Organization\u201d from 22 August 2017 to 20 December 2017, Planned on the 100th Anniversary of On Growth and Form by D\u2019Arcy Thompson. Available online: https:\/\/www.newton.ac.uk\/event\/gfs\/."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1134\/S0015462810060025","article-title":"Marker transport in a composite vortex","volume":"45","author":"Stepanova","year":"2010","journal-title":"Fluid Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1002\/andp.18872660202","article-title":"Uber die Schwingungen fallender Tropfen","volume":"30","author":"Lenard","year":"1887","journal-title":"Ann. Phys. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1098\/rspl.1879.0015","article-title":"On the capillary phenomena of jets","volume":"29","author":"Rayleigh","year":"1879","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1134\/S0001433818020123","article-title":"Surface oscillations of free falling droplet of ideal liquid","volume":"54","author":"Kistovich","year":"2018","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1098\/rsta.1897.0005","article-title":"Impact with a liquid surface, studied by the aid of instantaneous photography","volume":"189","author":"Worthington","year":"1897","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"ref_12","unstructured":"Worthington, A.M. (1908). A Study of Splashes, Longmans, Green and Co."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1126\/science.52.1343.295","article-title":"The sound of splashes","volume":"52","author":"Jones","year":"1920","journal-title":"Science"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1121\/1.1907831","article-title":"Splashes as sources of sound in liquids","volume":"31","author":"Franz","year":"1959","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1146\/annurev.fl.25.010193.003045","article-title":"The impact of drops on liquid surfaces and the underwater noise of rain","volume":"25","author":"Prosperetti","year":"1993","journal-title":"Ann. Rev. Fluid Mech."},{"key":"ref_16","first-page":"73","article-title":"Packets of capillary and acoustic waves of drop impact","volume":"1","author":"Chashechkin","year":"2021","journal-title":"Herald Bauman Moscow State Tech. Univ. Ser. Nat. Sci."},{"key":"ref_17","unstructured":"Edgerton, H.E., and Killian, J.R. (1939). Flash, Hale, Cushman and Flint."},{"key":"ref_18","first-page":"91","article-title":"The Generation of Vortex Ring","volume":"8","author":"Okabe","year":"1960","journal-title":"Kyushu Univ. Rep. Res. Inst. Appl. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1017\/S0022112078002189","article-title":"The large-scale structure of unsteady self-similar rolled-up vortex sheets","volume":"88","author":"Pullin","year":"1978","journal-title":"J. Fluid Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1063\/1.862606","article-title":"Vortex ring formation at tube and orifice openings","volume":"22","author":"Pullin","year":"1979","journal-title":"Phys. Fluids"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"067109","DOI":"10.1063\/5.0048813","article-title":"An extended model for orifice starting jets","volume":"33","author":"Limbourg","year":"2021","journal-title":"Phys. Fluids"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1146\/annurev.fl.24.010192.001315","article-title":"Vortex rings","volume":"24","author":"Sharif","year":"1992","journal-title":"Ann. Rev. Fluid Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1017\/S0022112072001041","article-title":"The structure and stability of vortex rings","volume":"51","author":"Maxworthy","year":"1972","journal-title":"J. Fluid Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1017\/S0022112074002370","article-title":"Turbulent vortex rings (with an appendix on an extended theory of laminar vortex rings)","volume":"64","author":"Maxworthy","year":"1974","journal-title":"J. Fluid Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1017\/S0022112077002171","article-title":"Some experimental studies of vortex rings","volume":"81","author":"Maxworthy","year":"1977","journal-title":"J. Fluid Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"A46","DOI":"10.1017\/jfm.2021.141","article-title":"An extension to the universal time scale for vortex ring formation","volume":"915","author":"Limbourg","year":"2021","journal-title":"J. Fluid Mech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/BF00918769","article-title":"Dynamics of laminar vortex rings in a stratified liquid","volume":"28","author":"Belyaev","year":"1987","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1007\/BF00906105","article-title":"Apparent internal waves in a fluid with exponential density distribution","volume":"22","author":"Makarov","year":"1981","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"ref_29","first-page":"147","article-title":"The generation of vortex rings, II","volume":"9","author":"Okabe","year":"1961","journal-title":"Rep. Res. Inst. Appl. Mech."},{"key":"ref_30","unstructured":"Batchelor, G.K. (1967). An Introduction to Fluid Dynamics, Cambridge University Press."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1063\/1.1708605","article-title":"Crater Depth in Fluid Impacts","volume":"37","author":"Engel","year":"1966","journal-title":"J. Appl. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1017\/S0022112067000709","article-title":"Formation of vortex rings from falling drops","volume":"29","author":"Chapman","year":"1967","journal-title":"J. Fluid Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/S0021-9797(85)80008-4","article-title":"Some drops don\u2019t splash","volume":"106","author":"Rodriguez","year":"1985","journal-title":"J. Colloid Interface Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0021-9797(88)90414-6","article-title":"The penetration of drop-formed vortex rings into pools of liquid","volume":"121","author":"Rodriguez","year":"1988","journal-title":"J. Colloid Interface Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1007\/BF00193914","article-title":"Penetration length and diameter development of vortex rings generated by impacting water drops","volume":"21","author":"Durst","year":"1996","journal-title":"Exp. Fluids"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s003480050062","article-title":"Vortex ring generation due to the coalescence of a water drop at a free surface","volume":"22","author":"Dooley","year":"1997","journal-title":"Exp. Fluids"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1098\/rsta.2005.1562","article-title":"Droplet impact on water layers: Post-impact analysis and computations","volume":"363","author":"Purvis","year":"2005","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1017\/jfm.2019.503","article-title":"Kinematics of vortex ring generated by a drop upon impacting a liquid pool","volume":"875","author":"Saha","year":"2019","journal-title":"J. Fluid Mech."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1538","DOI":"10.1088\/0957-0233\/6\/10\/014","article-title":"An apparatus to study drop-formed vortex rings","volume":"6","author":"Peck","year":"1995","journal-title":"Meas. Sci. Technol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1063\/1.868352","article-title":"The three-dimensional vortex structure of an impacting water drop","volume":"6","author":"Peck","year":"1994","journal-title":"Phys. Fluids"},{"key":"ref_41","first-page":"99","article-title":"Flow visualization in turbulent large-scale structure research","volume":"3","author":"Sigurdson","year":"1997","journal-title":"Flow Vis. Soc. Jpn. Atlas Vis."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Samimy, M., Breuer, K.S., Leal, L.G., and Steen, P.H. (2003). A Gallery of Fluid Motion, Cambridge University Press.","DOI":"10.1017\/CBO9780511610820"},{"key":"ref_43","first-page":"410","article-title":"Underwater ambient noise","volume":"7","author":"Knudsen","year":"1948","journal-title":"J. Mar. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.1121\/1.397353","article-title":"Underwater sound produced by individual drop impacts and rainfall","volume":"85","author":"Pumphrey","year":"1989","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1017\/S0022112090003378","article-title":"The entrainment of bubbles by drop impacts","volume":"220","author":"Pumphrey","year":"1990","journal-title":"J. Fluid Mech."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3555","DOI":"10.1121\/1.1910283","article-title":"Prediction of underwater sound levels from rain and wind","volume":"117","author":"Ma","year":"2005","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Liu, S., Li, Q., Shang, D., Tang, R., and Zhang, Q. (2021). Measurement of underwater acoustic energy radiated by single raindrops. Sensors, 21.","DOI":"10.3390\/s21082687"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Kathiravelu, G., Lucke, T., and Nichols, P. (2016). Rain Drop Measurement Techniques: A Review. Water, 8.","DOI":"10.3390\/w8010029"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1017\/S0022112096001267","article-title":"The transitional regime between coalescing and splashing drops","volume":"306","author":"Rein","year":"1996","journal-title":"J. Fluid Mech."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"082108","DOI":"10.1063\/1.4746793","article-title":"Bubble pinch-of and scaling during liquid drop impact on liquid pool","volume":"24","author":"Ray","year":"2012","journal-title":"Phys. Fluids"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1017\/jfm.2015.108","article-title":"Regimes during liquid drop impact on a liquid pool","volume":"768","author":"Ray","year":"2015","journal-title":"J. Fluid Mech."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1063\/1.1652061","article-title":"On some common features of drop impact on liquid surfaces","volume":"16","author":"Fedorchenko","year":"2004","journal-title":"Phys. Fluids"},{"key":"ref_53","unstructured":"Landau, L.D., and Lifshitz, E.M. (1987). Fluid Mechanics. V. Course of Theoretical Physics, Pergamon Press."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"092101","DOI":"10.1063\/1.4992124","article-title":"The regime of large bubble entrapment during a single drop impact on a liquid pool","volume":"29","author":"Deka","year":"2017","journal-title":"Phys. Fluids"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"036306","DOI":"10.1103\/PhysRevE.79.036306","article-title":"Drop impact onto a liquid layer of finite thickness: Dynamics of the cavity evolution","volume":"79","author":"Roisman","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"3639","DOI":"10.1016\/S0009-2509(02)00266-X","article-title":"Spreading of Inviscid Drop Impacting on Liquid Film","volume":"57","author":"Davidson","year":"2002","journal-title":"Chem. Eng. Sci."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2567","DOI":"10.1007\/s12206-011-0613-7","article-title":"A numerical analysis of drop impact on liquid film by using a level set method","volume":"25","author":"Lee","year":"2011","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"111211","DOI":"10.1016\/j.oceaneng.2022.111211","article-title":"Computational Fluid Dynamics modelling of hydrodynamic characteristics of oscillatory flow past a square cylinder using the level set method","volume":"253","author":"Dutta","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"012037","DOI":"10.1088\/1742-6596\/918\/1\/012037","article-title":"Simulation of droplet impact onto a deep pool for large Froude numbers in different open-source codes","volume":"918","author":"Korchagova","year":"2017","journal-title":"IOP Conf. Ser. J. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cjche.2016.05.018","article-title":"DNS analysis of incipient drop impact dynamics using an accurate level set method","volume":"25","author":"Chai","year":"2017","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s11012-019-00980-x","article-title":"Toward direct numerical simulation of high speed droplet impact","volume":"55","author":"Xavier","year":"2019","journal-title":"Meccanica"},{"key":"ref_62","unstructured":"Li, Z., Liu, C., and Wan, D. (2020, January 11\u201316). Numerical simulations of droplet impact onto a pool surface. Proceedings of the Thirtieth International Ocean and Polar Engineering Conference, Shanghai, China."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"033128","DOI":"10.1103\/PhysRevE.93.033128","article-title":"Vortex-ring-induced large bubble entrainment during drop impact","volume":"93","author":"Thoraval","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"073605","DOI":"10.1103\/PhysRevFluids.5.073605","article-title":"Marangoni effect on the impact of droplets onto a liquid-gas interface","volume":"5","author":"Feifei","year":"2020","journal-title":"Phys. Rev. Fluids"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"062108","DOI":"10.1063\/1.5099228","article-title":"Deformation characteristics and energy conversion during droplet impact on a water surface","volume":"31","author":"Ma","year":"2019","journal-title":"Phys. Fluids"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"027103","DOI":"10.1063\/1.4998675","article-title":"The shape and dynamics of the generation of the splash forms in single-phase systems after drop hitting","volume":"30","author":"Sochan","year":"2018","journal-title":"Phys. Fluids"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"032116","DOI":"10.1063\/5.0041087","article-title":"Viscous diffusion induced evolution of a vortex ring","volume":"33","author":"Manas","year":"2021","journal-title":"Phys. Fluids"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"R3","DOI":"10.1017\/jfm.2013.261","article-title":"Air entrainment during impact of droplets on liquid surfaces","volume":"726","author":"Tran","year":"2013","journal-title":"J. Fluid Mech."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.euromechflu.2016.11.009","article-title":"Experimental study of vortices and cavities from single and double drop impacts onto deep pools","volume":"62","author":"Santini","year":"2017","journal-title":"Eur. J. Mech. B Fluids"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"023601","DOI":"10.1103\/PhysRevFluids.2.023601","article-title":"Jet dynamics post drop impact on a deep pool","volume":"2","author":"Michon","year":"2017","journal-title":"Phys. Rev. Fluids"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"083602","DOI":"10.1103\/PhysRevFluids.2.083602","article-title":"Oblique drop impact onto a deep liquid pool","volume":"2","author":"Gielen","year":"2017","journal-title":"Phys. Rev. Fluids"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.ijmultiphaseflow.2018.01.015","article-title":"Droplet splashing on thin moving films at high Weber numbers","volume":"101","author":"Burzynski","year":"2018","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"073601","DOI":"10.1103\/PhysRevFluids.4.073601","article-title":"Role of surrounding gas in the outcome of droplet splashing","volume":"4","author":"Burzynski","year":"2019","journal-title":"Phys. Rev. Fluids"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1175\/JPO-D-17-0172.1","article-title":"Experimental study of drop impact on deep water surface in the presence of wind","volume":"48","author":"Liu","year":"2018","journal-title":"J. Phys. Oceanogr."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"1634","DOI":"10.1007\/s11434-008-0246-0","article-title":"Experiments on ring wave packet generated by water drop","volume":"53","author":"Zhu","year":"2008","journal-title":"Chin. Sci. Bull."},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Chashechkin, Y.D. (2021). Foundations of engineering mathematics applied for fluid flows. Axioms, 10.","DOI":"10.3390\/axioms10040286"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1134\/S102833582101002X","article-title":"Delay of cavity formation in the intrusive coalescence of a free falling drop with target fluid","volume":"66","author":"Chashechkin","year":"2021","journal-title":"Dokl. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1134\/S1028335821040078","article-title":"Drop decay into individual fibers at the boundary of the contact area with a target fluid","volume":"66","author":"Chashechkin","year":"2021","journal-title":"Dokl. Phys."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1134\/S1028335818070066","article-title":"Banded structures in the distribution pattern of a drop over the surface of the host liquid","volume":"63","author":"Chashechkin","year":"2018","journal-title":"Dokl. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1134\/S0001433819020026","article-title":"Evolution of the fine structure of the matter distribution of a free-falling droplet in mixing liquids","volume":"55","author":"Chashechkin","year":"2019","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"012107","DOI":"10.1063\/1.5064640","article-title":"Capillary surface wave formation and mixing of miscible liquids during droplet impact onto a liquid film","volume":"31","author":"Ersoy","year":"2019","journal-title":"Phys. Fluids"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1134\/S0001433822020037","article-title":"Transfer of the substance of a colored drop in a liquid layer with travelling plane gravity\u2013capillary waves","volume":"58","author":"Chashechkin","year":"2022","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1134\/S000143381205009X","article-title":"Structural stability of substance transport in a compound vortex","volume":"48","author":"Stepanova","year":"2012","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","unstructured":"Chashechkin, Y.D., and Ochirov, A.A. (2022). Periodic waves and ligaments on the surface of a viscous exponentially stratified fluid in a uniform gravity field. Axioms, 11.","DOI":"10.3390\/axioms11080402"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1134\/S1063771020040016","article-title":"Evolution of the structure of acoustic signals caused by the impact of a falling drop on a liquid","volume":"66","author":"Chashechkin","year":"2020","journal-title":"Acoust. Phys."},{"key":"ref_86","unstructured":"Feistel, R., Harvey, A.H., and Pawlowicz, R. (2016, January 1\u20135). International Association for the Properties of Water and Steam. Proceedings of the Advisory Note No. 6: Relationship between Various IAPWS Documents and the International Thermodynamic Equation of Seawater\u20132010 (TEOS-10), Dresden, Germany."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"471","DOI":"10.5194\/os-14-471-2018","article-title":"Thermodynamic properties of seawater, ice and humid air: TEOS-10, before and beyond","volume":"14","author":"Feistel","year":"2018","journal-title":"Ocean. Sci."},{"key":"ref_88","unstructured":"Popov, N.I., Fedorov, K.N., and Orlov, V.M. (1979). Morskaya Voda. Spravochnoye Rukovodstvo, Nauka. (In Russian)."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1134\/S000143701003001X","article-title":"Calculation of the sound velocity in stratified seawater based on a set of fundamental equations","volume":"50","author":"Bardakov","year":"2010","journal-title":"Oceanology"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1134\/S0001433807010112","article-title":"Dynamics of gravity-capillary waves on the surface of a nonuniformly heated fluid","volume":"43","author":"Kistovich","year":"2007","journal-title":"Izv. Atmos. Ocean. Phys."},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Chashechkin, Y.D. (2021). Conventional partial and new complete solutions of the fundamental equations of fluid mechanics in the problem of periodic internal waves with accompanying ligaments generation. Mathematics, 9.","DOI":"10.3390\/math9060586"},{"key":"ref_92","doi-asserted-by":"crossref","unstructured":"Eisenberg, D., and Kauzmann, W. (2005). The Structure and Properties of Water (Oxford Classic Texts in the Physical Sciences), Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198570264.001.0001"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s11224-011-9766-3","article-title":"Collective effects in diffusional motion of water molecules: Computer simulation","volume":"22","author":"Malenkov","year":"2011","journal-title":"Struct. Chem."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1017\/jfm.2018.383","article-title":"Early azimuthal instability during drop impact","volume":"848","author":"Li","year":"2018","journal-title":"J. Fluid Mech."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1134\/S001546282104008X","article-title":"Fine structure of the spreading pattern of a freely falling droplet in a fluid at rest","volume":"56","author":"Chashechkin","year":"2021","journal-title":"Fluid Dyn."},{"key":"ref_96","unstructured":"(2022, June 15). Hydrophysical Complex for Modeling Hydrodynamic Processes in the Environment and Their Impact on Underwater Technical Objects, as Well as the Distribution of Impurities in the Ocean and Atmosphere, Institute for Problems in Mechanics RAS. Available online: http:\/\/ipmnet.ru\/uniqequip\/gfk."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1134\/S0021894416030032","article-title":"Transformation of the bridge during separation of a droplet","volume":"57","author":"Chashechkin","year":"2016","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"ref_98","first-page":"801","article-title":"Complete Coalescence, Partial Bounce and Rebound: Different Regimes Resulting from the Interaction of a Free Falling Drop with a Target Fluid","volume":"16","author":"Chashechkin","year":"2020","journal-title":"Fluid Dyn. Mater. Process. FDMP"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1134\/S1028335820100067","article-title":"Multiple emissions of splashes upon drop impact","volume":"65","author":"Chashechkin","year":"2020","journal-title":"Dokl. Phys."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1134\/S1028335821080036","article-title":"Formation of a system of inclined loops in the flows of a drop impact","volume":"66","author":"Chashechkin","year":"2021","journal-title":"Dokl. Phys."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1134\/S0015462819070036","article-title":"Visualization of the Fine Perturbation Structure of a Liquid Surface by Flows Induced by a Drop Impact","volume":"54","author":"Chashechkin","year":"2019","journal-title":"Fluid Dyn."},{"key":"ref_102","unstructured":"Gibbs, J.W. (1902). Elementary Principles in Statistical Mechanics, Scribner\u2019s and Sons."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"243","DOI":"10.3103\/S1541308X08040018","article-title":"Study of Nanostructure of highly purified water by measuring scattering matrix elements of laser radiation","volume":"16","author":"Bunkin","year":"2008","journal-title":"Phys. Wave Phenom."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"3856","DOI":"10.1039\/b511257e","article-title":"Water molecule clusters measured at water\/air interfaces using atomic force microscopy","volume":"7","author":"Teschke","year":"2005","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1134\/S1028335821100013","article-title":"Visualization of media contact areas in drop impact flows with chemical reactions","volume":"66","author":"Chashechkin","year":"2021","journal-title":"Dokl. Phys."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/374\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:15:37Z","timestamp":1760123737000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/374"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,13]]},"references-count":105,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040374"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040374","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,13]]}}}