{"title":"考虑卡普托分数时间导数的漩涡容器中的液体漩涡形成","authors":"M. Turkyilmazoglu, A. S. Alofi","doi":"10.3390/fractalfract8040231","DOIUrl":null,"url":null,"abstract":"This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":3.6000,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo\",\"authors\":\"M. Turkyilmazoglu, A. S. Alofi\",\"doi\":\"10.3390/fractalfract8040231\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model.\",\"PeriodicalId\":12435,\"journal\":{\"name\":\"Fractal and Fractional\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.6000,\"publicationDate\":\"2024-04-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fractal and Fractional\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3390/fractalfract8040231\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fractal and Fractional","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3390/fractalfract8040231","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo
This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model.
期刊介绍:
Fractal and Fractional is an international, scientific, peer-reviewed, open access journal that focuses on the study of fractals and fractional calculus, as well as their applications across various fields of science and engineering. It is published monthly online by MDPI and offers a cutting-edge platform for research papers, reviews, and short notes in this specialized area. The journal, identified by ISSN 2504-3110, encourages scientists to submit their experimental and theoretical findings in great detail, with no limits on the length of manuscripts to ensure reproducibility. A key objective is to facilitate the publication of detailed research, including experimental procedures and calculations. "Fractal and Fractional" also stands out for its unique offerings: it warmly welcomes manuscripts related to research proposals and innovative ideas, and allows for the deposition of electronic files containing detailed calculations and experimental protocols as supplementary material.