{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:35:48Z","timestamp":1760150148504,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T00:00:00Z","timestamp":1698278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea (NRF)","doi-asserted-by":"publisher","award":["2020R1G1A1010247"],"award-info":[{"award-number":["2020R1G1A1010247"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Mathematical modeling and analysis of biologically inspired systems has been a fascinating research topic in recent years. In this work, we present the results obtained from the simulation of an elastic rod (that mimics a flagellum axoneme) rotational motion in a viscous fluid by using the lattice Boltzmann method (LBM) combined with an immersed boundary method (IBM). A finite element model consists of a set of beam and truss elements used to discretize the flagellum axoneme while the fluid flow is solved by the well-known LBM. The hydrodynamic coupling to maintain the no-slip boundary condition between the fluid and the elastic rod is conducted with the IBM. The rod is actuated with a torque applied at its base cross-section that acts as a driving motor of the axoneme. We simulated the rotational dynamics of the rod for three different rotational frequencies (low, medium, and high) of the motor. To compare with previous publication results, we chose the sperm number Sp=L(4\u03c0\u03bc\u03c9)\/(EI)1\/4 as the validation parameter. We found that at the low rotational frequency, f = 1.5 Hz, the rod performs stable twirling motion after attaining an equilibrium state (the rod undergoes rigid rotation about its axis). At the medium frequency, f = 2.65 Hz, the rod undergoes whirling motion, where the tip of the rod rotates about the central rotational axis of the driving motor. When the frequency increases further, i.e., when it reaches the critical value, fc\u00a0\u2248\u00a02.7 Hz, the whirling motion becomes over-whirling, where the tip of the filament falls back to the base and performs a steady crank-shafting motion. All three rotational dynamics, twirling, whirling, and over-whirling, and the critical value of rotational frequency are in good agreement with the previously published results. We also observed that our present simulation technique is computationally more efficient than previous works.<\/jats:p>","DOI":"10.3390\/axioms12111011","type":"journal-article","created":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T03:33:51Z","timestamp":1698377631000},"page":"1011","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Simulation of an Elastic Rod Whirling Instabilities by Using the Lattice Boltzmann Method Combined with an Immersed Boundary Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1438-1954","authenticated-orcid":false,"given":"Suresh","family":"Alapati","sequence":"first","affiliation":[{"name":"Department of Mechatronics Engineering, Kyungsung University, 309 Suyeong-ro, Nam-gu, Busan 48434, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wooseong","family":"Che","sequence":"additional","affiliation":[{"name":"Department of Mechatronics Engineering, Kyungsung University, 309 Suyeong-ro, Nam-gu, Busan 48434, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1308-7067","authenticated-orcid":false,"given":"Sunkara Srinivasa","family":"Rao","sequence":"additional","affiliation":[{"name":"Department of Electronics and Communication Engineering, Koneru Lakshmaiah Education Foundation, Bowrampet, Hyderabad 500043, Telangana, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1583-2622","authenticated-orcid":false,"given":"Giang T. T.","family":"Phan","sequence":"additional","affiliation":[{"name":"Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam"},{"name":"Faculty of Natural Sciences, Duy Tan University, Da Nang 550000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1038\/nrmicro2009","article-title":"Swimming with protists: Perception, motility and flagellum assembly","volume":"6","author":"Ginger","year":"2008","journal-title":"Nat. Rev. Microbiol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1590","DOI":"10.1103\/PhysRevLett.82.1590","article-title":"Self-organized beating and swimming of internally driven filaments","volume":"82","author":"Camalet","year":"1999","journal-title":"Phys. Rev. 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