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### Analytical Results of the Motion of Oscillating Dumbbell in a Viscous Fluid

Received: 31 December 2023    Accepted: 24 January 2024    Published: 5 February 2024
Abstract

The aim of this paper is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other is non-conducting sphere. As result, the oscillating dumbbell moves due to the induced flow oscillation of the surrounding fluid. The fluid flow past the spheres is described by the Stokes equation and the governing equation in the vector form for the oscillating dumbbell is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of the oscillating dumbbell is formulated. The trajectory of the oscillating dumbbell was found to be inversely proportional to the frequency of the external field, and the results demonstrated that the oscillating dumbbell moves in a circular path with a speed that decreases inversely with the length of the spring.

 Published in Mathematics and Computer Science (Volume 9, Issue 1) DOI 10.11648/mcs.20240901.11 Page(s) 1-11 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2024. Published by Science Publishing Group
Keywords

Fluid Dynamics, Low Reynolds Number, Oscillatory Motion, Stokes Equation, Two-Timing Method

References
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• APA Style

Al-Hatmi, M. M., Purnama, A. (2024). Analytical Results of the Motion of Oscillating Dumbbell in a Viscous Fluid. Mathematics and Computer Science, 9(1), 1-11. https://doi.org/10.11648/mcs.20240901.11

ACS Style

Al-Hatmi, M. M.; Purnama, A. Analytical Results of the Motion of Oscillating Dumbbell in a Viscous Fluid. Math. Comput. Sci. 2024, 9(1), 1-11. doi: 10.11648/mcs.20240901.11

AMA Style

Al-Hatmi MM, Purnama A. Analytical Results of the Motion of Oscillating Dumbbell in a Viscous Fluid. Math Comput Sci. 2024;9(1):1-11. doi: 10.11648/mcs.20240901.11

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author = {Mohammed Mattar Al-Hatmi and Anton Purnama},
title = {Analytical Results of the Motion of Oscillating Dumbbell in a Viscous Fluid},
journal = {Mathematics and Computer Science},
volume = {9},
number = {1},
pages = {1-11},
doi = {10.11648/mcs.20240901.11},
url = {https://doi.org/10.11648/mcs.20240901.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.mcs.20240901.11},
abstract = {The aim of this paper is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other is non-conducting sphere. As result, the oscillating dumbbell moves due to the induced flow oscillation of the surrounding fluid. The fluid flow past the spheres is described by the Stokes equation and the governing equation in the vector form for the oscillating dumbbell is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of the oscillating dumbbell is formulated. The trajectory of the oscillating dumbbell was found to be inversely proportional to the frequency of the external field, and the results demonstrated that the oscillating dumbbell moves in a circular path with a speed that decreases inversely with the length of the spring.},
year = {2024}
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AB  - The aim of this paper is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other is non-conducting sphere. As result, the oscillating dumbbell moves due to the induced flow oscillation of the surrounding fluid. The fluid flow past the spheres is described by the Stokes equation and the governing equation in the vector form for the oscillating dumbbell is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of the oscillating dumbbell is formulated. The trajectory of the oscillating dumbbell was found to be inversely proportional to the frequency of the external field, and the results demonstrated that the oscillating dumbbell moves in a circular path with a speed that decreases inversely with the length of the spring.
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Author Information
• Department of Basic Science, A’Sharqiyah University, Ibra, Oman

• Department of Mathematics, Sultan Qaboos University, Muscat, Oman

• Sections