Mathematics and Computer Science

Submit a Manuscript

Publishing with us to make your research visible to the widest possible audience.

Propose a Special Issue

Building a community of authors and readers to discuss the latest research and develop new ideas.

Research Article |

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

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.

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

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

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Alexander, G. P., and Yeomans, J. M. (2008). Dumb-bell swimmers. Europhysics Letters, 83(3), 34006.
2. Al-Hatmi, M. M., and Purnama, A. (2021). On the motion of two micro-spheres in a Stokes flow driven by an external oscillator field. Journal of Mathematics and Mathematical Sciences, 2021: 9211272.
3. Amaratunga, M., Rabenjafimanantsoa, H. A., and Time, R. W. (2021). Influence of low-frequency oscillatory motion on particle settling in Newtonian and shear- thinning non-Newtonian fluids. Journal of Petroleum Science and Engineering, 196: 107786.
4. Belovs, M. and Cebers, A. (2009). Ferromagnetic microswimmer. Physical Review E, 79(5): 051503.
5. Bogoliubov, N. N. and Mitropolskii, Y. A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations, 10. CRC Press.
6. Box, F., Han, E., Tipton, C.R., and Mullin, T. (2017). On the motion of linked spheres in a Stokes flow. Experiments in Fluids, 58: 1-10.
7. Box, F., Singh, K., and Mullin, T. (2018). The interaction between rotationally oscillating spheres and solid boundaries in a Stokes flow. Journal of Fluid Mechanics, 849: 834-859.
8. Castilla, R. (2022). Dynamics of a microsphere inside a spherical cavity with Newtonian fluid subjected to periodic contractions. Physics of Fluids, 34: 071901.
9. Dogangil, G., Ergeneman, O., Abbott, J. J., Pan, S., Hall, H., Muntwyler, S., and Nelson, B. J. (2008). Toward targeted retinal drug delivery with wireless magnetic microrobots. In 2008IEEE/RSJ InternationalConference on Intelligent Robots and Systems, 1921-1926.
10. Fusco, S., Chatzipirpiridis, G., Sivaraman, K. M., Ergeneman, O., Nelson, B. J., and Pan, S. (2013). Chitosan electrodeposition for microrobotic drug delivery. Advanced Healthcare Materials, 2(7): 1037- 1044.
11. Gilbert, A. D., Ogrin, F. Y., Petrov, P. G., and Wimlove, C. P. (2010). Theory of ferromagnetic microswimmers. Quarterly Journal of Mechanics and Applied Mathematics, 64(3): 239-263.
12. Grosjean, G., Hubert, M., Lagubeau, G., and Vandewalle, N. (2016). Realization of the Najafi-Golestanian microswimmer. Physical Review E, 94(2): 021101.
13. Happel, J. and Brenner, H. (2012). Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, 1. Springer Science & Business Media.
14. Ijspeert, A. J. (2014). Biorobotics: Using robots to emulate and investigate agile locomotion. Science, 346(6206): 196-203.
15. Landau, L., andLifshitz, E.(1987). CourseofTheoretical Physics. Fluid Mechanics, 6. Elsevier.
16. Lyubimova, T., Lyubimov, D., and Shardin, M. (2011). The interaction of rigid cylinders in a low Reynolds number pulsational flow. Microgravity Science and Technology, 23(3): 305-309.
17. Mathieu, J. B., Beaudoin, G., and Martel, S. (2006). Method of propulsion of a ferromagnetic core in the cardiovascular system through magnetic gradients generated by an MRI system. IEEE Transactions on Biomedical Engineering, 53(2): 292-299.
18. Mullin, T., Li, Y., Del Pino, C., and Ashmore, J. (2005). An experimental study of fixed points and chaos in the motion of spheres in a Stokes flow. Journal of Applied Mathematics, 70(5): 666-676.
19. Nayfeh, A. (1973). Perturbation Methods. John Wiley and Sons, New York.
20. Nelson, B. J., Kaliakatsos, I. K., and Abbott, J. J. (2010). Microrobots for minimally invasive medicine. Annual Review of Biomedical Engineering, 12: 55-85.
21. Norton, J. (1985). What was Einstein’s principle of equivalence?. Studies in History and Philosophy of Science Part A, 16(3): 203-246.
22. Ogrin, F. Y., Petrov, P. G., and Winlove, C. P. (2008). Ferromagnetic microswimmers. Physical Review Letters, 100(21): 218102.
23. Romanczuk, P., Bär, M., Ebeling, W., Lindner, B., and Schimansky-Geier, L. (2012). Active Brownian Particles. The European Physical Journal Special Topics, 202(1): 1-162.
24. Shapere, A., and Wilczek, F. (1989). Efficiencies of self- propulsion at low Reynolds number. Journal of Fluid Mechanics, 198:587-599.
25. Taylor, G. I. (1951). Analysis of the swimming of microscopic organisms. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 209(1099): 447-461.
26. Verhulst, F. (2007). Singular perturbation methods for slowfast dynamics. Journal of Nonlinear Dynamics, 50(4):747-753.
27. Wilson, H. J. (2005). An analytic form for the pair distribution function and rheology of a dilute suspension of rough spheres in plane strain flow. Journal of Fluid Mechanics, 534: 97-114.