The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

• Autorzy: Alexander Khapalov
• Języki publikacji: EN

Abstrakty

EN

We introduce and investigate the well-posedness of a model describing the self-propelled motion of a small abstract swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, typically associated with low Reynolds numbers. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by rotational and elastic Hooke forces. Models like this are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids.

Słowa kluczowe

EN

swimming models; coupled PDE/ODE systems; nonstationary Stokes equation

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2013
• Tom: 23
• Numer: 2
• Strony: 277-290

Daty

wydano

2013

otrzymano

2012-04-19

poprawiono

2012-10-18

poprawiono

2013-01-14

Twórcy

autor

Alexander Khapalov

• Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA

Bibliografia

• Alouges, F., DeSimone, A. and Lefebvre, A. (2008). Optimal Strokes for low Reynolds number swimmers: An example, Journal of Nonlinear Science 18: 27-302.
• Becker, L.E, Koehler, S.A. and Stone, H.A. (2003). On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer, Journal of Fluid Mechanics 490: 15-35.
• Belter, D. and Skrzypczyński, P (2010). A biologically inspired approach to feasible gait learning for a hexapod robot, International Journal of Applied Mathematics and Computer Science 20(1): 69-84, DOI: 10.2478/v10006-010-0005-7.
• Childress, S. (1981). Mechanics of Swimming and Flying, Cambridge University Press, Cambridge.
• Dal Maso, Gi., DeSimone, A. and Morandotti, M. (2011). An existence and uniqueness result for the motion of self-propelled micro-swimmers, SIAM Journal of Mathematical Analysis 43: 1345-1368.
• Fukuda, T., Kawamoto, A., Arai, F. and Matsuura, H. (1995). Steering mechanism and swimming experiment of micro mobile robot in water, Proceedings of Micro Electro Mechanical Systems (MEMS'95), Amsterdam, The Netherlands, pp. 300-305.
• Fauci, L. and Peskin, C.S. (1988). A computational model of aquatic animal locomotion, Journal of Computational Physics 77: 85-108.
• Fauci, L. (1993). Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics 141: 91-102.
• Galdi, G.P. (1999). On the steady self-propelled motion of a body in a viscous incompressible fluid, Archive for Rational Mechanics and Analysis 1448: 53-88.
• Galdi, G.P. (2002). On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in S. Friedlander and D. Serre (Eds.), Handbook of Mathematical Fluid Mechanics, Elsevier Science, Amsterdam, pp. 653-791.
• Gray, J. (1932). Study in animal locomotion IV: The propulsive power of the dolphin, Journal of Experimental Biology 10: 192-199.
• Gray, J. and Hancock, G.J. (1955). The propulsion of sea-urchin spermatozoa, Journal of Experimental Biology 32: 802.
• Guo, S. (2002). Afin type of micro-robot in pipe, Proceedings of the 2002 International Symposium on Micromechatronics and Human Science (MHS 2002), Ngoya, Japan, pp. 93-98.
• Gurtin, M.E. (1981). An Introduction to Continuum Mechanics, Academic Press, New York, NY.
• Hawthorne, M.F., Zink, J.I., Skelton, J.M., Bayer, M.J., Liu, Ch., Livshits, E., Baer, R. and Neuhauser, D. (2004). Electrical or photocontrol of rotary motion of a metallacarborane, Science 303: 1849.
• Happel, V. and Brenner, H. (1965). Low Reynolds Number Hydrodynamics, Prentice Hall, Upper Saddle River, NJ.
• Hirose, S. (1993). Biologically Inspired Robots: Snake-like Locomotors and Manipulators, Oxford University Press, Oxford.
• Kanso. E., Marsden, J.E., Rowley, C.W. and Melli-Huber, J. (2005). Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science 15: 255-289.
• Khapalov, A.Y. (1999). Approximate controllability properties of the semilinear heat equation with lumped controls, International Journal of Applied Mathematics and Computer Science 9(4): 751-765.
• Khapalov, A.Y. (2005). The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid, Technical Report 2005-5, Washington State University, Department of Mathematics, http://www.math.wsu.edu/TRS/2005-5.pdf.
• Khapalov, A.Y. and Eubanks, S. (2009). The wellposedness of a 2-D swimming model governed in the nonstationary Stokes fluid by multiplicative controls, Applicable Analysis 88: 1763-1783, DOI:10.1080/00036810903401222.
• Khapalov, A.Y. (2010). Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Mathematics Series, Vol. 1995, Springer-Verlag, Berlin/Heidelberg.
• Khapalov, A.Y. and Trinh, G. (2013). Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid, Discrete and Continuous Dynamical Systems Series A 33: 1513-1544.
• Khapalov, A.Y. (2013). Micro motions of a swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, arXiv: 1205.1088 [math.AP], (preprint).
• Koiller, J., Ehlers, F. and Montgomery, R. (1996). Problems and progress in microswimming, Journal of Nonlinear Science 6: 507-541.
• Ladyzhenskaya, O.A. (1963). The Mathematical Theory of Viscous Incompressible Flow, Cordon and Breach, New York, NY.
• Lighthill, M.J. (1975). Mathematics of Biofluid Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA.
• Mason, R. and Burdick, J.W. (2000). Experiments in carangiform robotic fish locomotion, Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, CA, USA, pp. 428-435.
• McIsaac, K.A. and Ostrowski, J.P. (2000). Motion planning for dynamic eel-like robots, Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, CA, USA, pp. 1695-1700.
• Martinez, S. and Cortes, J. (2001). Geometric control of robotic locomotion systems, Proceedings of the 10th Fall Workshop on Geometry and Physics, Miraflores de la Sierra, Spain, Vol. 4, pp. 183-198.
• Morgansen, K.A., Duindam, V., Mason, R.J., Burdick, J.W. and Murray, R.M. (2001). Nonlinear control methods for planar carangiform robot fish locomotion, Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, Korea, pp. 427-434.
• Peskin, C.S. (1977). Numerical analysis of blood flow in the heart, Journal of Computational Physics 25: 220-252.
• Peskin, C.S. and McQueen, D.M. (1994). A general method for the computer simulation of biological systems interacting with fluids, SEB Symposium on Biological Fluid Dynamics, Leeds, UK.
• San Martin, J., Takashi, T. and Tucsnak, M. (2007). A control theoretic approach to the swimming of microscopic organisms, Quarterly Applied Mathematics 65: 405-424.
• San Martin, J., Scheid, J.-F., Takashi, T. and Tucsnak, M. (2008). An initial and boundary value problem modeling of fish-like swimming, Archive of Rational Mechanics and Analysis 188: 429-455.
• Shapere, A. and Wilczeck, F. (1989). Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics 198: 557-585.
• Sigalotti, M. and Vivalda, J.-C. (2009). Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM: Control, Optimisation and Calculus of Variations 16: 1053-1076.
• Taylor, G.I. (1951). Analysis of the swimming of microscopic organisms, Proceeding of the Royal Society of London A 209: 447-461.
• Taylor, G.I. (1952). Analysis of the swimming of long and narrow animals, Proceedings of the Royal Society of London A 214(1117): 158-183.
• Temam, R. (1984). Navier-Stokes Equations, North-Holland, Amsterdam.
• Trintafyllou, M.S., Trintafyllou, G.S. and Yue, D.K.P. (2000). Hydrodynamics of fishlike swimming, Annual Review of Fluid Mechanics 32: 33-53.
• Tytell, E.D., Hsu, C.-Y, Williams, T.L., Cohen, A.H. and Fauci, L.J. (2010). Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming, Proceedings of the National Academy Sciences, USA 107: 19832-19837.
• Wu, T.Y. (1971). Hydrodynamics of swimming fish and cetaceans, Advances in Applied Mathematics 11: 1-63.

Identyfikatory

DOI

10.2478/amcs-2013-0021