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

Alouges, F., DeSimone, A. and Lefebvre, A. (2008). Optimal Strokes for low Reynolds number swimmers: An example, Journal of Nonlinear Science 18: 27-302.10.1007/s00332-007-9013-7Search in Google Scholar

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 FluidMechanics 490: 15-35.10.1017/S0022112003005184Search in Google Scholar

Belter, D. and Skrzypczy´nski, P (2010). A biologically inspired approach to feasible gait learning for a hexapod robot, InternationalJournal of Applied Mathematics and ComputerScience 20(1): 69-84, DOI: 10.2478/v10006-010-0005-7.10.2478/v10006-010-0005-7Search in Google Scholar

Childress, S. (1981). Mechanics of Swimming and Flying, Cambridge University Press, Cambridge.10.1017/CBO9780511569593Search in Google Scholar

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 MathematicalAnalysis 43: 1345-1368.10.1137/10080083XSearch in Google Scholar

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 MechanicalSystems (MEMS’95), Amsterdam, The Netherlands, pp. 300-305.Search in Google Scholar

Fauci, L. and Peskin, C.S. (1988). A computational model of aquatic animal locomotion, Journal of ComputationalPhysics 77: 85-108.10.1016/0021-9991(88)90158-1Search in Google Scholar

Fauci, L. (1993). Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics 141: 91-102.10.1090/conm/141/1212579Search in Google Scholar

Galdi, G.P. (1999). On the steady self-propelled motion of a body in a viscous incompressible fluid, Archive for RationalMechanics and Analysis 1448: 53-88.10.1007/s002050050156Search in Google Scholar

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 MathematicalFluid Mechanics, Elsevier Science, Amsterdam, pp. 653-791.10.1016/S1874-5792(02)80014-3Search in Google Scholar

Gray, J. (1932). Study in animal locomotion IV: The propulsive power of the dolphin, Journal of Experimental Biology 10: 192-199.10.1242/jeb.13.2.192Search in Google Scholar

Gray, J. and Hancock, G.J. (1955). The propulsion of sea-urchin spermatozoa, Journal of Experimental Biology 32: 802.10.1242/jeb.32.4.802Search in Google Scholar

Guo, S. (2002). Afin type of micro-robot in pipe, Proceedingsof the 2002 International Symposium on Micromechatronicsand Human Science (MHS 2002), Ngoya, Japan, pp. 93-98.Search in Google Scholar

Gurtin, M.E. (1981). An Introduction to Continuum Mechanics, Academic Press, New York, NY.Search in Google Scholar

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: 1849.10.1126/science.1093846Search in Google Scholar

Happel, V. and Brenner, H. (1965). Low Reynolds Number Hydrodynamics, Prentice Hall, Upper Saddle River, NJ.Search in Google Scholar

Hirose, S. (1993). Biologically Inspired Robots: Snake-likeLocomotors and Manipulators, Oxford University Press, Oxford.Search in Google Scholar

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: 255-289.10.1007/s00332-004-0650-9Search in Google Scholar

Khapalov, A.Y. (1999). Approximate controllability properties of the semilinear heat equation with lumped controls, InternationalJournal of Applied Mathematics and ComputerScience 9(4): 751-765.10.1051/cocv:1999104Search in Google Scholar

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.Search in Google Scholar

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 Analysis88: 1763-1783, DOI:10.1080/00036810903401222.10.1080/00036810903401222Search in Google Scholar

Khapalov, A.Y. (2010). Controllability of Partial DifferentialEquations Governed by Multiplicative Controls, Lecture Notes in Mathematics Series, Vol. 1995, Springer-Verlag, Berlin/Heidelberg.Search in Google Scholar

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 DynamicalSystems Series A 33: 1513-1544.10.3934/dcds.2013.33.1513Search in Google Scholar

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).Search in Google Scholar

Koiller, J., Ehlers, F. and Montgomery, R. (1996). Problems and progress in microswimming, Journal of Nonlinear Science6: 507-541.10.1007/BF02434055Search in Google Scholar

Ladyzhenskaya, O.A. (1963). The Mathematical Theory of ViscousIncompressible Flow, Cordon and Breach, New York, NY.Search in Google Scholar

Lighthill, M.J. (1975). Mathematics of Biofluid Dynamics, Society for Industrial and Applied Mathematics, Philadelphia, PA.Search in Google Scholar

Mason, R. and Burdick, J.W. (2000). Experiments in carangiform robotic fish locomotion, Proceedings of theIEEE International Conference on Robotics and Automation,San Francisco, CA, USA, pp. 428-435.Search in Google Scholar

McIsaac, K.A. and Ostrowski, J.P. (2000). Motion planning for dynamic eel-like robots, Proceedings of the IEEE InternationalConference on Robotics and Automation, San Francisco,CA, USA, pp. 1695-1700.Search in Google Scholar

Martinez, S. and Cortes, J. (2001). Geometric control of robotic locomotion systems, Proceedings of the 10th Fall Workshopon Geometry and Physics, Miraflores de la Sierra,Spain, Vol. 4, pp. 183-198.Search in Google Scholar

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 ofthe IEEE International Conference on Robotics and Automation,Seoul, Korea, pp. 427-434.Search in Google Scholar

Peskin, C.S. (1977). Numerical analysis of blood flow in the heart, Journal of Computational Physics 25: 220-252.10.1016/0021-9991(77)90100-0Search in Google Scholar

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.Search in Google Scholar

San Martin, J., Takashi, T. and Tucsnak, M. (2007). A control theoretic approach to the swimming of microscopic organisms, Quarterly Applied Mathematics 65: 405-424.10.1090/S0033-569X-07-01045-9Search in Google Scholar

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 andAnalysis 188: 429-455.10.1007/s00205-007-0092-2Search in Google Scholar

Shapere, A. and Wilczeck, F. (1989). Geometry of self-propulsion at low Reynolds number, Journal of FluidMechanics 198: 557-585.10.1017/S002211208900025XSearch in Google Scholar

Sigalotti, M. and Vivalda, J.-C. (2009). Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM: Control, Optimisationand Calculus of Variations 16: 1053-1076.10.1051/cocv/2009034Search in Google Scholar

Taylor, G.I. (1951). Analysis of the swimming of microscopic organisms, Proceeding of the Royal Society of London A209: 447-461.10.1098/rspa.1951.0218Search in Google Scholar

Taylor, G.I. (1952). Analysis of the swimming of long and narrow animals, Proceedings of the Royal Society of LondonA 214(1117): 158-183.10.1098/rspa.1952.0159Search in Google Scholar

Temam, R. (1984). Navier-Stokes Equations, North-Holland, Amsterdam.Search in Google Scholar

Trintafyllou, M.S., Trintafyllou, G.S. and Yue, D.K.P. (2000). Hydrodynamics of fishlike swimming, Annual Review ofFluid Mechanics 32: 33-53.10.1146/annurev.fluid.32.1.33Search in Google Scholar

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 NationalAcademy Sciences, USA 107: 19832-19837.10.1073/pnas.1011564107Search in Google Scholar

Wu, T.Y. (1971). Hydrodynamics of swimming fish and cetaceans, Advances in Applied Mathematics 11: 1-63.10.1016/S0065-2156(08)70340-5Search in Google Scholar