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Geometric aspects of transformations of forces acting upon a swimmer in a 3D incompressible fluid
1.  Department of Mathematics, Washington State University, Pullman, WA 991643113, United States 
References:
[1] 
F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low reynolds number Sswimmers: An example, J. Nonlinear Sci., 18 (2008), 27302. 
[2] 
J. M. Ball, J. E. Mardsen and M. Slemrod, Controllable for distributed bilinear systems, SIAM J. Control. Opt., (1982), 575597. 
[3] 
L. E. Becker, S. A. Koehler and H. A. Stone, On selfpropulsion of micromachines at low Reynolds number: Purcell's threelink swimmer, J. Fluid Mech., 490 (2003), 1535. 
[4] 
S. Childress, "Mechanics of Swimming and Flying," Cambridge University Press, 1981. 
[5] 
Gi. Dal Maso, A. DeSimone and M. Morandotti, An existence and uniqueness result for the motion of selfpropelled microswimmers, SIAM J. Math. Anal., 43 (2011), 13451368. doi: 10.1137/10080083X. 
[6] 
T. Fakuda, et al, Steering mechanism and swimming experiment of micro mobile robot in water, Proc. Micro Electro Mechanical Systems (MEMS'95), (1995), 300305. 
[7] 
L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion, J. Comp. Physics, 77 (1988), 85108. doi: 10.1016/00219991(88)901581. 
[8] 
L. J. Fauci, Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics, 141 (1993), 91102. doi: 10.1090/conm/141/1212579. 
[9] 
G. P. Galdi, On the steady selfpropelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 5388. doi: 10.1007/s002050050156. 
[10] 
G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in "Handbook of Mathematical Fluid Mechanics" (eds. S. Friedlander and D. Serre), Elsevier Science, (2002), 653791. 
[11] 
J. Gray, Study in animal locomotion IV  the propulsive power of the dolphin, J. Exp. Biology, 10 (1032), 192199. 
[12] 
J. Gray and G. J. Hancock, The propulsion of seaurchin spermatozoa, J. Exp. Biol., 32 802, 1955. 
[13] 
S. Guo, et al., Afin type of microrobot in pipe, Proc. of the 2002 Int. Symp. on micromechatronics and human science (MHS 2002), (2002), 9398. 
[14] 
M. E. Gurtin, "Introduction to Continuum Mechanics,'' Academic Press, 1981. 
[15] 
M. F. Hawthorne, J. I. Zink, J. M. Skelton, M. J. Bayer, Ch. Liu, E. Livshits, R. Baer and D. Neuhauser, Electrical or photocontrol of rotary motion of a metallacarborane, Science, 303, 1849, 2004. 
[16] 
V. Happel, and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media," Prentice Hall, 1965. 
[17] 
S. Hirose, "Biologically Inspired Robots: SnakeLike Locomotors and Manipulators," Oxford University Press, Oxford, 1993. 
[18] 
E. Kanso, J. E. Marsden, C. W. Rowley and J. MelliHuber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Science, 15 (2005), 255289. 
[19] 
A. Y. Khapalov, The wellposedness of a model of an apparatus swimming in the 2D Stokes fluid, Techn. Rep. 20055, Washington State University, Department of Mathematics, http://www.math.wsu.edu/TRS/20055.pdf). 
[20] 
A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Cont. Opt., 46 (2007), 655682. doi: 10.1137/050638424. 
[21] 
A. Y. Khapalov, Geometric aspects of controllability for a swimming phenomenon, Appl. Math. Optim., 57 (2008), 98124. doi: 10.1007/s002450079013x. 
[22] 
A. Y. Khapalov, Micro motions of a 2D swimming model governed by multiplicative controls, Nonlinear Analysis: Theory, Methods and Appl.: Special Issue: WCNA 2008, 71 (2009), 19701979. 
[23] 
A. Y. Khapalov and S. Eubanks, The wellposedness of a 2D swimming model governed in the nonstationary Stokes fluid by multiplicative controls, Applicable Analysis, 88 (2009), 17631783. doi: 10.1080/00036810903401222. 
[24] 
A. Y. Khapalov, "Controllability of Partial Differential Equations Governed by Multiplicative Controls,'' Lecture Notes in Mathematics Series, Vol. 1995, SpringerVerlag Berlin Heidelberg, 284p., 2010. 
[25] 
J. Koiller, F. Ehlers and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci., 6 (1996), 507541. 
[26] 
L. G. Leal, The Slow Motion of Slender RodLike Particles in a Second Order Fluid, J. Fluid Mech., 69 (1975), 305337. 
[27] 
M. J. Lighthill, "Mathematics of Biofluid Dynamics,'' Philadelphia, Society for Industrial and Applied Mathematics, 1975. 
[28] 
R. Mason and J. W. Burdick, Experiments in carangiform robotic fish locomotion, Proc. IEEE Int. Conf. Robotics and Automation, (2000), 428435. 
[29] 
K. A. McIsaac and J. P. Ostrowski, Motion planning for dynamic eellike robots, Proc. IEEE Int. Conf. Robotics and Automation, San Francisco, (2000), 16951700. 
[30] 
S. Martinez and J. Cort'es, Geometric control of robotic locomotion systems, Proc. X Fall Workshop on Geometry and Physics, Madrid, 2001, Publ. de la RSME, 4 (2001), 183198. 
[31] 
K. A. Morgansen, V. Duindam, R. J. Mason, J. W. Burdick and R. M. Murray, Nonlinear control methods for planar carangiform robot fish locomotion, Proc. IEEE Int. Conf. Robotics and Automation, (2001), 427434. 
[32] 
C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Physics, 25 (1977), 220252. doi: 10.1016/00219991(77)901000. 
[33] 
C. S. Peskin and D. M. McQueen, A general method for the computer simulation of biological systems interacting with fluids, SEB Symposium on biological fluid dynamics, Leeds, England, July 58, 1994. 
[34] 
J. San Martin, T. Takashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math., 65 (2007), 405424. 
[35] 
J. San Martin, J.F. Scheid, T. Takashi and M. Tucsnak, An initial and boundary value problem modeling of fishlike swimming, Arch. Ration. Mech. Anal., 188 (2008), 429455. doi: 10.1007/s0020500700922. 
[36] 
A. Shapere and F. Wilczeck, Geometry of selfpropulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557585. 
[37] 
M. Sigalotti and J.C. Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM: COCV, Published online August 11, 2009. 
[38] 
G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. R. Soc. Lond. A, 209 (1951), 447461. 
[39] 
G. I. Taylor, Analysis of the swimming of long and narrow animals, Proc. R. Soc. Lond. A, 214 (1952). 
[40]  
[41] 
M. S. Trintafyllou, G. S. Trintafyllou and D. K. P. Yue, Hydrodynamics of fishlike swimming, Ann. Rev. Fluid Mech., 32 (2000), 3353. doi: 10.1146/annurev.fluid.32.1.33. 
[42] 
E. D. Tytell, C.Y. Hsu, T. L. Williams, A. H. Cohen and L. J. Fauci, Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming, Proc. Natl Acad Sci USA, 107 (2010), 1983219837. doi: 10.1073/pnas.1011564107. 
[43] 
T. Y. Wu, Hydrodynamics of swimming fish and cetaceans, Adv. Appl. Math., 11 (1971), 163. 
show all references
References:
[1] 
F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low reynolds number Sswimmers: An example, J. Nonlinear Sci., 18 (2008), 27302. 
[2] 
J. M. Ball, J. E. Mardsen and M. Slemrod, Controllable for distributed bilinear systems, SIAM J. Control. Opt., (1982), 575597. 
[3] 
L. E. Becker, S. A. Koehler and H. A. Stone, On selfpropulsion of micromachines at low Reynolds number: Purcell's threelink swimmer, J. Fluid Mech., 490 (2003), 1535. 
[4] 
S. Childress, "Mechanics of Swimming and Flying," Cambridge University Press, 1981. 
[5] 
Gi. Dal Maso, A. DeSimone and M. Morandotti, An existence and uniqueness result for the motion of selfpropelled microswimmers, SIAM J. Math. Anal., 43 (2011), 13451368. doi: 10.1137/10080083X. 
[6] 
T. Fakuda, et al, Steering mechanism and swimming experiment of micro mobile robot in water, Proc. Micro Electro Mechanical Systems (MEMS'95), (1995), 300305. 
[7] 
L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion, J. Comp. Physics, 77 (1988), 85108. doi: 10.1016/00219991(88)901581. 
[8] 
L. J. Fauci, Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics, 141 (1993), 91102. doi: 10.1090/conm/141/1212579. 
[9] 
G. P. Galdi, On the steady selfpropelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 5388. doi: 10.1007/s002050050156. 
[10] 
G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in "Handbook of Mathematical Fluid Mechanics" (eds. S. Friedlander and D. Serre), Elsevier Science, (2002), 653791. 
[11] 
J. Gray, Study in animal locomotion IV  the propulsive power of the dolphin, J. Exp. Biology, 10 (1032), 192199. 
[12] 
J. Gray and G. J. Hancock, The propulsion of seaurchin spermatozoa, J. Exp. Biol., 32 802, 1955. 
[13] 
S. Guo, et al., Afin type of microrobot in pipe, Proc. of the 2002 Int. Symp. on micromechatronics and human science (MHS 2002), (2002), 9398. 
[14] 
M. E. Gurtin, "Introduction to Continuum Mechanics,'' Academic Press, 1981. 
[15] 
M. F. Hawthorne, J. I. Zink, J. M. Skelton, M. J. Bayer, Ch. Liu, E. Livshits, R. Baer and D. Neuhauser, Electrical or photocontrol of rotary motion of a metallacarborane, Science, 303, 1849, 2004. 
[16] 
V. Happel, and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media," Prentice Hall, 1965. 
[17] 
S. Hirose, "Biologically Inspired Robots: SnakeLike Locomotors and Manipulators," Oxford University Press, Oxford, 1993. 
[18] 
E. Kanso, J. E. Marsden, C. W. Rowley and J. MelliHuber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Science, 15 (2005), 255289. 
[19] 
A. Y. Khapalov, The wellposedness of a model of an apparatus swimming in the 2D Stokes fluid, Techn. Rep. 20055, Washington State University, Department of Mathematics, http://www.math.wsu.edu/TRS/20055.pdf). 
[20] 
A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Cont. Opt., 46 (2007), 655682. doi: 10.1137/050638424. 
[21] 
A. Y. Khapalov, Geometric aspects of controllability for a swimming phenomenon, Appl. Math. Optim., 57 (2008), 98124. doi: 10.1007/s002450079013x. 
[22] 
A. Y. Khapalov, Micro motions of a 2D swimming model governed by multiplicative controls, Nonlinear Analysis: Theory, Methods and Appl.: Special Issue: WCNA 2008, 71 (2009), 19701979. 
[23] 
A. Y. Khapalov and S. Eubanks, The wellposedness of a 2D swimming model governed in the nonstationary Stokes fluid by multiplicative controls, Applicable Analysis, 88 (2009), 17631783. doi: 10.1080/00036810903401222. 
[24] 
A. Y. Khapalov, "Controllability of Partial Differential Equations Governed by Multiplicative Controls,'' Lecture Notes in Mathematics Series, Vol. 1995, SpringerVerlag Berlin Heidelberg, 284p., 2010. 
[25] 
J. Koiller, F. Ehlers and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci., 6 (1996), 507541. 
[26] 
L. G. Leal, The Slow Motion of Slender RodLike Particles in a Second Order Fluid, J. Fluid Mech., 69 (1975), 305337. 
[27] 
M. J. Lighthill, "Mathematics of Biofluid Dynamics,'' Philadelphia, Society for Industrial and Applied Mathematics, 1975. 
[28] 
R. Mason and J. W. Burdick, Experiments in carangiform robotic fish locomotion, Proc. IEEE Int. Conf. Robotics and Automation, (2000), 428435. 
[29] 
K. A. McIsaac and J. P. Ostrowski, Motion planning for dynamic eellike robots, Proc. IEEE Int. Conf. Robotics and Automation, San Francisco, (2000), 16951700. 
[30] 
S. Martinez and J. Cort'es, Geometric control of robotic locomotion systems, Proc. X Fall Workshop on Geometry and Physics, Madrid, 2001, Publ. de la RSME, 4 (2001), 183198. 
[31] 
K. A. Morgansen, V. Duindam, R. J. Mason, J. W. Burdick and R. M. Murray, Nonlinear control methods for planar carangiform robot fish locomotion, Proc. IEEE Int. Conf. Robotics and Automation, (2001), 427434. 
[32] 
C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Physics, 25 (1977), 220252. doi: 10.1016/00219991(77)901000. 
[33] 
C. S. Peskin and D. M. McQueen, A general method for the computer simulation of biological systems interacting with fluids, SEB Symposium on biological fluid dynamics, Leeds, England, July 58, 1994. 
[34] 
J. San Martin, T. Takashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math., 65 (2007), 405424. 
[35] 
J. San Martin, J.F. Scheid, T. Takashi and M. Tucsnak, An initial and boundary value problem modeling of fishlike swimming, Arch. Ration. Mech. Anal., 188 (2008), 429455. doi: 10.1007/s0020500700922. 
[36] 
A. Shapere and F. Wilczeck, Geometry of selfpropulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557585. 
[37] 
M. Sigalotti and J.C. Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM: COCV, Published online August 11, 2009. 
[38] 
G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. R. Soc. Lond. A, 209 (1951), 447461. 
[39] 
G. I. Taylor, Analysis of the swimming of long and narrow animals, Proc. R. Soc. Lond. A, 214 (1952). 
[40]  
[41] 
M. S. Trintafyllou, G. S. Trintafyllou and D. K. P. Yue, Hydrodynamics of fishlike swimming, Ann. Rev. Fluid Mech., 32 (2000), 3353. doi: 10.1146/annurev.fluid.32.1.33. 
[42] 
E. D. Tytell, C.Y. Hsu, T. L. Williams, A. H. Cohen and L. J. Fauci, Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming, Proc. Natl Acad Sci USA, 107 (2010), 1983219837. doi: 10.1073/pnas.1011564107. 
[43] 
T. Y. Wu, Hydrodynamics of swimming fish and cetaceans, Adv. Appl. Math., 11 (1971), 163. 
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