Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid

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April 2013, 33(4): 1513-1544. doi: 10.3934/dcds.2013.33.1513

Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid

Alexander Khapalov ^1, and Giang Trinh ^1,

1.	Department of Mathematics, Washington State University, Pullman, WA 99164-3113, United States

Received October 2011 Revised April 2012 Published October 2012

Abstract
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Our goal in this paper is to investigate how the geometric shape of a swimmer affects the forces acting upon it in a 3-$D$ incompressible fluid, such as governed by the non-stationary Stokes or Navier-Stokes equations. Namely, we are interested in the following question: How will the swimmer's internal forces (i.e., not moving the center of swimmer's mass when it is not inside a fluid) ``transform'' their actions when the swimmer is placed into a fluid (thus, possibly, creating its self-propelling motion)?We focus on the case when the swimmer's body consists of either small parallelepipeds or balls. Such problems are of interest in biology and engineering application dealing with propulsion systems in fluids.

Keywords: Swimming models, Navier-Stokes equations, fluid mechanics, stokes equations, hybrid pde/integro-differential ode systems..

Mathematics Subject Classification: Primary: 35Q30, 76D99; Secondary: 70Q0.

Citation: Alexander Khapalov, Giang Trinh. Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1513-1544. doi: 10.3934/dcds.2013.33.1513

References:

[1]	F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low reynolds number Sswimmers: An example, J. Nonlinear Sci., 18 (2008), 27-302.
[2]	J. M. Ball, J. E. Mardsen and M. Slemrod, Controllable for distributed bilinear systems, SIAM J. Control. Opt., (1982), 575-597.
[3]	L. E. Becker, S. A. Koehler and H. A. Stone, On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer, J. Fluid Mech., 490 (2003), 15-35.
[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 self-propelled micro-swimmers, SIAM J. Math. Anal., 43 (2011), 1345-1368. 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), 300-305.
[7]	L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion, J. Comp. Physics, 77 (1988), 85-108. doi: 10.1016/0021-9991(88)90158-1.
[8]	L. J. Fauci, Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics, 141 (1993), 91-102. doi: 10.1090/conm/141/1212579.
[9]	G. P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 53-88. 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), 653-791.
[11]	J. Gray, Study in animal locomotion IV - the propulsive power of the dolphin, J. Exp. Biology, 10 (1032), 192-199.
[12]	J. Gray and G. J. Hancock, The propulsion of sea-urchin spermatozoa, J. Exp. Biol., 32 802, 1955.
[13]	S. Guo, et al., Afin type of micro-robot in pipe, Proc. of the 2002 Int. Symp. on micromechatronics and human science (MHS 2002), (2002), 93-98.
[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: Snake-Like Locomotors and Manipulators," Oxford University Press, Oxford, 1993.
[18]	E. Kanso, J. E. Marsden, C. W. Rowley and J. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Science, 15 (2005), 255-289.
[19]	A. Y. Khapalov, The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid, Techn. Rep. 2005-5, Washington State University, Department of Mathematics, http://www.math.wsu.edu/TRS/2005-5.pdf).
[20]	A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Cont. Opt., 46 (2007), 655-682. doi: 10.1137/050638424.
[21]	A. Y. Khapalov, Geometric aspects of controllability for a swimming phenomenon, Appl. Math. Optim., 57 (2008), 98-124. doi: 10.1007/s00245-007-9013-x.
[22]	A. Y. Khapalov, Micro motions of a 2-D swimming model governed by multiplicative controls, Nonlinear Analysis: Theory, Methods and Appl.: Special Issue: WCNA 2008, 71 (2009), 1970-1979.
[23]	A. Y. Khapalov and S. Eubanks, The wellposedness of a 2-D swimming model governed in the nonstationary Stokes fluid by multiplicative controls, Applicable Analysis, 88 (2009), 1763-1783. doi: 10.1080/00036810903401222.
[24]	A. Y. Khapalov, "Controllability of Partial Differential Equations Governed by Multiplicative Controls,'' Lecture Notes in Mathematics Series, Vol. 1995, Springer-Verlag Berlin Heidelberg, 284p., 2010.
[25]	J. Koiller, F. Ehlers and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci., 6 (1996), 507-541.
[26]	L. G. Leal, The Slow Motion of Slender Rod-Like Particles in a Second- Order Fluid, J. Fluid Mech., 69 (1975), 305-337.
[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), 428-435.
[29]	K. A. McIsaac and J. P. Ostrowski, Motion planning for dynamic eel-like robots, Proc. IEEE Int. Conf. Robotics and Automation, San Francisco, (2000), 1695-1700.
[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), 183-198.
[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), 427-434.
[32]	C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Physics, 25 (1977), 220-252. doi: 10.1016/0021-9991(77)90100-0.
[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 5-8, 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), 405-424.
[35]	J. San Martin, J.-F. Scheid, T. Takashi and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming, Arch. Ration. Mech. Anal., 188 (2008), 429-455. doi: 10.1007/s00205-007-0092-2.
[36]	A. Shapere and F. Wilczeck, Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557-585.
[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), 447-461.
[39]	G. I. Taylor, Analysis of the swimming of long and narrow animals, Proc. R. Soc. Lond. A, 214 (1952).
[40]	R. Temam, "Navier-Stokes Equations," North-Holland, 1984.
[41]	M. S. Trintafyllou, G. S. Trintafyllou and D. K. P. Yue, Hydrodynamics of fishlike swimming, Ann. Rev. Fluid Mech., 32 (2000), 33-53. 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), 19832-19837. doi: 10.1073/pnas.1011564107.
[43]	T. Y. Wu, Hydrodynamics of swimming fish and cetaceans, Adv. Appl. Math., 11 (1971), 1-63.

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), 27-302.
[2]	J. M. Ball, J. E. Mardsen and M. Slemrod, Controllable for distributed bilinear systems, SIAM J. Control. Opt., (1982), 575-597.
[3]	L. E. Becker, S. A. Koehler and H. A. Stone, On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer, J. Fluid Mech., 490 (2003), 15-35.
[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 self-propelled micro-swimmers, SIAM J. Math. Anal., 43 (2011), 1345-1368. 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), 300-305.
[7]	L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion, J. Comp. Physics, 77 (1988), 85-108. doi: 10.1016/0021-9991(88)90158-1.
[8]	L. J. Fauci, Computational modeling of the swimming of biflagellated algal cells, Contemporary Mathematics, 141 (1993), 91-102. doi: 10.1090/conm/141/1212579.
[9]	G. P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 53-88. 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), 653-791.
[11]	J. Gray, Study in animal locomotion IV - the propulsive power of the dolphin, J. Exp. Biology, 10 (1032), 192-199.
[12]	J. Gray and G. J. Hancock, The propulsion of sea-urchin spermatozoa, J. Exp. Biol., 32 802, 1955.
[13]	S. Guo, et al., Afin type of micro-robot in pipe, Proc. of the 2002 Int. Symp. on micromechatronics and human science (MHS 2002), (2002), 93-98.
[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: Snake-Like Locomotors and Manipulators," Oxford University Press, Oxford, 1993.
[18]	E. Kanso, J. E. Marsden, C. W. Rowley and J. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Science, 15 (2005), 255-289.
[19]	A. Y. Khapalov, The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid, Techn. Rep. 2005-5, Washington State University, Department of Mathematics, http://www.math.wsu.edu/TRS/2005-5.pdf).
[20]	A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Cont. Opt., 46 (2007), 655-682. doi: 10.1137/050638424.
[21]	A. Y. Khapalov, Geometric aspects of controllability for a swimming phenomenon, Appl. Math. Optim., 57 (2008), 98-124. doi: 10.1007/s00245-007-9013-x.
[22]	A. Y. Khapalov, Micro motions of a 2-D swimming model governed by multiplicative controls, Nonlinear Analysis: Theory, Methods and Appl.: Special Issue: WCNA 2008, 71 (2009), 1970-1979.
[23]	A. Y. Khapalov and S. Eubanks, The wellposedness of a 2-D swimming model governed in the nonstationary Stokes fluid by multiplicative controls, Applicable Analysis, 88 (2009), 1763-1783. doi: 10.1080/00036810903401222.
[24]	A. Y. Khapalov, "Controllability of Partial Differential Equations Governed by Multiplicative Controls,'' Lecture Notes in Mathematics Series, Vol. 1995, Springer-Verlag Berlin Heidelberg, 284p., 2010.
[25]	J. Koiller, F. Ehlers and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci., 6 (1996), 507-541.
[26]	L. G. Leal, The Slow Motion of Slender Rod-Like Particles in a Second- Order Fluid, J. Fluid Mech., 69 (1975), 305-337.
[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), 428-435.
[29]	K. A. McIsaac and J. P. Ostrowski, Motion planning for dynamic eel-like robots, Proc. IEEE Int. Conf. Robotics and Automation, San Francisco, (2000), 1695-1700.
[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), 183-198.
[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), 427-434.
[32]	C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Physics, 25 (1977), 220-252. doi: 10.1016/0021-9991(77)90100-0.
[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 5-8, 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), 405-424.
[35]	J. San Martin, J.-F. Scheid, T. Takashi and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming, Arch. Ration. Mech. Anal., 188 (2008), 429-455. doi: 10.1007/s00205-007-0092-2.
[36]	A. Shapere and F. Wilczeck, Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557-585.
[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), 447-461.
[39]	G. I. Taylor, Analysis of the swimming of long and narrow animals, Proc. R. Soc. Lond. A, 214 (1952).
[40]	R. Temam, "Navier-Stokes Equations," North-Holland, 1984.
[41]	M. S. Trintafyllou, G. S. Trintafyllou and D. K. P. Yue, Hydrodynamics of fishlike swimming, Ann. Rev. Fluid Mech., 32 (2000), 33-53. 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), 19832-19837. doi: 10.1073/pnas.1011564107.
[43]	T. Y. Wu, Hydrodynamics of swimming fish and cetaceans, Adv. Appl. Math., 11 (1971), 1-63.

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