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

1. 

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

Received  October 2011 Revised  April 2012 Published  October 2012

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.
Citation: Alexander Khapalov, Giang Trinh. Geometric aspects of transformations of forces acting upon a swimmer in a 3-D incompressible fluid. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

J. M. Ball, J. E. Mardsen and M. Slemrod, Controllable for distributed bilinear systems,, SIAM J. Control. Opt., (1982), 575.   Google Scholar

[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.   Google Scholar

[4]

S. Childress, "Mechanics of Swimming and Flying,", Cambridge University Press, (1981).   Google Scholar

[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.  doi: 10.1137/10080083X.  Google Scholar

[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.   Google Scholar

[7]

L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion,, J. Comp. Physics, 77 (1988), 85.  doi: 10.1016/0021-9991(88)90158-1.  Google Scholar

[8]

L. J. Fauci, Computational modeling of the swimming of biflagellated algal cells,, Contemporary Mathematics, 141 (1993), 91.  doi: 10.1090/conm/141/1212579.  Google Scholar

[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.  doi: 10.1007/s002050050156.  Google Scholar

[10]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in, (2002), 653.   Google Scholar

[11]

J. Gray, Study in animal locomotion IV - the propulsive power of the dolphin,, J. Exp. Biology, 10 (1032), 192.   Google Scholar

[12]

J. Gray and G. J. Hancock, The propulsion of sea-urchin spermatozoa,, J. Exp. Biol., 32 (1955).   Google Scholar

[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.   Google Scholar

[14]

M. E. Gurtin, "Introduction to Continuum Mechanics,'', Academic Press, (1981).   Google Scholar

[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).   Google Scholar

[16]

V. Happel, and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media,", Prentice Hall, (1965).   Google Scholar

[17]

S. Hirose, "Biologically Inspired Robots: Snake-Like Locomotors and Manipulators,", Oxford University Press, (1993).   Google Scholar

[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.   Google Scholar

[19]

A. Y. Khapalov, The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid,, Techn. Rep. 2005-5, (): 2005.   Google Scholar

[20]

A. Y. Khapalov, Local controllability for a "swimming'' model,, SIAM J. Cont. Opt., 46 (2007), 655.  doi: 10.1137/050638424.  Google Scholar

[21]

A. Y. Khapalov, Geometric aspects of controllability for a swimming phenomenon,, Appl. Math. Optim., 57 (2008), 98.  doi: 10.1007/s00245-007-9013-x.  Google Scholar

[22]

A. Y. Khapalov, Micro motions of a 2-D swimming model governed by multiplicative controls,, Nonlinear Analysis: Theory, 71 (2009), 1970.   Google Scholar

[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.  doi: 10.1080/00036810903401222.  Google Scholar

[24]

A. Y. Khapalov, "Controllability of Partial Differential Equations Governed by Multiplicative Controls,'', Lecture Notes in Mathematics Series, (1995).   Google Scholar

[25]

J. Koiller, F. Ehlers and R. Montgomery, Problems and progress in microswimming,, J. Nonlinear Sci., 6 (1996), 507.   Google Scholar

[26]

L. G. Leal, The Slow Motion of Slender Rod-Like Particles in a Second- Order Fluid,, J. Fluid Mech., 69 (1975), 305.   Google Scholar

[27]

M. J. Lighthill, "Mathematics of Biofluid Dynamics,'', Philadelphia, (1975).   Google Scholar

[28]

R. Mason and J. W. Burdick, Experiments in carangiform robotic fish locomotion,, Proc. IEEE Int. Conf. Robotics and Automation, (2000), 428.   Google Scholar

[29]

K. A. McIsaac and J. P. Ostrowski, Motion planning for dynamic eel-like robots,, Proc. IEEE Int. Conf. Robotics and Automation, (2000), 1695.   Google Scholar

[30]

S. Martinez and J. Cort'es, Geometric control of robotic locomotion systems,, Proc. X Fall Workshop on Geometry and Physics, 4 (2001), 183.   Google Scholar

[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.   Google Scholar

[32]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comp. Physics, 25 (1977), 220.  doi: 10.1016/0021-9991(77)90100-0.  Google Scholar

[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, (1994), 5.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00205-007-0092-2.  Google Scholar

[36]

A. Shapere and F. Wilczeck, Geometry of self-propulsion at low Reynolds number,, J. Fluid Mech., 198 (1989), 557.   Google Scholar

[37]

M. Sigalotti and J.-C. Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms,, ESAIM: COCV, (2009).   Google Scholar

[38]

G. I. Taylor, Analysis of the swimming of microscopic organisms,, Proc. R. Soc. Lond. A, 209 (1951), 447.   Google Scholar

[39]

G. I. Taylor, Analysis of the swimming of long and narrow animals,, Proc. R. Soc. Lond. A, 214 (1952).   Google Scholar

[40]

R. Temam, "Navier-Stokes Equations,", North-Holland, (1984).   Google Scholar

[41]

M. S. Trintafyllou, G. S. Trintafyllou and D. K. P. Yue, Hydrodynamics of fishlike swimming,, Ann. Rev. Fluid Mech., 32 (2000), 33.  doi: 10.1146/annurev.fluid.32.1.33.  Google Scholar

[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.  doi: 10.1073/pnas.1011564107.  Google Scholar

[43]

T. Y. Wu, Hydrodynamics of swimming fish and cetaceans,, Adv. Appl. Math., 11 (1971), 1.   Google Scholar

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.   Google Scholar

[2]

J. M. Ball, J. E. Mardsen and M. Slemrod, Controllable for distributed bilinear systems,, SIAM J. Control. Opt., (1982), 575.   Google Scholar

[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.   Google Scholar

[4]

S. Childress, "Mechanics of Swimming and Flying,", Cambridge University Press, (1981).   Google Scholar

[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.  doi: 10.1137/10080083X.  Google Scholar

[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.   Google Scholar

[7]

L. J. Fauci and C. S. Peskin, A computational model of aquatic animal locomotion,, J. Comp. Physics, 77 (1988), 85.  doi: 10.1016/0021-9991(88)90158-1.  Google Scholar

[8]

L. J. Fauci, Computational modeling of the swimming of biflagellated algal cells,, Contemporary Mathematics, 141 (1993), 91.  doi: 10.1090/conm/141/1212579.  Google Scholar

[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.  doi: 10.1007/s002050050156.  Google Scholar

[10]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in, (2002), 653.   Google Scholar

[11]

J. Gray, Study in animal locomotion IV - the propulsive power of the dolphin,, J. Exp. Biology, 10 (1032), 192.   Google Scholar

[12]

J. Gray and G. J. Hancock, The propulsion of sea-urchin spermatozoa,, J. Exp. Biol., 32 (1955).   Google Scholar

[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.   Google Scholar

[14]

M. E. Gurtin, "Introduction to Continuum Mechanics,'', Academic Press, (1981).   Google Scholar

[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).   Google Scholar

[16]

V. Happel, and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media,", Prentice Hall, (1965).   Google Scholar

[17]

S. Hirose, "Biologically Inspired Robots: Snake-Like Locomotors and Manipulators,", Oxford University Press, (1993).   Google Scholar

[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.   Google Scholar

[19]

A. Y. Khapalov, The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid,, Techn. Rep. 2005-5, (): 2005.   Google Scholar

[20]

A. Y. Khapalov, Local controllability for a "swimming'' model,, SIAM J. Cont. Opt., 46 (2007), 655.  doi: 10.1137/050638424.  Google Scholar

[21]

A. Y. Khapalov, Geometric aspects of controllability for a swimming phenomenon,, Appl. Math. Optim., 57 (2008), 98.  doi: 10.1007/s00245-007-9013-x.  Google Scholar

[22]

A. Y. Khapalov, Micro motions of a 2-D swimming model governed by multiplicative controls,, Nonlinear Analysis: Theory, 71 (2009), 1970.   Google Scholar

[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.  doi: 10.1080/00036810903401222.  Google Scholar

[24]

A. Y. Khapalov, "Controllability of Partial Differential Equations Governed by Multiplicative Controls,'', Lecture Notes in Mathematics Series, (1995).   Google Scholar

[25]

J. Koiller, F. Ehlers and R. Montgomery, Problems and progress in microswimming,, J. Nonlinear Sci., 6 (1996), 507.   Google Scholar

[26]

L. G. Leal, The Slow Motion of Slender Rod-Like Particles in a Second- Order Fluid,, J. Fluid Mech., 69 (1975), 305.   Google Scholar

[27]

M. J. Lighthill, "Mathematics of Biofluid Dynamics,'', Philadelphia, (1975).   Google Scholar

[28]

R. Mason and J. W. Burdick, Experiments in carangiform robotic fish locomotion,, Proc. IEEE Int. Conf. Robotics and Automation, (2000), 428.   Google Scholar

[29]

K. A. McIsaac and J. P. Ostrowski, Motion planning for dynamic eel-like robots,, Proc. IEEE Int. Conf. Robotics and Automation, (2000), 1695.   Google Scholar

[30]

S. Martinez and J. Cort'es, Geometric control of robotic locomotion systems,, Proc. X Fall Workshop on Geometry and Physics, 4 (2001), 183.   Google Scholar

[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.   Google Scholar

[32]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comp. Physics, 25 (1977), 220.  doi: 10.1016/0021-9991(77)90100-0.  Google Scholar

[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, (1994), 5.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00205-007-0092-2.  Google Scholar

[36]

A. Shapere and F. Wilczeck, Geometry of self-propulsion at low Reynolds number,, J. Fluid Mech., 198 (1989), 557.   Google Scholar

[37]

M. Sigalotti and J.-C. Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms,, ESAIM: COCV, (2009).   Google Scholar

[38]

G. I. Taylor, Analysis of the swimming of microscopic organisms,, Proc. R. Soc. Lond. A, 209 (1951), 447.   Google Scholar

[39]

G. I. Taylor, Analysis of the swimming of long and narrow animals,, Proc. R. Soc. Lond. A, 214 (1952).   Google Scholar

[40]

R. Temam, "Navier-Stokes Equations,", North-Holland, (1984).   Google Scholar

[41]

M. S. Trintafyllou, G. S. Trintafyllou and D. K. P. Yue, Hydrodynamics of fishlike swimming,, Ann. Rev. Fluid Mech., 32 (2000), 33.  doi: 10.1146/annurev.fluid.32.1.33.  Google Scholar

[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.  doi: 10.1073/pnas.1011564107.  Google Scholar

[43]

T. Y. Wu, Hydrodynamics of swimming fish and cetaceans,, Adv. Appl. Math., 11 (1971), 1.   Google Scholar

[1]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[2]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[3]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[4]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[5]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[6]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[8]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[9]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[10]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[11]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[12]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[13]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[14]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[15]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[16]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[17]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[18]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[19]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[20]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]