Skip to main content
SpringerLink
Log in

On the Self-propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Boundary Data

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

Consider a rigid body, \({\mathscr {B}}\), constrained to move by translational motion in an unbounded viscous liquid. The driving mechanism is a given distribution of time-periodic velocity field, \({\varvec{v}}_*\), at the interface body-liquid, of magnitude \(\delta \) (in appropriate function class). The main objective is to find conditions on \({\varvec{v}}_*\) ensuring that \({\mathscr {B}}\) performs a non-zero net motion, namely, \({\mathscr {B}}\) can cover any given distance in a finite time. The approach to the problem depends on whether the averaged value of \({\varvec{v}}_*\) over a period of time is (case (b)) or is not (case (a)) identically zero. In case (a) we solve the problem in a relatively straightforward way, by showing that, for small \(\delta \), it reduces to the study of a suitable and well-investigated time-independent Stokes (linear) problem. In case (b), however, the question is much more complicated, because we show that it cannot be brought to the study of a linear problem. Therefore, in case (b), self-propulsion is a genuinely nonlinear issue that we solve directly on the nonlinear system by a contradiction argument. In this way, we are able to give, also in case (b), sufficient conditions for self-propulsion (for small \(\delta \)). Finally, we demonstrate, by means of counterexamples, that such conditions are, in general, also necessary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. A detailed analysis of homogeneous Sobolev spaces including their main properties can be found in [8, Section II.6].

  2. We use summation convention over repeated indices, unless confusion may arise.

  3. Possibly, by modifying \(\tau \) by adding to it a suitable function of time.

  4. For simplicity, we set \(\varvec{\xi }_{{\varvec{u}}_n}\equiv \varvec{\xi }_n\), \(\varvec{\chi }_{{\varvec{w}}_n}\equiv \varvec{\chi }_n\).

  5. Notice that \({\mathcal V}_\sharp ^{2,\frac{3q}{3-q}}\subset {\mathcal V}_\sharp ^{2,q}\).

References

  1. Čanić, S.: Moving boundary problems. Bull. Am. Math. Soc. (2020). https://doi.org/10.1090/bull/1703

    Article  Google Scholar 

  2. Court, S.: Existence of 3D strong solutions for a system modeling a deformable solid inside a viscous incompressible fluid. J. Dyn. Differ. Equ. 29, 737–782 (2017)

    Article  MathSciNet  Google Scholar 

  3. De Martino, A., Passerini, A.B.: Existence and nonlinear stability of convective solutions for almost compressible fluids in Bénard problem. J. Math. Phys. 60(11), 113101–16 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  4. Galdi, G.P.: On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148, 53–88 (1999)

    Article  MathSciNet  Google Scholar 

  5. Galdi, G.P.: An introduction to the Navier–Stokes initial-boundary value problem. In: Galdi, G.P., Heywood, M.I., Rannacher, R. (eds.) Fundamental Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics, pp. 1–70. Birkhäuser, Basel (2000)

    Google Scholar 

  6. Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. In: Serre, D., Friedlander, S. (eds.) Handbook of Mathematical Fluid Dynamics, vol. I, pp. 653–791. North-Holland, Amsterdam (2002)

    Chapter  Google Scholar 

  7. Galdi, G.P.: A steady-state exterior Navier–Stokes problem that is not well-posed. Proc. Am. Math. Soc. 137, 679–684 (2009)

    Article  MathSciNet  Google Scholar 

  8. Galdi, G.P.: An introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problems. Springer Monographs in Mathematics, Second edn. Springer, New York (2011)

    MATH  Google Scholar 

  9. Galdi, G.P.: On time-periodic flow of a viscous liquid past a moving cylinder. Arch. Ration. Mech. Anal. 210, 451–498 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. Galdi, G.P., Kyed, M.: Time-periodic flow of a viscous liquid past a body. In: Fefferman, C.L., Robinson, J.C., Rodrigo, J.L., Diez, J.L.R. (eds.) Partial Differential Equations in Fluid Mechanics, London Mathematical Society Lecture Note series, vol. 452, pp. 20–49. Cambridge University Press, Cambridge (2018)

    Google Scholar 

  11. Galdi, G.P., Kyed, M.: Time-periodic solutions to the Navier–Stokes equations. In: Giga, Y., Novotný, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 509–578. Springer, Cham (2018)

    Chapter  Google Scholar 

  12. Galdi, G.P., Silvestre, A.L.: Existence of time-periodic solutions to the Navier–Stokes equations around a moving body. Pac. J. Math. 223, 251–267 (2006)

    Article  MathSciNet  Google Scholar 

  13. Galdi, G.P., Silvestre, A.L.: On the motion of a rigid body in a Navier–Stokes liquid under the action of a time-periodic force. Indiana Univ. Math. J. 58, 2805–2842 (2009)

    Article  MathSciNet  Google Scholar 

  14. Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Prentice-Hall Inc, Englewood Cliffs (1965)

    MATH  Google Scholar 

  15. Heywood, J.G.: The Navier–Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)

    Article  MathSciNet  Google Scholar 

  16. Hishida, T., Silvestre, A.L., Takahashi, T.: A boundary control problem for the steady self-propelled motion of a rigid body in a Navier–Stokes fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 1507–1541 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  17. Kozono, H., Sohr, H.: On stationary Navier–Stokes equations in unbounded domains. Ricerche Mat. 42, 69–86 (1993)

    MathSciNet  MATH  Google Scholar 

  18. Mácha, V., Nečasová, Š.: Self-propelled motion in a viscous compressible fluid. Proc. R. Soc. Edinb. Sect. A 146, 415–433 (2016)

    Article  MathSciNet  Google Scholar 

  19. Mácha, V., Nečasová, Š.: Self-propelled motion in a viscous compressible fluid-unbounded domains. Math. Models Methods Appl. Sci. 26, 627–643 (2016)

    Article  MathSciNet  Google Scholar 

  20. Nečasová, Š., Takahashi, T., Tucsnak, M.: Weak solutions for the motion of a self-propelled deformable structure in a viscous incompressible fluid. Acta Appl. Math. 116, 329–352 (2011)

    Article  MathSciNet  Google Scholar 

  21. Raymond, J.-P., Vanninathan, M.: A fluid-structure model coupling the Navier–Stokes equations and the Lamé system. J. Math. Pures Appl. 102, 546–596 (2014)

    Article  MathSciNet  Google Scholar 

  22. San Martín, J., Scheid, J.-F., Takahashi, T., Tucsnak, M.: An initial and boundary value problem modeling of fish-like swimming. Arch. Ration. Mech. Anal. 188, 429–455 (2008)

    Article  MathSciNet  Google Scholar 

  23. Solonnikov, V.A.: Estimates of the solutions of the nonstationary Navier–Stokes system. Boundary value problems of mathematical physics and related questions in the theory of functions. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38, 153–231 (1973)

    MathSciNet  Google Scholar 

  24. Starovoitov, V.N.: Solvability of the problem of the self-propelled motion of several rigid bodies in a viscous incompressible fluid. Comput. Math. Appl. 53, 413–435 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank Mr. Jan A. Wein for sharing several conversations on the topic of self-propulsion.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

Authors
  1. Giovanni P. Galdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni P. Galdi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galdi, G.P. On the Self-propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Boundary Data. J. Math. Fluid Mech. 22, 61 (2020). https://doi.org/10.1007/s00021-020-00537-z

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-020-00537-z

Search

Navigation