Skip to main content
SpringerLink
Log in

Steady solutions to the problem of a ball dynamics in a Stokes–Poiseuille flow

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

The existence is proved of steady solutions to the problem of motion of a rigid ball in a cylindrical pipe filled with a viscous incompressible fluid. The cross section of the pipe has an arbitrary form and the fluid flow is governed by the Stokes equations. At infinity, the velocity profile tends to that of the Poiseuille flow. It is established that a steady solution exists for an arbitrary position of the ball in the pipe. The ball performs a rectilinear motion along the generators of the pipe, and the linear and angular velocities depend on the position of the ball center in the cross section of the cylinder.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Segréand A. Silberberg, “Behavior of Macroscopic Rigid Spheres in Poiseuille Flow. P. 2: Experimental Results and Interpretation,” J. FluidMech. 14 (1), 136–157 (1962).

    Article  Google Scholar 

  2. P. G. Saffman, “The Lift on a Small Sphere in a Slow Shear Flow,” J. Fluid Mech. 22 (2), 385–400 (1965).

    Article  MATH  Google Scholar 

  3. D. Serre, “Chute libre d’un solide dans un fluide visqueux incompressible. Existence,” Japan J. Appl. Math. 4 (1), 99–110 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  4. G. P. Galdi and A. Vaidya, “Translational Steady Fall of Symmetric Bodies in a Navier–Stokes Liquid with Application to Particle Sedimentation,” J. Math. Fluid Mech. 3 (1), 183–211 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  5. G. P. Galdi, “On theMotion of a Rigid Body in a Viscous Fluid: a Mathematical Analysis with Applications,” in Handbook of Mathematical Fluid Dynamics, Vol. 1 (Elsevier, Chicago, Lyon, 2002), pp. 653–791.

    Chapter  Google Scholar 

  6. M. Hillairet and D. Serre, “Free Steady Fall of a Solid in a Fluid along a Ramp,” Ann. Institut H. Poincaré. Anal. Non Linear. 20 (5), 779–803 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. L. Silvestre, “On the Existence of Steady Flows of a Navier–Stokes Liquid around aMoving Rigid Body,” Math. Methods Appl. Sci. 27 (12), 1399–1409 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Necasova, “Asymptotic Properties of the Steady Fall of a Body in Viscous Fluids,” Math. Methods Appl. Sci. 27 (17), 1969–1995 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Necasova, “Steady Fall of a Rigid Body in Viscous Fluid,” Nonlinear Analysis: Theory, Methods and Applications 63 (5–7), e2113–e2119 (2005).

    Article  MATH  Google Scholar 

  10. M. Hillairet and P. Wittwer, “Existence of Stationary Solutions of the Navier–Stokes Equations in Two Dimensions in the Presence of a Wall,” J. Evolution Equations 9, 675–706 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Takahashi and M. Tucsnak, “Global Strong Solutions for the Two-Dimensional Motion of an Infinite Cylinder in a Viscous Fluid,” J. Math. Fluid Mech. 6 (1), 53–77 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  12. K.-H. Hoffmann and V. N. Starovoitov, “On aMotion of a Solid Body in a Viscous Fluid. Two-Dimensional Case,” Adv. in Math. Sci. Appl. 9 (2), 633–648 (1999).

    MATH  MathSciNet  Google Scholar 

  13. K.-H. Hoffmann and V. N. Starovoitov, “Zur Bewegung einer Kugel in einer zähen Flüssigkeit,” Documenta Mathematica 5, 15–21 (2000).

    MATH  MathSciNet  Google Scholar 

  14. V. N. Starovoitov, Non-Regular Problems in Hydrodynamics, Doctorate Dissertation in Physics and Mathematics (Novosibirsk, 2000).

    Google Scholar 

  15. V. N. Starovoitov, “Behavior of a Rigid Body in an Incompressible Viscous Fluid near a Boundary,” in Free Boundary Problems (Birkhäuser, Basel, 2003), pp. 313–327.

    Chapter  Google Scholar 

  16. G. V. Alekseev, “Optimization in Problems of Material–Body Cloaking Using the Wave–Flow Method,” Dokl. Akad. Nauk 449 (6), 652–656 (2013) [Dokl. Phys. 58 (4), 147–151 (2013)].

    MathSciNet  Google Scholar 

  17. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed. (Springer, New York, 2013)

    Book  MATH  Google Scholar 

  18. P. I. Plotnikov, V. N. Starovoitov, and B. N. Starovoitova, “Properties of the Solution of a Variational Problem of the Motion of a Rigid Body in a Stokes Fluid,” Dokl. Akad. Nauk 446 (2), 131–137 (2012) [Dokl. Math. 86 (2), 615–621 (2012)].

    MathSciNet  Google Scholar 

  19. V. N. Starovoitov, “Optimal Control of a Cylinder Rotating in a Viscous Fluid,” Sibirsk. Zh. Industr. Mat. 16 (1), 95–105 (2013) [J. Appl. Indust.Math. 7 (2), 259–268 (2013)].

    MathSciNet  Google Scholar 

  20. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady- State Problems (Springer, New York, 2011).

    Book  Google Scholar 

  21. G. V. Alekseev and D. A. Tereshko, Analysis and Optimization in Hydrodynamics of a Viscous Fluid (Dal’nauka, Vladivostok, 2008) [in Russian].

    Google Scholar 

  22. O. A. Ladyzhenskaya and V. A. Solonnnikov, “Determination of the Solutions of Boundary Value Problems for Stationary Stokes and Navier-Stokes EquationsHaving anUnboundedDirichlet Integral,” Zap.Nauchn. Sem. Leningrad. Otdel.Mat. Inst. Steklov. (LOMI) 96, 117–160 (1980).

    MATH  MathSciNet  Google Scholar 

  23. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (SIAM, 1976; Mir, Moscow, 1979).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lavrent’ev Institute of Hydrodynamics, pr. Akad. Lavrent’eva 15, Novosibirsk, 630090, Russia

    V. N. Starovoitov

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    V. N. Starovoitov

Authors
  1. V. N. Starovoitov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. N. Starovoitov.

Additional information

Original Russian Text © V.N. Starovoitov, 2015, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2015, Vol. XVIII, No. 3, pp. 76–85.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Starovoitov, V.N. Steady solutions to the problem of a ball dynamics in a Stokes–Poiseuille flow. J. Appl. Ind. Math. 9, 588–597 (2015). https://doi.org/10.1134/S1990478915040158

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478915040158

Keywords

Search

Navigation