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Flows with closed lines of flow and motion of droplets at high Reynolds numbers

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Abstract

A general asymptotic method is proposed for the description of flows of a viscous incompressible fluid with closed lines of flow at high Reynolds numbers. The method makes it possible to calculate the unknown constant in the Prandtl-Batchelor theorem for a broad class of problems. The problem is considered of the motion of a spherical droplet in a fluid. The equations of the boundary layer inside the droplet are also obtained and solved. It is shown that the velocity field inside the droplet tends with increase in the Reynolds number to the flow velocity of the Hill vortex. On the basis of the solutions to the equations of the boundary layer, an equation is derived for the constant strength of the vortex inside the droplet which confirms the general relationship obtained in the study. A comparison is given of the asymptotic theory with the numerical calculations of various authors. A law of similarity is established for fluid droplets with respect to two criteria (in place of three in the general case) with a relatively slow internal motion. This case usually holds for fluid droplets moving in gases at a large Reynolds number.

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Literature cited

  1. G. K. Batchelor, “On steady laminar flow with closed streamlines at large Reynolds number,” J. Fluid Mech.,1, 177 (1956).

    Google Scholar 

  2. V. V. Pukhnachev, Nonclassical Problems in Boundary Layer Theory [in Russian], Izd. Novosib. Un., Novosibirsk (1979).

    Google Scholar 

  3. G. K. Batchelor, Introduction to Fluid Dynamics, Cambridge University Press (1967).

  4. M. A. Lavrent'ev and B. V. Shabat, Problems of Hydrodynamics and their Mathematical Models [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  5. W. W. Wood, “Boundary layers whose streamlines are closed,” J. Fluid Mech.,2, 77 (1957).

    Google Scholar 

  6. J. F. Harper and D. W. Moore, “The motion of a spherical liquid drop at high Reynolds number,” J. Fluid Mech.,32, 367 (1968).

    Google Scholar 

  7. B. P. Le Clair, A. E. Hamielec, H. R. Pruppacher, and W. D. Hall, “A theoretical and experimental study of the internal circulation in water drops falling at terminal velocity in air,” J. Atmos. Sci.,29, 728 (1972).

    Google Scholar 

  8. G. Rimon and S. I. Cheng, “Numerical solution of a uniform flow over a sphere at intermediate Reynolds Numbers,” Phys. Fluids,12, 949 (1969).

    Google Scholar 

  9. S. Tomotica and I. Imai, “The distribution of laminar skin friction on a sphere placed in a uniform stream,” Proc. Phys. Math. Soc. Jpn.,20, 288 (1936).

    Google Scholar 

  10. V. G. Levich, Physicochemical Hydrodynamics [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

  11. D. W. Moore, “The boundary layer on a spherical gas bubble,” J. Fluid Mech.,16, 161 (1963).

    Google Scholar 

  12. A. G. Petrov, “The Lagrange function for vortex flows and the dynamics of deformed droplets,” Prikl. Mat. Mekh.,41, 79 (1977).

    Google Scholar 

  13. R. Kronig and J. Brink, “On the theory of extraction from falling droplets,” Appl. Sci. Res. Sec. A:2, 142 (1951).

    Google Scholar 

  14. J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Mass. (1968).

    Google Scholar 

  15. A. M. Golovin and A. F. Zhivotyagin, “Unsteady convective mass transfer inside a droplet in the presence of a volume chemical reaction,” Prikl. Mat. Mekh.,47, 771 (1983).

    Google Scholar 

  16. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  17. V. Ya. Rivkind and G. M. Ryskin, “Structure of flow in the motion of a spherical droplet in a fluid medium in the region of transition Reynolds numbers,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1, 8 (1976).

    Google Scholar 

  18. V. Ya. Rivkind, G. M. Ryskin, and G. A. Fishbein, “Flow round a spherical droplet in the transition region of Reynolds numbers,” Prikl. Mat. Mekh.,40, 741 (1976).

    Google Scholar 

  19. L. I. Sedov, Methods of Similarity and Dimensionality in Mechanics [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  20. A. L. Gonor and V. Ya. Rivkind, “Dynamics of a droplet,” Itogi Nauki Tekh. Ser. Mekh. Zhidk. Gaza,17, 86 (1982).

    Google Scholar 

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Authors and Affiliations

  1. Moscow

    O. V. Voinov & A. G. Petrov

Authors
  1. O. V. Voinov
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  2. A. G. Petrov
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 61–70, September–October, 1987.

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Voinov, O.V., Petrov, A.G. Flows with closed lines of flow and motion of droplets at high Reynolds numbers. Fluid Dyn 22, 708–717 (1987). https://doi.org/10.1007/BF01051691

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  • DOI: https://doi.org/10.1007/BF01051691

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