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Motion of a small sphere in a nonhomogeneous flow of an incompressible liquid

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Abstract

We consider the motion of a small sphere in an arbitrary potential flow of an ideal liquid. For the general case we obtain an integral of the equations of motion and a particular solution. We find flows in which the force acting on the sphere is central. We also obtain exact solutions of the equations of motion of the sphere for the cases of stationary flows around a cylinder and around a body of revolution when the forces are noncentral. N. E. Zhukovskii [1] calculated the force acting on a fixed sphere in an arbitrary nonstationary flow. Kelvin [2] obtained the equations of motion of a sphere in a stationary flow of a liquid circulating through a hole in a solid. A formula for the force F, acting on a fixed small body of volume V in a stationary flow with speed v, was obtained by Taylor [3]: F = (∂T 0 / ∂v)Vv + 1/2ϱV v 2 Here T0 is the kinetic energy of an unbounded liquid in which a body moves with velocity v. This problem was solved in [3] through a direct integration of the pressure forces over the surface of the body in a flow defined by multipoles of the first and second orders at infinity.

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Literature cited

  1. N. E. Zhukovskii, “Generalization of a problem of Bjerkness concerning the hydrodynamic forces acting on pulsating or oscillating bodies inside a liquid mass,” in: Collected Works, Vol. 2, Hydrodynamics [in Russian], Gostekhteoretizdat, Moscow-Leningrad (1949).

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  2. W. Kelvin, “On the motion of rigid solids in a liquid circulating irrotationally through perforations in them or a fixed solid,” Philos. Mag.,45, (1873).

  3. G. I. Taylor, “The forces on a body placed in a curved or converging stream of fluid,” Proc. Roy. Soc., Ser. A,120, No. 785 (1928).

  4. V. V. Voinov, O. V. Voinov, and A. G. Petrov, “Hydrodynamic interaction of bodies in an ideal incompressible liquid and their motion in nonhomogeneous flows,” Prikl. Matem, i Mekhan.,37, No. 4 (1973).

  5. O. V. Voinov and A. G. Petrov, “The Lagrange function for a gas bubble in a nonhomogeneous flow,” Dokl. Akad. Nauk SSSR,210, No. 5 (1973).

  6. O. V. Voinov, “On the force acting on a sphere in a nonhomogeneous flow of an ideal incompressible liquid,” Zh. Prikl. Mekhan. i Tekh. Fiz., No. 4 (1973).

  7. V. I. Il'ichev, A. A. Kanzeba, G. N. Kuznetsov, and A. T. Listrov, “Motion of a gas bubble in the hydrodynamic flow field over a body,” Trudy Akust. Inst., No. 6 (1969).

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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 Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 57–61, September–October, 1973.

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Voinov, O.V., Petrov, A.G. Motion of a small sphere in a nonhomogeneous flow of an incompressible liquid. J Appl Mech Tech Phys 14, 650–653 (1973). https://doi.org/10.1007/BF00856876

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  • DOI: https://doi.org/10.1007/BF00856876

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