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IssuesArchive of Issues2001-3pp.95-104

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Yu. K. Bivin, "Motion of a body near a free surface of a fluid or plastic medium," Mech. Solids. 36 (3), 95-104 (2001)
Year 2001 Volume 36 Number 3 Pages 95-104
Title Motion of a body near a free surface of a fluid or plastic medium
Author(s) Yu. K. Bivin (Moscow)
Abstract Various cases of motion of a body near a free surface of a fluid have been thoroughly studied in great detail in connection with the problem of finding the forces that act on such a body, provided that its motion is known [1]. Another problem for a free body moving near the border of a medium is to describe the character of its motion under the action of these forces. The ricochet off free surfaces of fluids or plastic media possessing strength has been thoroughly examined for spherical and conical bodies; see, for instance, [2-4]. An investigation of the trajectory of a spherical body fired parallel to the free surface of a plastic medium has been performed in [5].

Below, we consider the results of an experimental investigation of the inertial motion of spherical, conical, and cylindrical bodies whose initial velocity vector is either parallel to the free surface of a fluid (or plastic medium) or is directed at a small angle to that surface, so that the conditions are close to those leading to ricochet. We find out to what extent the free surface affects the motion of the body and examine the conditions of ricochet for bodies of different geometry. For spherical bodies, these are compared with the widely used conditions proposed by Birkhoff [6]. A notable fact is that for spherical bodies entering water at an angle, the immersion trajectory at a critical entrance angle is actually independent of the entrance speed within the range for which the critical angle does not vary. But this trajectory depends on the ratio of the specific densities of the fluid and the sphere, because this ratio affects the critical entrance angle. The critical angle ωc is that separating the angles for which there is ricochet or not. In the case of the initial motion parallel to the free surface, it is shown that the path traveled by the body in a plastic medium depends on its initial depth and velocity, as well as on the shape of the body and its mass. Moreover, the path does not exceed the penetration depth for the body entering the free surface in the orthogonal direction. For a body entering a fluid, the dependence of its path along the free surface on the initial velocity and the mass of the body is not so pronounced. If the angle between the velocity vector of a ball and the free surface of plasticine is smaller than the critical angle, the trace of its contact with plasticine is longer than the normal penetration depth and depends on the impact angle, the velocity, and the strength of the medium. We consider the motion of cylindrical and conical bodies near a free surface of a continuous medium and point out that their behavior differs in that they swivel in different directions around the transverse axis. Possibilities of controlling this motion are examined.

Experiments were performed for bodies of diameter 10 mm. The body was accelerated to the desired velocity by an air-gun. The velocity of the flying body was registered by photocells located at fixed intervals along the axis of shooting. The photocell signal was passed through a converter and then fed to a frequency meter, whose readings, together with the distances between the photocells, were used to determine the velocity of the body entering the medium.

The barrel of the air-gun was set up parallel to the horizon to within several angular minutes. With the same accuracy, the free surface of plasticine was set up parallel to the horizon, while the projectiles were shot in a parallel or an oblique direction to the free surface. When firing at an angle to the free surface, the precision of setting up the gun was about 10 minutes. In order to investigate the effect of the free surface of a fluid, a tank was used with a hole of diameter 20 mm in its vertical wall. Prior to filling the tank, the hole was closed by a polyethylene film of thickness 0.02 mm. The tank was made of sheet organic glass of thickness 25 mm and its dimensions were 200×200 mm in the plane orthogonal to the shooting axis and its length was 600 mm. The inlet was located at 160 mm above the bottom of the tank and equally distanced from its lateral walls, so that the influence of the bottom and the lateral walls on the motion of the body would be much weaker than that of the free surface. It turned out that the effect of the free surface could not be felt at distances larger than 40 mm for bodies used in the experiments. When dealing with plasticine, blocks of dimensions similar to those of the water tank were used and the temperature of the plasticine was 25°C.
References
1.  G. V. Logvinovich, Hydrodynamics of Flows with Free Boundaries [in Russian], Naukova Dumka, Kiev, 1969.
2.  W. Johnson and S. R. Reid, "Ricochet of spheres off water," J. Mech. Eng. Sci., Vol. 17, No. 2, pp. 71-81, 1975.
3.  Yu. K. Bivin, "Oblique impact of a solid body on soil or water," Izv. AN SSSR. MTT [Mechanics of Solids], No. 6, pp. 185-189, 1989.
4.  G. H. Daneshi and W. Johnson, "The ricochet of spherical projectiles off sand," Intern. J. Mech. Sci., Vol. 19, No. 8, pp. 491-497, 1977.
5.  W. Johnson and G. H. Daneshi, "The trajectory of a projectile when fired parallel and near to the free surface of a plastic solid," Intern. J. Mech. Sci., Vol. 20, pp. 255-263, 1977.
6.  G. Birkhoff, G. D. Birkhoff, W. E. Bliek, E. H. Handler, F. D. Murnaghan, and T. L. Smith, "Ricochet off water," A.M.P. Memo, 42 4M, 1944.
Received 03 February 1999
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