 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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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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