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ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, Nos. 1­2, pp. 100­117. c Pleiades Publishing, Ltd., 2013.

The Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid
Evgeny V. Vetchanin1* , Ivan S. Mamaev2,
1

3, 4 **

, and Valentin A. Tenenev1

***

Izhevsk State Technical University ul. Studencheskaya 7, Izhevsk, 426069 Russia Institute of Computer Science, Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
3 2

A.A. Blagonravov Mechanical Engineering Research Institute of RAS ul. Bardina 4, Moscow, 117334 Russia Institute of Mathematics and Mechanics of the Ural Branch of RAS ul. S.Kovalevskoy 16, Ekaterinburg, 620990 Russia
Received July 11, 2012; accepted January 16, 2013

4

Abstract--An investigation of the characteristics of motion of a rigid bo dy with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier ­ Stokes equations and equations of motion for a rigid bo dy. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped bo dy in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a bo dy in an arbitrary given direction is shown. MSC2010 numbers: 70Hxx, 70G65 DOI: 10.1134/S1560354713010073 Keywords: finite-volume numerical metho d, Navier­Stokes equations, variable internal mass distribution, motion control

INTRODUCTION The investigation of motion of bodies in a fluid is a classical area of hydrodynamics. The pioneering work in this field was concerned with the motion in an ideal fluid. If the ideal fluid is incompressible and possesses a single-valued potential, the equations of motion of a rigid body are a system of six ordinary differential equations and separate partial differential equations governing the motion of the fluid. In Lagrangian and Hamiltonian forms the equations of motion of a rigid body under the above-mentioned conditions were obtained and studied by Kirchhoff. Now they constitute a separate sub ject area in rigid body dynamics [1]. Stationary helical motions of rigid bodies in an ideal fluid were investigated by Lamb [4]. The motion of a body in a gravitational field was studied by Chaplygin and Steklov. Chaplygin also explored the motion of a body in a fluid in the presence of circulation. We refer the reader to [1] and [2, 3, 29] for the modern view on various analytical and numerical aspects of motion of a rigid body in a fluid as well as for a detailed list of references. One of the first works devoted to the motion of bodies in a viscous fluid is that of British (Irish-born) scientist G. G. Stokes (1851). He was the first to obtain an analytical solution of the
* ** ***

E-mail: eugene186@mail.ru E-mail: mamaev@rcd.ru E-mail: tenenev@istu.ru

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Navier ­ Stokes equations for the creeping flow of a viscous incompressible fluid near a sphere. His formula for resistance (drag) force is 1 (1) F = V 2 SCD , 2 where V is the velocity, is the density of the fluid, S is the midsection and CD is the coefficient of resistance. According to Stokes, the coefficient of resistance CD is given by the expression 24 . (2) CD = Re This solution was obtained by completely neglecting inertial forces acting on a fluid near a body, and a standard curve of resistance shows that this law is valid for Re < 1 [5]. Stokes' solution was elaborated upon by C. W. Oseen (1910). He pointed out that one cannot neglect inertial terms at a large distance from a sphere, even if the velocity of the sphere relative to an unperturbed flow is small [6]. The inertial terms were partially taken into account by introducing small perturbations into the solution of the initial problem and by neglecting terms of second order of smallness. As a result, the Oseen law takes the form 3 24 1+ Re . (3) CD = Re 16 The applicability of the law (3) is limited to the condition Re < 5 [5]. Lamb showed that Oseens's solution is "worse" near the surface of a sphere [6]. The conditions of flow over a sphere at various values of Re were examined in [7] using the method of direct numerical simulation. An analysis of the topology of flow at various values of Re was made and the laminar-turbulent transition was reproduced for Re > 105 . Experimental results on the interaction of a moving sphere with a vortex ring are contained in [8]. A theoretical analysis of the motion of a sphere in a fluid and of its interaction with vortex structures is presented in [9]. We note that modern computer technology and numerical methods make it possible to solve completely the Navier ­ Stokes equations for bodies with a complex geometry and various boundary conditions relative to a fluid. The problem of motion of a body with a deformable outer surface in a fluid is more complicated. This problem is due to Hicks but was initiated again by mathematical biology and biomechanics when the characteristics of motion of fishes and microorganisms were explored. We mention here just a few of the large number of publications devoted to this problem (including a number of robotized prototypes, e.g., Robotuna). The problem of the self-propulsion of a deformable body in an ideal medium was studied in [10]. Questions of the stability of motion of a deformable body in an ideal fluid, as applied to the problem of N spheres, were considered in [11]. These N spheres are regarded as a deformable body. It is shown that the stability of motion of such a system can be controlled by an appropriate deformation. An application to the investigation of the phenomenon of clustering of emerging gas bubbles in suspension is presented. The hydrodynamics and stability of a variable body in an ideal medium near a rigid boundary was discussed in [12]. The possibilities of the method used are demonstrated by considering the problem of pulsation of a spherical bubble near a rigid spherical surface. The now classical work of Lighthill [13] dealt with the motion of a deformable sphere at small Reynolds numbers on the basis of Stokes' formula. Various types of deformation were considered: radia compression ­ expansion, change from a sphere to an ellipsoid and tangential motion of points of a sphere. In [14, 15] the motion of microorganisms under conditions of large viscosity is considered. The velocity of motion due to the propagation of small amplitude waves is evaluated. The rectilinear motion of bodies due to oscillations of attached links (analogs of fins) is considered in [16­18]. The investigation is carried out using the phenomenological dissipation model in which the resistance forces of a fluid are taken to be proportional to the square of velocity with a constant coefficient of resistance. The mass of the attached movable links is taken to be negligibly small. The experimental investigation of motion of the fluid surrounding a microorganism was carried out in [19]. The measurements revealed the velocity properties of the fluid and the value of mechanical energy dissipated by the organism during the motion.
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The immersed boundary method is in widespread use to carry out numerical investigations of motion. In [20], the method of numerical investigation of the hydrodynamics of a floating body is proposed. Results of numerical simulation of the flow around a worm on the basis of the three-dimensional Navier ­ Stokes equations are presented. In [21], a computational investigation of motion of medusas featuring axisymmetric solutions of the Navier ­ Stokes equations on deformable grids is carried out. In [22], the planar motion of a three-link body is studied by the vortex particle method proposed in [23]. In this paper we investigate another type of locomotion (which is apparently not used in nature) executed by moving internal masses according to some given laws. The characteristics of the motion of internal masses can be chosen such that the body moves in the prescribed direction. The main advantage of such systems is a minimized environmental impact due to the absence of external propulsion devices such as screws, blades or fins. This means of setting in motion can be used in robotics to create noiseless, maneuverable means of locomotion. Such a means of locomotion is possible both on the surface of a fluid and inside it. The controlling masses must be set in motion by a vibrodrive, which is, as stated in [24], simple and highly reliable. Systems with internal masses with different laws of friction were studied in [25]. Optimal periodic motions of controlling bodies were obtained. As in [16, 17], viscous friction was taken to be proportional to the square of velocity. The authors of [26] consider problems of motion of a mobile robot moving due to periodic oscillations of internal masses built into its body. The interaction of the robot with the medium in which it moves is implemented by means of four supporting floats with controllable change of the inclination angle. The viscous resistance forces are calculated using methods obtained experimentally. The hydrodynamics of a body with internal mass and a rigid shell was studied in the recent paper [27]. Using the solution of the Navier­Stokes equations in the form "stream functionvorticity", the patterns of flow around a body in a fluid, dependence of viscous forces, pressure forces and maximal velocities on the Reynolds number were obtained. Analogous results were obtained for a body with internal mass and a deformable shell. The change of the form was given according to some law without regard to the influence of hydrodynamical forces on the process of deformation. Two-dimensional problems of motion of bodies of elliptical and circular form with internal masses in an infinite volume of a fluid were studied in [28]. The Navier­Stokes equations, along with dynamic rigid body equations, were solved numerically by the pro jection technique. Phase tra jectories of forces and momenta were obtained, and differences of the motion of bodies in a viscous fluid from the motion in an ideal fluid were shown. The problem of controlling the motion of a body with internal masses was discussed in [30]. To ensure that the body moves near the prescribed tra jectory, the problem of optimal control of Lagrange type was solved by a real-coded genetic algorithm. To avoid a multiple solution of the Navier­Stokes equations, the control problem was solved on the basis of a series of parametric calculations. As a result, approximate dependencies were obtained for the forces acting on the body; the dependencies were built using Data Mining technology. Here we consider the three-dimensional viscous hydrodynamics of motion of a body with variable mass distribution. Section 2 presents a mathematical model describing the motion of a body with internal masses and a fluid surrounding it. Section 3 describes the numerical method used. Section 4 shows the results of verifying the calculation method. Section 5 presents the results of calculations for the motion of a body with zero buoyancy due to displacement of internal masses. Section 6 shows the possibility of overcoming the gravity force by means of internal mass oscillations. 1. EQUATIONS OF MOTION Consider the spatial motion of a body with variable mass geometry. The body has a material shell of invariable form and mass M . Inside the shell, material points with masses mk , k = 1,N move according to the prescribed laws of motion k (t), k (t). The motion of the body and the system of material points relates to two Cartesian coordinate systems: the fixed Oxy z and the moving O1 (Fig. 1). The point O1 is the center of mass of the shell of the body.
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The equations of motion governing the motion of a system of material points are presented in [31]. dx dy dr0 = v1 = , v2 = , We introduce the notation: U = dt dt dt dz T is the vector of velocity of the point O1 ; v3 = dt T = 1 ,2 ,3 is the angular velocity vector of rotation of d d d T ,, are the vectors of motion of the body; and = dt dt dt material points. The vector equation for the momentum P of the body moving in a viscous medium [32] is:
Fig. 1. Coordinate systems dP + â P1 = F R t , (1.1) dt where F is the vector of the force acting on the rigid shell. For the angular momentum K of the body we have the equation dK + â K1 + U â P1 = GR t , (1.2) dt where G is the moment of force acting on the rigid shell. The expressions for the momentum of the system of bodies P1 and the angular momentum K1 have the form
N N N

P1 = M +
k=1

m

k

U+
k=1

mk k +
k=1

mk â k ,

(1.3) (1.4)

P = P1 +1 U,
N N N

K1 = D +
k=1

mk k â ( â k )+
k=1

mk k â k +
k=1

mk k â U,

(1.5) (1.6)

K = K1 + 2 ,

where D is the tensor of the moments of inertia of the rigid body, 1 is the tensor of added masses, 2 is the tensor of added moments of inertia. The body is assumed to start from rest, i.e. P(0) = 0, K(0) = 0. The system of ordinary differential equations (1.1), (1.2) and algebraic equations (1.4), (1.6) is solved numerically. To determine the coordinates of the position of the center of mass in a fixed coordinate system, the system of equations of the form d dxc = U, , + â = 0, dt dt d dyc = U, , + â = 0, dt dt d dzc = U, , +â =0 dt dt is additionally solved with the initial conditions (0) = (1, 0, 0)T , (0) = (0, 1, 0)T , (0) = (0, 0, 1)T , xc (0) = xc0 , yc (0) = yc0 , zc (0) = zc0 , where , , are the unit vectors directed along the axes x, y, z. The right-hand sides of Eqs. (1.1) and (1.2) contain the force with which the fluid acts on the rigid shell, and its moment. To calculate them, a complete system of the Navier­Stokes equations must be solved. In the moving coordinate system attached to the body, the hydrodynamical equations are [32] · V = 0, (1.7) V + (V - W)V = -p + 2 V - â V, (1.8) t
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where V = (u, v , w)T is the absolute velocity vector of the fluid and p, and are the pressure, density and kinematic coefficient of viscosity, respectively. In Eqs. (1.7) and (1.8) all vectors V, W and have been written in pro jections onto the moving coordinate system. For the system of equations (1.7) and (1.8) the boundary conditions V · n = U + â rs · n, n â V = n â U + â rs , p = 0, n V were given on the body's surface at the outer boundary of the region p = 0, = x Here W = U + â r1 is the transport velocity vector of the particle of the fluid The reaction forces of the fluid acting on the body, and the moment of forces are the surface integrals [32] F
R

V V = = 0. y z at the point r1 . determined by (1.9) (1.10)

t=
S

-pI + V + V

T

· n ds,
T

GR t =
S

rs â -pI + V + V

· n ds.

2. THE NUMERICAL METHOD For the numerical solution of Eqs. (1.1) and (1.2) the explicit Runge ­ Kutta method was used. Equations (1.7) and (1.8) were solved on a spherical finite-volume unstaggered grid using the method proposed in [32, 33]. The hydromechanical parameters were determined in the centers of finite volumes: Vik - Vik-1 = k N Vik-1 + k N Vik-2 t (2.1) 1 pn + k L Vik - Vik-1 , + k + k L Vik-1 - xi where Vi0 = V NV
k i n i

and the operators N and L have the form Vjk xj xi - 2 Vjk xi xj LV - xj = Vik - W Vjk + Vik V
k j

=

xj

- â V,

(2.2)

k i

V k i. xj xj 1 , 6 3 = - 5 . 12

(2.3)

According to [34], the coefficients k , k , k and k are: 4 1 1 = 1 = , 2 = 2 = , 3 = 3 = 15 15 8 5 3 17 1 = , 2 = , 3 = , 1 = 0, 2 = - , 15 12 4 60 The predicted values Vi3 are used to calculate the potential 1 Vi3 2 = , t xi x2 i by means of which the velocity and pressure fields are adjusted pn V
n+1 i +1

(2.4)

= pn + , = Vi3 - t . xi
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n+1

The resistance force and its moment (1.9), (1.10) are calculated from the values pn+1 and V i . The values of the resistance force and moment thus obtained are used to determine the translational and rotational velocities of the body at time tn+1 . In order for the velocity and pressure fields to satisfy the kinematic boundary conditions at the next time instant, the Laplace equation must be solved for the adjustment p with the boundary condition on the body's surface [32] (2.7) n · p = - n · V, t where n is the outer unit vector normal to the surface of the moving body. The new adjusted parameters are calculated as follows: pn V
n+1 i +1

=p

n+1

=V

n+1 i

+ p, p - t . xi

(2.8) (2.9)

To prevent nonphysical oscillations, an interpolation procedure involving pressure gradients [35] T is used. We denote the velocities on the faces of finite volumes by Uf = Ux ,Uy ,Uz . These values are calculated using the following averaging procedure: V = V +t p p ,
cc

(2.10) (2.11) , (2.12)

Uf = I Vi , Vj , Uf = Uf - t
fc

where I is the operator of interpolation of the values from neighboring cells to the common face; the indices cc, fc denote the pressure gradients calculated in the center of the cell and on the face, respectively; and i, j are the numbers of cells of the grid. The introduction of an additional velocity array on the faces of finite volumes allows one to simulate the staggered grid and to determine the basic hydromechanical parameters in the cells of the grid. The pressure gradient in the cells is calculated using the Gauss formula [36, 37] applied to a specific finite volume 1 nij sij ij , (2.13) i = Vi
j

where Vi the size of the i-the cell, sij is the face between the i-th and j -th cells of the grid, nij is the outer unit vector normal to the face sij and ij is the value of on the ij -th face. For a uniform Cartesian grid, the formula (2.13) leads to the central difference, which adversely affects the process of calculations. At high Reynolds numbers Re, even when the interpolation (2.10)­(2.12) is used, it is possible to obtain a nonphysical staggered distribution of hydromechanical parameters [38]. To eliminate these problems, the gradients are calculated from the least squares formula [37] Gi =
j 2 wij dij dij ,

(2.14) (2.15)

i =
j

wij G-1 · dij j - i , i

where dij is the vector directed from the i-th cell into the j -th cell and wij is the weight function equal to 1 dij . For elongated cells the formula (2.15) gives better results than the Gauss formula. For nonelongated cells both formulae give approximately the same results.
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(a)

(b)

Fig. 2. The velocity field at t = 20 and t = 40.

3. VERIFICATION OF THE SOLUTION METHOD The method applied has second order accuracy in time and space coordinates. The method was tested using the problem of motion of a sphere in an incompressible fluid at Re = 1 000. The sphere has a constant velocity V = (0, 0, 1). Fig. 2 shows the vector field of the flow near the sphere. Figure 2, a corresponds to the time t = 20. The flow near the sphere is symmetric and regular. A ring vortex is formed in the rear part. In the process of evolution of the flow the symmetry is violated. Figure 2, b shows the flow at time t = 40 and the separation of vortices from the rear part of the body. The vortex separations entail oscillations of the resistance force in all three coordinate directions. Figure 3 shows the isosurfaces of vorticity â V near the sphere at different time instants after the violation of the flow symmetry. As can be seen from Fig. 3, the flow behind the sphere is non-regular due to the vortex separations. The separations occur in both transverse directions.

(a)

(b)

(c)

(d)

Fig. 3. The vorticity isosurfaces at t = 40 (a), t = 50 (b), t = 60 (c) and t = 70 (d).

Figure 4, a shows the components of the resistance force. The corresponding coefficients of resistance and lifting force Fl =
2 2 Fx + Fy are shown in Fig. 4, b.

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(a)

(b)

Fig. 4. The force components (a) and the resistance coefficients (b).

The coefficient of resistance on the regular segment of the process is Cd = 0.43, which corresponds to the experimental data [5]. The Strouhal number St = fd V , where f is the frequency of shedding of vortex rings, d is the diameter of the sphere and V is the velocity of motion, is calculated from oscillations of the lifting force. It is assumed that one shedding of a vortex ring corresponds to two opposite peaks Fl . The vortex separations start approximately at time t = 30, St = 0.17, and the mean coefficient of resistance is C d = 0.53, which is in agreement with the data presented in [7]. The resistance force oscillations occur with variable frequency in all spatial directions. This is caused by the three-dimensional irregular nature of hydrodynamics, which points to the necessity of simulation of a three-dimensional flow even for the axisymmetric problem. Such force fluctuations occur, for example, when a body falls in water. Due to vortex separations the tra jectory of the falling body deviates from a straight line. The fall of a ball (radius 0.1 and density 1 100) under the action of the gravity force against the axis Ox was simulated. The tra jectory of motion is shown in Fig. 5.

(a)

(b)

Fig. 5. The tra jectory within 60 s in the planes yOx and zOx (a), yOz (b).

The motion of a body can be strongly influenced by the Magnus force [39] arising from rotations of the sphere. The rotation of the sphere is caused by asymmetric vortex separations leading to an asymmetric distribution of viscous forces and to the appearance of a torque. Figure 6 shows the components of the angular velocity of the body expressed in the fixed coordinate system. The forces acting on the falling ball are shown in Fig. 7. Thus, when the ball sinks slowly with density differing only slightly from that of the fluid, a deviation of the motion from the vertical tra jectory is possible.
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Fig. 6. The components of the angular velocity of a body.

Fig. 7. The forces acting on the ball.

4. THE MOTION OF A BALL WITH VARIABLE MASS GEOMETRY WITHOUT GRAVITY FORCE Consider a spherical body with mass M and with internal mass can be given in the form 1 (t) = 1 (t) - sin2 ( /k ), 1 1 (t ) = cos2 (/k - 21 t T0 , T0 = =t- T0 zero buoyancy. The law of motion of the T 0 , 0, 0 , where k , 21 k ), , 21 1 1+ k, 2 1 < (4.1)

where k regulates the oscillation frequency, 1 is the coefficient of asymmetry of oscillations and 0 is the displacement. In an ideal fluid the body will execute an oscillatory motion with constant amplitude relative to the original position. In a viscous fluid the resistance force of the fluid changes nonlinearly due to viscous friction as the body moves with variable velocity. For 1 > 1 the oscillations will be asymmetric and the body will travel some distance for each oscillation period. Figure 8 shows the motion of the body along the axis x in the cases of an ideal and viscous fluid for 1 = 4 and different k . The spatial variables of the motion in this figure have been non-dimensionalized using the diameter of the body. The motion of the body was considered in water; we present the values of calculated parameters for this medium. The kinematic coefficient of viscosity is = 10-6 , the density of the medium is
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Fig. 8. The motion of a body.

= 1 000, the radius of the sphere is r = 0.1, the mass of the main body is M = 2.188 and the mass of the internal material point is m1 = 2. As can be seen from Fig. 8, an increase in the maximum displacement occurs within each oscillation period. At a higher oscillation frequency the body moves faster. The flow pattern near an oscillating body is shown in Fig. 9. During the motion to the left an encircling vortex (Fig. 9, a) is formed around the body. During braking and change of the direction of motion the encircling vortex is displaced by the newly formed vortex moving in the opposite direction (Fig. 9, b, c, d). As the flow evolves, the displaced vortex stretches and decays away from the body.

(a)

(b)

(c)

(d)

Fig. 9. The flow pattern at times 0.1 (a), 0.197 (b), 0.198 (c) and 0.199 (d). REGULAR AND CHAOTIC DYNAMICS Vol. 18 Nos. 1­2 2013


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Let us consider the pressure distribution on a sphere at different instants of motion (Fig. 10). At time t = 0.097 the body reaches the maximum velocity in the negative direction (-0.162611 m/s). After that the body starts braking, the pressure in the rear part (180 ) increases due to the inflow of the perturbed fluid and that in the front part (0 ) decreases due to the outflow of the fluid from the body. At time t = 0.107 the velocity of the body is -0.160722 m/s and at t = 0.15 it becomes -0.108252 m/s.

Fig. 10. The pressure during braking of a body.

Figure 11 shows the pressure curves after a change of the direction of motion at times t = 0.197 (0.0021449 m/s), t = 0.198 (0.00219088 m/s) and t = 0.199 (0.00238052 m/s). During speeding-up in the positive direction the moving body must push the fluid through, which leads to an increased pressure at the point 180 and a decreased pressure at the point 0 , but as the body gains speed the fluid is set in motion, and the pressure exerted by it on the above-mentioned points decreases.

Fig. 11. The pressure under a change of the direction of motion.

Of interest are non-spherical (for example, drop-shaped) bodies (Fig. 12). For such a body the resistance forces depend on the direction of motion. The coefficients of resistance for a drop moving f b in the direction of the pointy end (CD ) and in the direction of the base (CD ) are indicated in s is indicated. Table 1. For comparison, the coefficient of resistance of the sphere CD It can be seen from Table 1 that at high values of Re the coefficients of resistance vary depending on the direction of motion. Note that the coefficient of resistance is lower when the body moves with the base ahead.
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Fig. 12. A drop-shaped body.

Table 1. Co efficients of resistance Re C C C
f D b D s D

0.9 43.13 43.13 30

9 5.58 5.64 5

90 1.4 1.4 1.2

900 0.635 0.599 0.5

9 000 0.476 0.367 0.4

90 000 0.425 0.305 0.42

A calculation of the motion of a drop-shaped body with radius of the base rb = 0.045 and mass 0.626 was performed. The internal mass equal to 0.3 moved according to the law (4.1) with 1 = 4, k = 0.2, r1 = 0.09 and 0 = 0.045. Figure 13 shows a comparison of the motions of the drop-shaped body and the spherical one with equal mass and midsection. The displacements are normalized using the diameter of the respective bodies.

Fig. 13. The motions of a spherical body and drop-shaped bodies.

The curve called a "drop with the pointy end ahead" corresponds to the case where the first phase of the law (4.1) is executed in the direction of the pointy end and the second phase in the direction of the base. Accordingly, for the curve "drop with the base ahead" the first phase is executed in the direction of the base and the second phase in the direction of the pointy end. As evident from Fig. 13, the qualitative pattern of motion has not changed. On the initial segment the curves for the motion with the base and the pointy end ahead coincide. However, the drop moving with the pointy end ahead gradually begins to outstrip the drop moving with the base ahead. Since the curves pass very closely, we consider the enlarged fragments of graph 13 shown in Fig. 14. It can be seen from Fig. 14, a that the drop-shaped body moving in the direction of the pointy end rolls back faster than the spherical body. However, as the drop advances, it outstrips the sphere by a small distance, as is evident from Fig. 14, b. The large rollback of the drop and the large advance are due to the resistance forces acting on the bodies from the fluid. The pattern of the resistance forces is shown in Fig. 15 and Fig. 16.
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(a)

(b)

Fig. 14. The motions of a spherical body and drop-shaped bodies: (a) rollback of the bodies, (b) advance of the bodies.

Fig. 15. Pressure force.

Fig. 16. Viscous force.

The total resistance force is shown in Fig. 17. Under a change of the direction of motion or at the start of motion there arise peak values of the pressure force. The pressure exerted on the spherical body is stronger than that exerted on the drop-shaped body (Fig. 15). The pressure creates a larger resistance when the internal mass moves fast and the body has a large velocity. The viscous force changes in time more smoothly than the
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Fig. 17. Total resistance.

pressure force (Fig. 16). The peak values of the viscous forces are an order smaller in magnitude than the peak values of the pressure forces. The change of the translational velocity in time is shown in Fig. 18.

Fig. 18. The pattern of velocity changes of a body.

k in the 21 formula (4.1)) the body acquires a high velocity. During the second phase, when the point slowly k in the formula (4.1)), the body moves in returns into the initial position (the condition 21 the opposite direction with small velocity. It can be seen from Fig. 18 that during the first phase of motion (the condition < 5. THE MOTION OF A BALL WITH VARIABLE MASS GEOMETRY IN A GRAVITATIONAL FIELD The effect of motion of a body with variable mass geometry in a viscous fluid can be used to overcome the gravity force. Consider a ball with density b and radius r = 0.1 in a liquid medium with density = 1 000 and kinematic coefficient of viscosity = 10-6 . The mass of the internal material point is 2. The law of motion of the internal mass is given by (4.1), 1 = 4. For several values of b and oscillation frequencies of the internal mass, the graphs of the motion of the body are shown below. The displacements are normalized using the diameter of the body.
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Fig. 19. The motion of a body in a gravitational field (b = 1001, 2.5 Hz).

Fig. 20. The motion of a body in a gravitational field (b = 1003, 2.5 Hz).

Fig. 21. The motion of a body in a gravitational field (b = 1003, 5 Hz).

In an ideal fluid the body sinks, as expected. When b = 1 001 and the frequency is 2.5 Hz, the body gradually begins to rise to the surface. When b = 1 003 and the frequency is 2.5 Hz it is not possible to counteract the gravity force. When the oscillation frequency of the internal mass increases to 5 Hz, the body rises to the surface. The following conclusion may be drawn: the larger the mass that must be lifted, the higher must be the oscillation frequency of the internal mass. Of interest is the motion of the body not only against the gravity force but also in other directions. Consider a body with b = 1 100. The gravity force is directed along the axis Ox of
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the fixed coordinate system, and the internal mass performs oscillations along the axis O with frequency 2.5 Hz. The tra jectory of motion is shown in Fig. 22.

Fig. 22. The tra jectory of a body.

As evident from Fig. 22, the body sinks and advances in the direction perpendicular to the gravity force but with a change of the direction of motion. This is due to the rotation of the body under the action of the pair of forces "gravity ­ buoyancy force". The angular velocity of rotation is shown in Fig. 23.

Fig. 23. The angular velocity of a body.

The oscillations of the angular velocity gradually decay because the body is acted upon by the moment of viscous resistance force. There is no damping of rotation during the motion of the body in an ideal fluid as opposed to the motion in a viscous fluid. Thus, the motion of the body in a gravitational field requires control. 6. DISCUSSION AND PROBLEM STATEMENT The results of numerical simulation of the motion of a rigid sphere in a fluid have shown the possibility of directional self-propulsion of the body by changing the mass geometry. It is shown that as the oscillation frequency of the internal material point increases, the displacement of the body (and thus the average velocity of self-propulsion) increases. The pattern of dependence of resistance forces for the back-and-forth motion of the internal mass has been obtained. When the sphere moves with a buoyancy close to zero, it can overcome the gravity force. During oscillations of the internal mass the body perpendicular to the gravity force begins to rotate strongly, and the tra jectory becomes unpredictable. For bodies with variable mass distribution the following problems remain open.
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Fig. 24. The angular velocity of a body.

1. Experimental investigation of motion and verification of computational investigations using experimental data. Exploration of the dependence of motion on various external factors. 2. Three-dimensional viscous hydrodynamics near the boundary between media and a rigid surface. 3. Numerical simulation of motion for the turbulization of the boundary layer by large-eddy simulation and direct numerical simulation. 4. Control of the three-dimensional motion along a given tra jectory in cases of asymmetric vortex separations and nonzero buoyancy. 5. The motion of a light body on a water surface. ACKNOWLEDGMENTS This research was supported by the Presidential grant of leading scientific schools NSh2519.2012.1., the Target Programme "Development of Scientific Potential of Higher Schools" (State contract 1.2953.2011, 2012­2014); the Federal Target Programme "Scientific and ScientificPedagogical Personnel of Innovative Russia" (State contract 14.B37.21.1935, 2009­2013); grant of leading scientific schools NSh-2519.2012.1. REFERENCES
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