SECTION 2
SUBSECTION 2.1. THIN-WALLED ELEMENTS
IN MODELING OF BIOMECHANICAL SYSTEMS
RADIATIVE FORCE ACTING ON A PARTICLE IN A FLUID-FILLED
CYLINDRICAL SHELL UNDER PROPAGATION SPHERICAL
ACOUSTIC WAVE
Kubenko V. D., Zhuk O. P.
Institute of Mechanics of National academy of Sciences of Ukraine
03057, 3, Nesterov Str., Kyjiv, Ukraine
[email protected]
Introduction. A periodic sound pressure and constant pressure of sound radiation usually referred to as the radiation pressure are induced by the harmonic wave
propagation in acoustic medium. The radiation pressure exerts by an acoustic field
provided that an average impulse transferred by the wave for the period of oscillation varies in some volume of medium resulting in the generation of the stationary
component of the sound pressure acting over this volume. Analysis of the paper is
founded on classical work by King ([1], see also [2]): evaluation of acoustic pressure accounting for the terms of the second order of magnitude can be performed
with the use of the velocity field potential obtained from the solution of a linear
problem of a primary wave scattering on an obstacle.
Determination of acoustic pressure and radiation force acting on a spherical
rigid particle located in a fluid-filled cylindrical shell are based on solution of problem of diffraction of a spherical harmonic wave that is excited by point source, at a
spherical body. The problem is solved by the variables separation method. To satisfy the boundary conditions on cylindrical and spherical surfaces, the mutual expansions of spherical wave functions over the cylindrical ones and vice versa are used.
Required constants in the solution are calculated from an infinite system of the algebraic equations which is solved by a truncation method. As a numerical example
propagation of a plane wave in fluid-filled rigid cavity with spherical obstacle is
considered. It is established that value of the radiation force is affected significantly
by the presence of the cylindrical boundary surface. Depending on the frequency of
an incident wave, the radiation force can change its direction. It is established that
there are peak values of the force at some discrete values of the frequency. The
mentioned peculiarities can stipulate specific localization effects and influence the
motion of the particles in the cylindrical cavity under action of the acoustic wave
Linear problem for velocity potential determination. Cylindrical coordinate
system  , z  is introduced in such a manner that its origin O coincides with the
spherical particle center and its axis Oz runs along the shell axis (axisymmetrical
case). Spherical system  r,  is connected with the particle. At l distance from the
particle embedded at point O1 wave source excites spherical harmonic wave with fre-
128
quency  and potential inc  Wr11eir1 , where α – wave number, r1 – distance from
O1 , W – amplitude. Time factor eit is omitted. Wave field in liquid describes by
wave equation and its general solutions in cylindrical and spherical coordinates are

sph  r ,     An hn   r  Pn cos    ,  cyl  , z  

 B  J0
1
n 0



2   2 eiz d  .
The common potential is   inc  sph  cyl . Linear Kirchoffe-Love equations of shells is used. To satisfy boundary conditions at the shell surface (equality
of normal velocities of the shell and liquid) and at the spherical particle surface
(liquid normal velocity diminishes) it is necessary to rewrite the potential  both in
spherical and cylindrical coordinates. It is made with help of next relations [24]:

eir1
  i  2n  1 J n  r  hn  l  Pn cos    ,
r1
n 0
hn  r  Pn cos    


i n    
 Pn    H 0
2 


2  2  eiz d  ,


eiz J 0  2  2   i n  2n  1 Pn   jn  r  Pn cos    .
 
n 0
Here hn , jn – spherical Hankel and Bessel functions of n-th order, H n , J n – cylindrical functions, Pn – Legendre polinome.
Infinitely system of algebraic equations for coefficients An is obtained from
boundary conditions. The system is solved by truncation and thus the wave potential
has been determined.
Radiative forces determination. Hydrodynamic force has the following relation:

Fz  2r02  p sin    cos    d  ,
0
where pressure of acoustic waves on the spherical particle surface p we will compute to within quantities of the second order
2
2
2
 1   
1    1   
p
 
  2
  
 .
t 2  r  2r0    2  t 
If it uses the evaluated wave potential it can write next radiation force expression ( r0  radius of the particle)

n 1
n  n  2   r022  Rn Rn1  Sn Sn1  ,
n0  2n  1 2n  3
Fz  2 


Sn   1  4n  1 J 2n  r0   Re An J n  r0   Im An yn  r0   Re Bn J n  r0  ,
n
Rn   1  4n  3 J 2n1  r0   Im Bn J n  r0   Re An yn  r0   Im An J n  r0  .
n
129
Numerical example. In particular case point of source is placed at long distance so we may consider propagation of a plane harmonic wave. Then surface of
the shell is assumed as rigid one.
The analysis of the performed computation allows drawing the following conclusions:
– the proposed technique allows to calculate with the preset accuracy the value of
the acoustic pressure and radiation force depending on frequency of the incident
wave for a considerable range of frequencies and radii ratio between a cylindrical
cavity and a spherical particle;
– for the frequencies of the incident plane wave which correspond to the eigenvalues of the liquid ring formed by the cross-section of the cavity passing through the
center of the sphere, the significant rise of the acoustic pressure in the liquid take
place;
– radiation force acting on a rigid spherical particle located in the unbounded fluid
is tended along the incident wave direction and increases monotonically as the frequency rises;
– dependence of the radiation force on the frequency for the case of spherical particle located in the cylindrical cavity is much more complex due to complexity of diffraction field inside the cavity;
– depending on the frequency, the radiation force has the same or opposite direction
to the direction of the incident wave propagation for the particle located in the cylindrical cavity;
– for the particle located in a cylindrical cavity, the sharp peaks of the radiation
force are observed in the vicinity of some frequencies. These maxima are attributed
to the resonant effects that occur at the frequencies close to the natural frequencies
of the liquid ring formed by the cross-section of the cavity passing through the center of the spherical particle.
1.
2.
3.
4.
5.
References
King L. V. On the acoustic radiation pressure on spheres // Proc. Roy. Soc. Ser. A. –
1934. – Vol. 147. – P.246 – 265.
Guz A. N., Zhuk A. P. Motion of solid particles in a liquid under the action of an
acoustic field: the mechanism of radiation pressure // Int. Appl. Mech. – 2004. – Vol.
40. – P. 246 – 265.
Ivanov E. A. Diffraction of electromagnetic waves on two bodies. Minsk: Nauka i
tehnika. – 1968. – 584 p. (in Russian).
Erofeenko V. T. Relations between main solutions of Helmholtz and Laplace equationsin cylindrical and spherical coordinates // Izv. АS BSSr. Ser. Phys.-math. –
1972, – No. 4. – P. 42 – 46. (in Russian)
Kubenko, V. D. Diffraction of steady waves at a set of spherical and cylindrical bodies
in an acoustic medium // Int. Appl. Mech. – 1987. – Vol. 23. – P.605 – 610.
130