The acoustic radiation force in lossless fluids in Eulerian and Lagrangian coordinates

Journal of the Acoustical Society of America

Volumn 103, Issue 5 I, 1998, Pages 2321-2332

  Beissner, K ^a  

^a PHYSIKALISCH TECHNISCHE BUNDESANSTALT   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTICS; ARTICLE; CALCULATION; FORCE; MATHEMATICAL ANALYSIS; PRIORITY JOURNAL; THEORY;

EID: 0031840629     PISSN: 00014966     EISSN: None     Source Type: Journal    
DOI: 10.1121/1.422751     Document Type: Article

References (75)

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17. B.-T. Chu and R. E. Apfel, J. Acoust. Soc. Am. 72, 1673-1687 (1982). As far as the Langevin radiation force is concerned, their claim to deal with an exact, three-dimensional theory is expressed in particular in the "Notes added in proof" on p. 1686.
18. W. L. Nyborg and J. A. Rooney, J. Acoust. Soc. Am. 75, 263-264 (1984); B.-T. Chu and R. E. Apfel, J. Acoust. Soc. Am. 75, 1003-1004 (1984).
19. W. L. Nyborg and J. A. Rooney, J. Acoust. Soc. Am. 75, 263-264 (1984); B.-T. Chu and R. E. Apfel, J. Acoust. Soc. Am. 75, 1003-1004 (1984).
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23. Cf., however, the discussion of the influence of radiation stresses (not on targets but) on flows as, for example, by D. G. Andrews and M. E. McIntyre, J. Fluid Mech. 89, 609-646 (1978).
24. He stated that the closed surface integrals of the two versions of radiation stresses are both zero (in the no-target case) and then concluded that the radiation stresses themselves are equal to each other, but this conclusion is not necessarily true (zero-divergence quantities are not necessarily equal to one another; cf., for example, the various zero-divergence quantities appearing in this paper, including the unit matrix and, in addition, the zero matrix, too).
25. K. Beissner, Acustica 57, 1-4 (1985).
26. K. Beissner, J. Sound Vib. 93, 537-548 (1984); cf. also J. Acoust. Soc. Am. 76, 1505-1510 (1984).
27. K. Beissner, J. Sound Vib. 93, 537-548 (1984); cf. also J. Acoust. Soc. Am. 76, 1505-1510 (1984).
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29. K. Beissner, J. Acoust. Soc. Am. 79, 1610-1612 (1986).
30. E. B. Miller and D. G. Eitzen, IEEE Trans. Sonics Ultrason. 26, 28-37 (1979).
31. Z.-Y. Jiang and J. F. Greenleaf, J. Acoust. Soc. Am. 100, 741-747 (1996).
32. H. Jeffreys, Cartesian Tensors (Cambridge U.P., Cambridge, 1931).
33. R. R. Long, Mechanics of Solids and Fluids (Prentice-Hall, Englewood Cliffs, NJ, 1961).
34. S. F. Borg, Matrix-Tensor Methods in Continuum Mechanics (World Scientific, Singapore, 1990).
35. J. Serrin, in Handbuch der Physik (Encyclopedia of Physics), edited by S. Flügge and C. Truesdell (Springer-Verlag, Berlin, 1959), Vol. VIII/1, pp. 125-263.
36. This does not automatically imply that all these quantities are general tensors in curvilinear coordinates, i.e., in the broader sense of tensor analysis.
37. D. Mintzer, P. Tamarkin, and R. B. Lindsay, in American Institute of Physics Handbook, edited by D. E. Gray (McGraw-Hill, New York, 1972), 3rd ed., pp. 2-2-2-18.
38. In Ref. 15 and obviously also in Ref. 16, the stress indices are understood in the reverse sense.
39. j.
40. I. N. Bronshtein and K. A. Semendyayev, Handbook of Mathematics (Harri Deutsch, Frankfurt/Van Nostrand Reinhold, New York, 1985), 3rd ed.
41. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).
42. It should, however, be noted that the curl here often appears in combination with another curl or with a vector product, cf. Eqs. (14), (15), (17), and others. So this restriction might not be necessary.
43. J. L. Ericksen, in Handbuch der Physik (Encyclopedia of Physics), edited by S. Flügge (Springer-Verlag, Berlin, 1960), Vol III/1, pp. 794-858.
44. H. Lamb, Hydrodynamics (Dover, New York, First American Edition 1945).
45. Conditions for the applicability of the Stokes integral theorem are that the matrix and its curl exist, and are continuous, in a region containing the surface under consideration and its border line. The theorem is applied here twice, namely separately to both hemispheres (or generally, to both parts of the closed surface), and not to the whole sphere and its entire interior. So the rule mentioned is valid even with a target in the interior of the sphere (or generally, of the closed surface).
46. P. J. Westervelt, J. Acoust. Soc. Am. 29, 26-29 (1957).
47. K. Beissner, Acustica 62, 255-263 (1987).
48. Nota bene: as long as the radiation stress is a zero-divergence quantity which, in general, is not the case in lossy fluids, but this need not be discussed here as lossy fluids are not dealt with in this paper.
49. The divergence of the radiation stress is assumed to be nonzero in the interior of the target or at its surface. The target usually represents a discontinuity with respect to the sound-propagating medium and its acoustic properties. The case that the divergence of the radiation stress is zero even throughout the target and its surface is trivial and is equivalent to the no-target case.
50. K. Beissner and S. N. Makarov, J. Acoust. Soc. Am. 97, 898-905 (1995). K. Beissner, J. Acoust. Soc. Am. 99, 1244-1247 (1996). K. Beissner, in Proc. Ultrasonics World Congress 1995, edited by J. Herbertz (GEFAU, Duisburg, 1995), pp. 41-49.
51. K. Beissner and S. N. Makarov, J. Acoust. Soc. Am. 97, 898-905 (1995). K. Beissner, J. Acoust. Soc. Am. 99, 1244-1247 (1996). K. Beissner, in Proc. Ultrasonics World Congress 1995, edited by J. Herbertz (GEFAU, Duisburg, 1995), pp. 41-49.
52. K. Beissner and S. N. Makarov, J. Acoust. Soc. Am. 97, 898-905 (1995). K. Beissner, J. Acoust. Soc. Am. 99, 1244-1247 (1996). K. Beissner, in Proc. Ultrasonics World Congress 1995, edited by J. Herbertz (GEFAU, Duisburg, 1995), pp. 41-49.
53. F. E. Fox and W. A. Wallace, J. Acoust. Soc. Am. 26, 994-1006 (1954). As far as Langevin's relations, Eqs. (37) and (73), are concerned, the Fox-Wallace nonlinearity parameter appeals in their third-order approximation only or, on the other hand, it may appear in the constant which depends on the boundary conditions. When the Langevin case (where the constant is zero or can be neglected) is considered up to the second order, the nonlinearity parameter is irrelevant.
54. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1963), 2nd imp.
55. If it were understood as pointing inwards, a minus sign would appear as, for example, in Ref. 20.
56. In the literature it is often stated that the potential acoustic energy density is equal to the kinetic one (their difference being zero) or in a more indirect way, that the total acoustic energy density is twice the kinetic one and so on. This is an indication that only plane waves and not general fields are dealt with in these cases.
57. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968). The wave-stress tensor appears as the wave momentum flux density in their Eq. (6.2.17), but then they speak of a radiation pressure (a vector in their notation) exerted by the wave, and associate it with the wave momentum density. Already Brillouin criticized Rayleigh for attributing the radiation forces to the momentum density and not to the momentum flux density. [In the present paper, the momentum refers to the fluid and not to the wave; cf. also M. E. McIntyre, J. Fluid Mech. 106, 331-347 (1981)].
58. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968). The wave-stress tensor appears as the wave momentum flux density in their Eq. (6.2.17), but then they speak of a radiation pressure (a vector in their notation) exerted by the wave, and associate it with the wave momentum density. Already Brillouin criticized Rayleigh for attributing the radiation forces to the momentum density and not to the momentum flux density. [In the present paper, the momentum refers to the fluid and not to the wave; cf. also M. E. McIntyre, J. Fluid Mech. 106, 331-347 (1981)].
59. K. Yosioka and Y. Kawasima, Acustica 5, 167-173 (1955); T. Hasegawa and K. Yosioka, J. Acoust. Soc. Am. 46, 1139-1143 (1969).
60. K. Yosioka and Y. Kawasima, Acustica 5, 167-173 (1955); T. Hasegawa and K. Yosioka, J. Acoust. Soc. Am. 46, 1139-1143 (1969).
61. L. P. Gor'kov, Sov. Phys. Dokl. 6, 773-775 (1962).
62. P. B. Nagy, J. Acoust. Soc. Am. 79, 1794-1797 (1986).
63. In a lossless fluid the instantaneous contact force is of course normal to the instantaneous surface under consideration, but the orientation of a material surface is in general time dependent, and the time-average force is, therefore, not necessarily normal to the time-average surface, cf. Rets. 2 and 22 and J. Awatani, J. Acoust. Soc. Am. 27, 278-281 (1955). Note in particular that the acoustic pressure is an alternating quantity. Consider the average of a positive force vector and a negative one, both being of the same order of magnitude but with a slightly different orientation. The superposition of these vectors leads to a small resultant vector whose direction depends on the details of the situation but can even be tangent to the surface (the radiation force is generally small in comparison with the force amplitudes of the instantaneous field).
64. E. M. J. Herrey, J. Acoust. Soc. Am. 27, 891-896 (1955).
65. K. Beissner, in Proceedings of the Ultrasonics World Congress 1997, edited by K. Takagi, distributed by S. Ueha (Tokyo Institute of Technology, Yokohama, 1997), pp. 198-199.
66. Grad x to be distinguished from grad x=U and from Grad X=U.
67. Equivalents of Eqs. (52) and (53/54) in a more generalized tensor notation can be found as Eqs. (20.10) and (20.8) in Ref. 15. Equivalents of Eqs. (56) and (63) in a more generalized tensor notation can be found as Eqs. (210.4) and (210.8) in Ref. 15, but here a different nomenclature and the different definition of the stress indices should be taken into account.
68. Strictly speaking, this is one solution but not the only one. Any matrix K′ for which dA·K′=0 is valid can be added to K. However, we are interested here in the resulting force values, and as these other matrices give no force contribution they can be ignored here.
69. The symbols F and H have been adopted from Ref. 16. The symbols K and B stand for "Kirchhoff" and "Boussinesq." The matrix symbols H and B here have nothing to do with the electromagnetic field vectors.
70. In addition, the following may occur in an acoustic streaming: Fluid particles which were close together in the reference state may get more and more distant from one another with time. This leads to the Lagrangian spatial derivatives assuming increasingly higher values and therefore, F and H are not in a steady state.
71. It may be, however, that in the temporal average of Eq. (35), the deviation of v X curl v from zero in an acoustic streaming is only a higher-order effect, but this need not be discussed here.
72. In the older literature the lateral stresses have sometimes been considered important for the "boundary conditions" mentioned in the Introduction. Note, however, that all results obtained here refer to the "relaxed" condition as set out in the discussions in Ref. 19 and that in this context the lateral stresses are unimportant for the radiation force on the target in the cases considered.
73. It might appear that with the velocity potential being in a steady state, its time derivative, the acoustic pressure, should generally by zero on the time average. It should, however, be noted that Eq. (83) (left) is only valid in the first-order approximation and that in fact Eq. (37) [as well as Eq. (73)] does not contain first-order contributions to the time-average acoustic pressure.
74. 0 are unimportant for the radiation force as they either have the form of a curl or vanish in the time average, due to the steady-state condition. The similar applies to ψ and Eq. (96).
75. Cf. Eqs. (A23) and (A26) of Ref. 23 (where, however, the minus signs do not appear as the surface element vector is understood there to point inwards).