LETTERS TO THE EDITOR This Letters section is for publishing (a) brief acoustical research or applied acoustical reports, (b) comments on articles or letters previously published in this Journal, and (c) a reply by the article author to criticism by the Letter author in (b). Extensive reports should be submitted as articles, not in a letter series. Letters are peer-reviewed on the same basis as articles, but usually require less review time before acceptance. Letters cannot exceed four printed pages (approximately 3000–4000 words) including figures, tables, references, and a required abstract of about 100 words.
Acoustic radiation torque on small objects in viscous fluids and connection with viscous dissipation (L) Likun Zhanga) Department of Physics and Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712
Philip L. Marston Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814
(Received 16 June 2014; revised 7 October 2014; accepted 8 October 2014) This analysis reports a formula for torque and viscous power dissipation in scattering of orthogonal waves and vortex beams by a small compressible solid sphere in a slightly viscous fluid. The analysis is based on a viscous correction to far-field scattering, together with beam superposition. The analysis revels the relation between the torque and dissipation. The torque in a heavy sphere limit agrees with a prior analysis by Busse and Wang using boundary flow analysis. The results are applicable to arbitrary sound fields with proper phase distribution, and are extended to other small axiC 2014 Acoustical Society of America. symmetric obstacles such as circular disks and cylinders. V [http://dx.doi.org/10.1121/1.4900441] PACS number(s): 43.25.Qp, 43.25.Uv, 43.25.Nm, 43.35.Ty [OAS] I. INTRODUCTION
There has been interest in acoustically rotating axisymmetric obstacles using either standing orthogonal waves1–6 or vortex beams with an azimuthal phase dependence.7–12 The rotation is due to radiation torque associated with the viscous dissipation in the wavefield. This paper analyzes the viscous dissipation and torque on a compressible obstacle of small size in a slightly viscous fluid due to standing orthogonal waves and vortex beams. Torque produced by two orthogonal standing waves with a phase difference xt0 , p0 ¼ px sinðkxÞ expðixtÞ þ py sinðkyÞ exp½ixðt t0 Þ;
(1)
was analyzed by Busse and Wang,2 where the analysis was based on a solution of flow in the viscous boundary layer. The torque analysis was limited to heavy obstacles. The viscous dissipation and its connection with the torque was not available. Torque exerted by vortex beams with an azimuthal phase dependence expðim/Þ, where the integer m is named as a topological charge,7 was analyzed by Zhang and Marston.9 The analysis on axial torque exerted by a vortex beam p0 ¼ p exp½iðm/ xtÞ
Pages: 2917–2921
with axisymmetric fields p on an axisymmetric object on the axis immersed in an ideal surrounding fluid, indicates that the torque is proportional to the absorption by the object and a factor m=x. Situations where spheres are attracted to the axis of acoustical vortices have been noted by various authors (see, e.g., Refs. 13 and 14). The factor m=x is associated the ratio of angular momentum transport and energy transport.7,9,11 The torque-absorption relation could further give a first approximation for a viscous surrounding fluid, provided that the boundary layer adjacent to the object is small and the acoustic streaming is weak.8 The absorption includes the viscous dissipation in the boundary layer and in the obstacle if the object is absorptive. The absorption can be obtained from a viscous correction to scattering theory.9,15–17 This paper applies prior formulations, together with beam superposition, to analyze the viscous dissipation and torque from (1) and (2) on a small obstacle immersed in a slightly viscous fluid. Section II presents a connection of the vortex beams and two orthogonal waves. The torque analysis is presented in Sec. III, followed by the recovery of the dense sphere limit in Sec. IV. The viscous dissipation and its connection with the torque are analyzed in Secs. V and VI. Extension to general situations of axisymmetrical obstacles and wave fields is made in Sec. VII.
(2) II. VORTEX BEAMS AND ORTHOGONAL WAVES
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 136 (6), December 2014
This section concerns the connection of the orthogonal waves in (1) and the vortex beams in (2). For the interest of acting on an object with jkaj 1 (a is the object size),
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considering only fields in the small jkrj regime (where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 ), the orthogonal waves (1) in the kr 1 limit approximates in cylindrical coordinates to p0 ¼ ðpx cos / þ py sin / expðixt0 ÞÞ 2J1 ðkrÞ expðixtÞ: (3) The terms in this expression can be regrouped as p0 ¼ pþ þ p ; p6 ¼ ðpx 7 ipy expðixt0 ÞÞ J1 ðkrÞ exp½ið6/ xtÞ:
(4)
(5)
where b ¼ 90 gives the standing-wave limit, and pi ¼ px 7 ipy expðixt0 Þ; m ¼ 61:
(6)
The amplitudes of the two Bessel standing waves equal each other when the phase difference is xt0 ¼ 0. For arbitrary phase difference xt0 and amplitude ratio px =py , the two Bessel standing waves generally have different amplitude. The difference is maximized when px ¼ py with xt0 ¼ p=2. For this special case the small-kr approximation (3) reduces to one single vortex Bessel beam, p0 ¼ 2px;y J1 ðkrÞ exp½ið/ xtÞ:
This section applies above beam superposition to analyze the torque from the vortex Bessel beam and the two orthogonal waves on a small, compressible solid sphere. For a sphere placed on the axis of the vortex Bessel beam (5), following from the torque-absorption relation,9 the torque can be written as T ¼ pa2 ðjpi j2 =2qcÞðm=xÞQabs ;
This expression represents the orthogonal waves in the small-kr limit as a superposition of two vortex beams p6 where the topological charge is m ¼ 61 [cf. (2)]. The vortex beam components p6 can be further identified as a standingwave limit of traveling vortex Bessel beams, p0 ¼ pi Jm ðkr sin bÞ expðikz cos b þ im/ ixtÞ;
III. TORQUE ON A COMPRESSIBLE SOLID SPHERE
(7)
The approximation is illustrated in Fig. 1 by a comparison of their amplitude and phase.
(8)
where q the density of surrounding fluid, and c the sound speed in the fluid, and the factor Qabs is the dimensionless absorption efficiency15 Qabs ¼
1 X
ðn mÞ! ðn þ mÞ! ðkaÞ n¼jmj 2 1 jsn j2 ; Pm n ðcos bÞ 1
2
ð2n þ 1Þ
(9)
where Pm n is associated Legendre function and the functions sn are the coefficients of the far-field scattering, consisting with that of plane wave incidence. The case of an ideal sphere immersed in an inviscid fluid gives jsn j ¼ 1 without absorption such that there is no torque T ¼ 0. Since small particles are considered, the first two terms n ¼ 0; 1 of the spherical harmonic decomposition are dominant and hence the series in (9) can be truncated at n ¼ 1. Moreover, as shown by (9), the monopole scattering coefficient does not play a role since the series begin at n ¼ 1 for a spherical particle located on the axis of a vortex of charge one. Consequently, the monopole scattering coefficient relating to the compressibility contrast between particle and ambient fluid is not present. In the low-ka limit, the coefficient of the dipole term (n ¼ 1) is commonly written as17,18 ðs1 1Þ ¼ ði=3ÞðkaÞ3 f2 ;
(10)
where f2 depends on the sphere’s properties. It is known in the inviscid case that f2 ¼ 2ðqi qÞ=ðq þ 2qi Þ, where qi is the object’s density and q the surrounding fluid’s density. For the scattering in a slightly viscous fluid, the obstacle function f2 is corrected by the viscous effect to be a complex value16 f2 ’ 2ðqi qÞ=ðq þ 2qi Þ þ Að1 þ iÞðd=aÞ;
(11)
where A is a function of the density ratio qi =q,
FIG. 1. (Color online) The two orthogonal waves (1) with px ¼ py and xt0 ¼ p=2 in the regime jkrj 1 is approximated by a vortex standing wave (7), as illustrated by the comparison of their normalized amplitude (top) and phase (bottom). The black circle corresponds to kr ¼ 1. The orthogonal waves of arbitrary amplitude and phase in small ka region is approximated by the superposition of two vortex standing waves [cf. Eq. (4)]. 2918
J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
A ¼ 6½ðqi qÞ=ðq þ 2qi Þ2 ; (12) pffiffiffiffiffiffiffiffiffiffiffi and d ¼ 2=x is the thickness of viscous boundary layer (with the kinematic viscosity of the surrounding fluid). Equation (11) has assumed that d=a 1 where the correction is up to the first order of d=a. The correction agrees with a relevant analysis for a heavy sphere.19 Restrictions concerning (11) and (12) for spheres are examined elsewhere.16,19,20 Together (8)–(11), where ½P11 ðcos bÞ2 ¼ sin2 b, gives17 T ¼ Adðjpi j2 =2qc2 Þpa2 sin2 b;
(13)
L. Zhang and P. L. Marston: Letters to the Editor
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which is the torque from the m ¼ 1 Bessel vortex beam (5) on a small compressible solid sphere of finite mass in a slightly viscous fluid. The torque on a compressible solid sphere in a viscous fluid due to the incidence of the two orthogonal waves (1) can be obtained from the torque from the vortex Bessel beam. The torque from the orthogonal waves Torthogonal equals to the summation of the torques produced by the individual incidence of vortex beams pþ and p , denoted by Tþ and T , respectively. This summation follows from the orthognathy of the vortex beams, or could be understood in terms of the conservation of angular momentum. The standing-wave limit of (13), i.e., b ¼ 90 so sin2 b ¼ 1, gives T6 ¼ Ad½ðp2x þ p2y 6 2px py sin xt0 Þ=2qc2 pa2 ;
(14)
where the wave amplitude (6) has been used. It then follows from the summation Torthogonal ¼ Tþ þ T that Torthogonal ¼ Adðpx py =2qc2 Þ4pa2 sinðxt0 Þ;
(15)
which is the radiation torque from the two orthogonal waves on a small compressible solid sphere of finite mass. The balance of this radiation torque with the viscous drag torque17 8pqa3 X on the rotating sphere results in a steady rotation rate, X ¼ Aðd=aÞðpx py =4q2 c2 Þ sinðxt0 Þ:
(16)
The dependence on sphere’s properties is through the coefficient A given in (12). IV. RECOVERY OF TORQUE ON A HEAVY SPHERE
The result (15) for torque from two orthogonal waves on a compressible solid sphere in the limit of qi =q ! 1 reduces to the torque on a heavy sphere, where the coefficient A reduces to A ¼ 3=2 [cf. (12)]. The reduced torque, Torthogonal ¼ ð3=2Þdðpx py =2qc2 Þ4pa2 sinðxt0 Þ;
(17)
agrees with Eq. (30) in Busse and Wang2 using the analysis of boundary flow. The agreement in this limiting case serves as a validation of our analysis using the approach of the viscous correction of the far-field scattering and beam superposition. Our analysis extends Busse and Wang’s analysis to give the torque on a compressible solid sphere of finite mass.
T ¼ ðm=xÞPabs :
(19)
Recall that the standing-wave limit is sin2 b ¼ 1 with b ¼ 90 . The dissipation for the orthogonal waves can be obtained from this result for the vortex Bessel beam. According to the orthogonality of vortex beams, the total dissipation Pabs in the incidence of the orthogonal waves (1) equals to the summation of the dissipation Pabsþ and Pabs– that are associated with the individual incidence of vortex beam components, pþ and p , respectively. Together (18) and (6) gives Pabs6 ¼ Adxpa2 ðp2x þ p2y 6 2px py sin xt0 Þ=2qc2 :
(20)
It follows from Pabs ¼ Pabsþ þ Pabs that Pabs ¼ Adxpa2 ðp2x þ p2y Þ=qc2 ;
(21)
which is the absorption due to the incidence of the two orthogonal waves on a compressible solid sphere of small size in a slightly viscous fluid. This dissipation does not depend on the phase difference xt0 between the two waves. The viscous dissipation is proportional to the energy of the orthogonal waves, the boundary layer thickness d, and an additional scaling of sound frequency x. VI. CONNECTION OF TORQUE AND VISCOUS DISSIPATION
The torque and viscous dissipation in the case of a vortex beam follows the torque-absorption relation (19). For the case of the two orthogonal waves illuminating on a sphere of small ka, the torque expression (21) and dissipation expression (15) give the torque-absorption relation, 2px py sinðxt0 Þ a : (22) Torthogonal ¼ Pabs with a ¼ p2x þ p2y x The dimensionless factor a is a parameter of the two orthogonal waves. It can be read as an effective topological charge of the two orthogonal waves. The effective topological charge a as a function of amplitude ratio px =py and phase difference xt0 is illustrated in Fig. 2. It has jaj 1, and only when
V. VISCOUS DISSIPATION
The analysis using the viscous correction to the far-field scattering allows us to give an analytical solution for the viscous dissipation associated with the torque. The dissipation due to the incidence of the m ¼ 1 Bessel beam on a compressible solid sphere of small size in a slightly viscous fluid is Pabs ¼ Apa2 dxðjpi j2 =2qc2 Þ sin2 b;
(18)
which follows from the torque expression (13) and the torque-absorption relation,9 J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
FIG. 2. (Color online) The dimensionless factor a in the torque-dissipation relation [cf. (22)] as a function of amplitude ratio px =py and phase difference xt0 characterizes the efficiency of torque for given viscous dissipation. The maximum efficiency occurs when px =py ¼ 1 and xt0 ¼ 6p=2 (black crosses). The small and large px =py values approach to the limit of a plane standing wave. L. Zhang and P. L. Marston: Letters to the Editor
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sinðxt0 Þ ¼ 61 and px ¼ py does jaj ¼ 1. In other words, for given dissipation, the torque is most efficient when the two waves are out of 90 phase and have identical amplitude. This corresponds to the situation that the amplitude difference of the two vortex beam components p6 is maximized. The torque is zero when xt0 ¼ 0 corresponding to the situation that the two components p6 have the same amplitude. In that case, the two vortices are coaxial, equal amplitude, and opposite charges. We know that two such vortices are not stable and cancel each other. Hence, the beam should not carry any angular momentum. It is easy to check with (4) that the interference of the two beams cancel the azimuthal dependence of the phase, i.e., only its real amplitude depends on /. VII. APPLICABILITY TO GENERAL SITUATIONS
This section considers the applicability of the above analysis to general situations of orthogonal waves and axisymmetrical obstacles of other shapes. The torque from two orthogonal waves on arbitrary shapes of axisymmetric objects still follows the orthogonality argument, Torthogonal ¼ Tþ þ T , where T6 ¼ ð61=xÞPabs6 according to the torque-power relation. In terms of a proportionality with the absorption, Torthogonal ¼ ða=xÞPabs , it has a ¼ ðPabsþ Pabs Þ=Pabs . Even the dissipation here is in general unknown for other shapes of objects, the coefficient a still follows (22) by recalling the dissipation orthogonality, Pabs ¼ Pabsþ þ Pabs , and noticing the proportionality of dissipation Pabs6 with beam intensity, i.e., Pabs6 / jp6 j2 , where p6 is given in (4). Therefore, the torque-absorption relation (22) for a sphere is applicable to any axisymmetrical object of small size when illuminated by the two orthogonal waves. The relation (22) can be used to obtain the dissipation in the scattering of two orthogonal waves by a circular disk and cylinder of small size, provided that the corresponding torque was given in the analysis of mean boundary flow.2 The torque and resulting dissipation are identical to (15) and (21), respectively, where a is the radius of the disk and cylinder (ka 1), and the coefficient is replaced by Adisk ¼ ðka=4Þ2 ; Acylinder ¼ 1=2:
(23)
Because when px ¼ py and xt0 ¼ p=2 the two orthogonal waves in small-kr region is approximated by a vortex Bessel standing wave, the replacement of coefficient A is also applicable to (13) and (18) for the case when the cylinder and disk of small ka is illuminated by a vortex Bessel standing wave. VIII. SUMMARY AND DISCUSSION
This paper has analyzed the torque on a compressible solid sphere of small size produced by vortex beams and standing orthogonal waves in viscous fluids and the connection with the viscous dissipation. The results are (15) and (21) for the orthogonal waves in (1), and (13) and (18) for the m ¼ 1 vortex Bessel beam in (5). The coefficient A given in (12) contains the material dependence. The torque relates to the dissipation through a linear proportionality (22) where the scale factor is a function of the phase difference and the 2920
J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
amplitude ratio of the two orthogonal waves. For a given dissipation, the torque is most efficient when the two waves are out of phase 90 and have identical amplitude. The viscous dissipation is independent on the phase difference. The analysis uses prior formulations9,15 for the firstorder vortex beams (m ¼ 1) where the viscous correction to the dipole scattering coefficient is introduced.16 Notice that because the absorption is taken to be dominated by the dipole field when ka is small [cf. (9)], the torque does not depend explicitly on the compressibility in this approximation. In the limit of a heavy solid sphere, the torque from the orthogonal waves agrees with prior analysis using boundary mean flow.2 Some generalizations from combining the approaches of viscous correction and boundary mean flow are examined for other axisymmetrical obstacles such as a heavy cylinder and disk where the coefficient A is replaced by (23). The results can be used for rotation rate prediction in acoustophoresis. The results are applicable to arbitrary standing waves or vortex beams with proper phase distribution because at small-kr all fields have the same form as first-order vortex wave fields,21 where the profile approaches to first-order Bessel function at small-kr region. ACKNOWLEDGMENTS
L.Z. acknowledges the support of the 2013–2014 F. V. Hunt Postdoctoral Research Fellowship in Acoustics. P.L.M. acknowledges support from ONR. 1
T. G. Wang, H. Kanber, and I. Rudnick, “First-order torques and solidbody spinning velocities in intense sound fields,” Phys. Rev. Lett. 38, 128–130 (1977). 2 F. H. Busse and T. G. Wang, “Torque generated by orthogonal acoustic waves—Theory,” J. Acoust. Soc. Am. 69, 1634–1638 (1981). 3 T. G. Wang, E. H. Trinh, A. P. Croonquist, and D. D. Elleman, “Shapes of rotating free drops: Spacelab experimental results,” Phys. Rev. Lett. 56, 452–455 (1986). 4 K. Ohsaka and E. H. Trinh, “Three-lobed shape bifurcation of rotating liquid drops,” Phys. Rev. Lett. 84, 1700–1703 (2000). 5 J. T. Wang and J. Dual, “Theoretical and numerical calculations for the timeaveraged acoustic force and torque acting on a rigid cylinder of arbitrary size in a low viscosity fluid,” J. Acoust. Soc. Am. 129, 3490–3501 (2011). 6 D. Foresti and D. Poulikakos, “Acoustophoretic contactless elevation, orbital transport and spinning of matter in air,” Phys. Rev. Lett. 112, 024301 (2014). 7 B. T. Hefner and P. L. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999). 8 L. K. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum (L),” J. Acoust. Soc. Am. 129, 1679–1680 (2011). 9 L. K. Zhang and P. L. Marston, “Angular momentum flux of nonparaxial acoustic vortex beams and torques on axisymmetric objects,” Phys. Rev. E 84, 065601 (2011). 10 K. Volke-Sep ulveda, A. O. Santillan, and R. R. Boullosa, “Transfer of angular momentum to matter from acoustical vortices in free space,” Phys. Rev. Lett. 100, 024302 (2008). 11 C. E. M. Demore, Z. Yang, A. Volovick, S. Cochran, M. P. MacDonald, and G. C. Spalding, “Mechanical evidence of the orbital angular momentum to energy ratio of vortex beams,” Phys. Rev. Lett. 108, 194301 (2012). 12 A. Anhauser, R. Wunenburger, and E. Brasselet, “Acoustic rotational manipulation using orbital angular momentum transfer,” Phys. Rev. Lett. 109, 034301 (2012). 13 D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam tweezers,” J. Appl. Phys. 113, 184901 (2013). 14 C. R. P. Courtney, B. W. Drinkwater, C. E. M. Demore, S. Cochran, A. Grinenko, and P. D. Wilcox, “Dexterous manipulation of microparticles L. Zhang and P. L. Marston: Letters to the Editor
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using Bessel-function acoustic pressure fields,” Appl. Phys. Lett. 102, 123508 (2013). 15 L. K. Zhang and P. L. Marston, “Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres,” Phys. Rev. E 84, 035601 (2011). 16 M. Settnes and H. Bruus, “Forces acting on a small particle in an acoustical field in a viscous fluid,” Phys. Rev. E 85, 016327 (2012). 17 P. L. Marston, “Viscous contributions to low-frequency scattering, power absorption, radiation force, and radiation torque for spheres in acoustic beams,” Proc. Meet. Acoust. 19, 045005 (2013).
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18
L. P. Gorkov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl. 6, 773–775 (1962). 19 A. A. Doinikov, “Acoustic radiation force on a spherical particle in a viscous heat-conducting fluid. II. force on a rigid sphere,” J. Acoust. Soc. Am. 101, 722–730 (1997). 20 S. D. Danilov and M. A. Mironov, “Mean force on a small sphere in a sound field in a viscous fluid,” J. Acoust. Soc. Am. 107, 143–153 (2000). 21 J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
L. Zhang and P. L. Marston: Letters to the Editor
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Acoustic radiation torque on small objects in viscous fluids and connection with viscous dissipation (L) Likun Zhanga) Department of Physics and Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712
Philip L. Marston Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814
(Received 16 June 2014; revised 7 October 2014; accepted 8 October 2014) This analysis reports a formula for torque and viscous power dissipation in scattering of orthogonal waves and vortex beams by a small compressible solid sphere in a slightly viscous fluid. The analysis is based on a viscous correction to far-field scattering, together with beam superposition. The analysis revels the relation between the torque and dissipation. The torque in a heavy sphere limit agrees with a prior analysis by Busse and Wang using boundary flow analysis. The results are applicable to arbitrary sound fields with proper phase distribution, and are extended to other small axiC 2014 Acoustical Society of America. symmetric obstacles such as circular disks and cylinders. V [http://dx.doi.org/10.1121/1.4900441] PACS number(s): 43.25.Qp, 43.25.Uv, 43.25.Nm, 43.35.Ty [OAS] I. INTRODUCTION
There has been interest in acoustically rotating axisymmetric obstacles using either standing orthogonal waves1–6 or vortex beams with an azimuthal phase dependence.7–12 The rotation is due to radiation torque associated with the viscous dissipation in the wavefield. This paper analyzes the viscous dissipation and torque on a compressible obstacle of small size in a slightly viscous fluid due to standing orthogonal waves and vortex beams. Torque produced by two orthogonal standing waves with a phase difference xt0 , p0 ¼ px sinðkxÞ expðixtÞ þ py sinðkyÞ exp½ixðt t0 Þ;
(1)
was analyzed by Busse and Wang,2 where the analysis was based on a solution of flow in the viscous boundary layer. The torque analysis was limited to heavy obstacles. The viscous dissipation and its connection with the torque was not available. Torque exerted by vortex beams with an azimuthal phase dependence expðim/Þ, where the integer m is named as a topological charge,7 was analyzed by Zhang and Marston.9 The analysis on axial torque exerted by a vortex beam p0 ¼ p exp½iðm/ xtÞ
Pages: 2917–2921
with axisymmetric fields p on an axisymmetric object on the axis immersed in an ideal surrounding fluid, indicates that the torque is proportional to the absorption by the object and a factor m=x. Situations where spheres are attracted to the axis of acoustical vortices have been noted by various authors (see, e.g., Refs. 13 and 14). The factor m=x is associated the ratio of angular momentum transport and energy transport.7,9,11 The torque-absorption relation could further give a first approximation for a viscous surrounding fluid, provided that the boundary layer adjacent to the object is small and the acoustic streaming is weak.8 The absorption includes the viscous dissipation in the boundary layer and in the obstacle if the object is absorptive. The absorption can be obtained from a viscous correction to scattering theory.9,15–17 This paper applies prior formulations, together with beam superposition, to analyze the viscous dissipation and torque from (1) and (2) on a small obstacle immersed in a slightly viscous fluid. Section II presents a connection of the vortex beams and two orthogonal waves. The torque analysis is presented in Sec. III, followed by the recovery of the dense sphere limit in Sec. IV. The viscous dissipation and its connection with the torque are analyzed in Secs. V and VI. Extension to general situations of axisymmetrical obstacles and wave fields is made in Sec. VII.
(2) II. VORTEX BEAMS AND ORTHOGONAL WAVES
a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 136 (6), December 2014
This section concerns the connection of the orthogonal waves in (1) and the vortex beams in (2). For the interest of acting on an object with jkaj 1 (a is the object size),
0001-4966/2014/136(6)/2917/5/$30.00
C 2014 Acoustical Society of America V
2917
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.83.63.20 On: Thu, 04 Dec 2014 23:59:38
considering only fields in the small jkrj regime (where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þ y2 ), the orthogonal waves (1) in the kr 1 limit approximates in cylindrical coordinates to p0 ¼ ðpx cos / þ py sin / expðixt0 ÞÞ 2J1 ðkrÞ expðixtÞ: (3) The terms in this expression can be regrouped as p0 ¼ pþ þ p ; p6 ¼ ðpx 7 ipy expðixt0 ÞÞ J1 ðkrÞ exp½ið6/ xtÞ:
(4)
(5)
where b ¼ 90 gives the standing-wave limit, and pi ¼ px 7 ipy expðixt0 Þ; m ¼ 61:
(6)
The amplitudes of the two Bessel standing waves equal each other when the phase difference is xt0 ¼ 0. For arbitrary phase difference xt0 and amplitude ratio px =py , the two Bessel standing waves generally have different amplitude. The difference is maximized when px ¼ py with xt0 ¼ p=2. For this special case the small-kr approximation (3) reduces to one single vortex Bessel beam, p0 ¼ 2px;y J1 ðkrÞ exp½ið/ xtÞ:
This section applies above beam superposition to analyze the torque from the vortex Bessel beam and the two orthogonal waves on a small, compressible solid sphere. For a sphere placed on the axis of the vortex Bessel beam (5), following from the torque-absorption relation,9 the torque can be written as T ¼ pa2 ðjpi j2 =2qcÞðm=xÞQabs ;
This expression represents the orthogonal waves in the small-kr limit as a superposition of two vortex beams p6 where the topological charge is m ¼ 61 [cf. (2)]. The vortex beam components p6 can be further identified as a standingwave limit of traveling vortex Bessel beams, p0 ¼ pi Jm ðkr sin bÞ expðikz cos b þ im/ ixtÞ;
III. TORQUE ON A COMPRESSIBLE SOLID SPHERE
(7)
The approximation is illustrated in Fig. 1 by a comparison of their amplitude and phase.
(8)
where q the density of surrounding fluid, and c the sound speed in the fluid, and the factor Qabs is the dimensionless absorption efficiency15 Qabs ¼
1 X
ðn mÞ! ðn þ mÞ! ðkaÞ n¼jmj 2 1 jsn j2 ; Pm n ðcos bÞ 1
2
ð2n þ 1Þ
(9)
where Pm n is associated Legendre function and the functions sn are the coefficients of the far-field scattering, consisting with that of plane wave incidence. The case of an ideal sphere immersed in an inviscid fluid gives jsn j ¼ 1 without absorption such that there is no torque T ¼ 0. Since small particles are considered, the first two terms n ¼ 0; 1 of the spherical harmonic decomposition are dominant and hence the series in (9) can be truncated at n ¼ 1. Moreover, as shown by (9), the monopole scattering coefficient does not play a role since the series begin at n ¼ 1 for a spherical particle located on the axis of a vortex of charge one. Consequently, the monopole scattering coefficient relating to the compressibility contrast between particle and ambient fluid is not present. In the low-ka limit, the coefficient of the dipole term (n ¼ 1) is commonly written as17,18 ðs1 1Þ ¼ ði=3ÞðkaÞ3 f2 ;
(10)
where f2 depends on the sphere’s properties. It is known in the inviscid case that f2 ¼ 2ðqi qÞ=ðq þ 2qi Þ, where qi is the object’s density and q the surrounding fluid’s density. For the scattering in a slightly viscous fluid, the obstacle function f2 is corrected by the viscous effect to be a complex value16 f2 ’ 2ðqi qÞ=ðq þ 2qi Þ þ Að1 þ iÞðd=aÞ;
(11)
where A is a function of the density ratio qi =q,
FIG. 1. (Color online) The two orthogonal waves (1) with px ¼ py and xt0 ¼ p=2 in the regime jkrj 1 is approximated by a vortex standing wave (7), as illustrated by the comparison of their normalized amplitude (top) and phase (bottom). The black circle corresponds to kr ¼ 1. The orthogonal waves of arbitrary amplitude and phase in small ka region is approximated by the superposition of two vortex standing waves [cf. Eq. (4)]. 2918
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A ¼ 6½ðqi qÞ=ðq þ 2qi Þ2 ; (12) pffiffiffiffiffiffiffiffiffiffiffi and d ¼ 2=x is the thickness of viscous boundary layer (with the kinematic viscosity of the surrounding fluid). Equation (11) has assumed that d=a 1 where the correction is up to the first order of d=a. The correction agrees with a relevant analysis for a heavy sphere.19 Restrictions concerning (11) and (12) for spheres are examined elsewhere.16,19,20 Together (8)–(11), where ½P11 ðcos bÞ2 ¼ sin2 b, gives17 T ¼ Adðjpi j2 =2qc2 Þpa2 sin2 b;
(13)
L. Zhang and P. L. Marston: Letters to the Editor
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which is the torque from the m ¼ 1 Bessel vortex beam (5) on a small compressible solid sphere of finite mass in a slightly viscous fluid. The torque on a compressible solid sphere in a viscous fluid due to the incidence of the two orthogonal waves (1) can be obtained from the torque from the vortex Bessel beam. The torque from the orthogonal waves Torthogonal equals to the summation of the torques produced by the individual incidence of vortex beams pþ and p , denoted by Tþ and T , respectively. This summation follows from the orthognathy of the vortex beams, or could be understood in terms of the conservation of angular momentum. The standing-wave limit of (13), i.e., b ¼ 90 so sin2 b ¼ 1, gives T6 ¼ Ad½ðp2x þ p2y 6 2px py sin xt0 Þ=2qc2 pa2 ;
(14)
where the wave amplitude (6) has been used. It then follows from the summation Torthogonal ¼ Tþ þ T that Torthogonal ¼ Adðpx py =2qc2 Þ4pa2 sinðxt0 Þ;
(15)
which is the radiation torque from the two orthogonal waves on a small compressible solid sphere of finite mass. The balance of this radiation torque with the viscous drag torque17 8pqa3 X on the rotating sphere results in a steady rotation rate, X ¼ Aðd=aÞðpx py =4q2 c2 Þ sinðxt0 Þ:
(16)
The dependence on sphere’s properties is through the coefficient A given in (12). IV. RECOVERY OF TORQUE ON A HEAVY SPHERE
The result (15) for torque from two orthogonal waves on a compressible solid sphere in the limit of qi =q ! 1 reduces to the torque on a heavy sphere, where the coefficient A reduces to A ¼ 3=2 [cf. (12)]. The reduced torque, Torthogonal ¼ ð3=2Þdðpx py =2qc2 Þ4pa2 sinðxt0 Þ;
(17)
agrees with Eq. (30) in Busse and Wang2 using the analysis of boundary flow. The agreement in this limiting case serves as a validation of our analysis using the approach of the viscous correction of the far-field scattering and beam superposition. Our analysis extends Busse and Wang’s analysis to give the torque on a compressible solid sphere of finite mass.
T ¼ ðm=xÞPabs :
(19)
Recall that the standing-wave limit is sin2 b ¼ 1 with b ¼ 90 . The dissipation for the orthogonal waves can be obtained from this result for the vortex Bessel beam. According to the orthogonality of vortex beams, the total dissipation Pabs in the incidence of the orthogonal waves (1) equals to the summation of the dissipation Pabsþ and Pabs– that are associated with the individual incidence of vortex beam components, pþ and p , respectively. Together (18) and (6) gives Pabs6 ¼ Adxpa2 ðp2x þ p2y 6 2px py sin xt0 Þ=2qc2 :
(20)
It follows from Pabs ¼ Pabsþ þ Pabs that Pabs ¼ Adxpa2 ðp2x þ p2y Þ=qc2 ;
(21)
which is the absorption due to the incidence of the two orthogonal waves on a compressible solid sphere of small size in a slightly viscous fluid. This dissipation does not depend on the phase difference xt0 between the two waves. The viscous dissipation is proportional to the energy of the orthogonal waves, the boundary layer thickness d, and an additional scaling of sound frequency x. VI. CONNECTION OF TORQUE AND VISCOUS DISSIPATION
The torque and viscous dissipation in the case of a vortex beam follows the torque-absorption relation (19). For the case of the two orthogonal waves illuminating on a sphere of small ka, the torque expression (21) and dissipation expression (15) give the torque-absorption relation, 2px py sinðxt0 Þ a : (22) Torthogonal ¼ Pabs with a ¼ p2x þ p2y x The dimensionless factor a is a parameter of the two orthogonal waves. It can be read as an effective topological charge of the two orthogonal waves. The effective topological charge a as a function of amplitude ratio px =py and phase difference xt0 is illustrated in Fig. 2. It has jaj 1, and only when
V. VISCOUS DISSIPATION
The analysis using the viscous correction to the far-field scattering allows us to give an analytical solution for the viscous dissipation associated with the torque. The dissipation due to the incidence of the m ¼ 1 Bessel beam on a compressible solid sphere of small size in a slightly viscous fluid is Pabs ¼ Apa2 dxðjpi j2 =2qc2 Þ sin2 b;
(18)
which follows from the torque expression (13) and the torque-absorption relation,9 J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
FIG. 2. (Color online) The dimensionless factor a in the torque-dissipation relation [cf. (22)] as a function of amplitude ratio px =py and phase difference xt0 characterizes the efficiency of torque for given viscous dissipation. The maximum efficiency occurs when px =py ¼ 1 and xt0 ¼ 6p=2 (black crosses). The small and large px =py values approach to the limit of a plane standing wave. L. Zhang and P. L. Marston: Letters to the Editor
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sinðxt0 Þ ¼ 61 and px ¼ py does jaj ¼ 1. In other words, for given dissipation, the torque is most efficient when the two waves are out of 90 phase and have identical amplitude. This corresponds to the situation that the amplitude difference of the two vortex beam components p6 is maximized. The torque is zero when xt0 ¼ 0 corresponding to the situation that the two components p6 have the same amplitude. In that case, the two vortices are coaxial, equal amplitude, and opposite charges. We know that two such vortices are not stable and cancel each other. Hence, the beam should not carry any angular momentum. It is easy to check with (4) that the interference of the two beams cancel the azimuthal dependence of the phase, i.e., only its real amplitude depends on /. VII. APPLICABILITY TO GENERAL SITUATIONS
This section considers the applicability of the above analysis to general situations of orthogonal waves and axisymmetrical obstacles of other shapes. The torque from two orthogonal waves on arbitrary shapes of axisymmetric objects still follows the orthogonality argument, Torthogonal ¼ Tþ þ T , where T6 ¼ ð61=xÞPabs6 according to the torque-power relation. In terms of a proportionality with the absorption, Torthogonal ¼ ða=xÞPabs , it has a ¼ ðPabsþ Pabs Þ=Pabs . Even the dissipation here is in general unknown for other shapes of objects, the coefficient a still follows (22) by recalling the dissipation orthogonality, Pabs ¼ Pabsþ þ Pabs , and noticing the proportionality of dissipation Pabs6 with beam intensity, i.e., Pabs6 / jp6 j2 , where p6 is given in (4). Therefore, the torque-absorption relation (22) for a sphere is applicable to any axisymmetrical object of small size when illuminated by the two orthogonal waves. The relation (22) can be used to obtain the dissipation in the scattering of two orthogonal waves by a circular disk and cylinder of small size, provided that the corresponding torque was given in the analysis of mean boundary flow.2 The torque and resulting dissipation are identical to (15) and (21), respectively, where a is the radius of the disk and cylinder (ka 1), and the coefficient is replaced by Adisk ¼ ðka=4Þ2 ; Acylinder ¼ 1=2:
(23)
Because when px ¼ py and xt0 ¼ p=2 the two orthogonal waves in small-kr region is approximated by a vortex Bessel standing wave, the replacement of coefficient A is also applicable to (13) and (18) for the case when the cylinder and disk of small ka is illuminated by a vortex Bessel standing wave. VIII. SUMMARY AND DISCUSSION
This paper has analyzed the torque on a compressible solid sphere of small size produced by vortex beams and standing orthogonal waves in viscous fluids and the connection with the viscous dissipation. The results are (15) and (21) for the orthogonal waves in (1), and (13) and (18) for the m ¼ 1 vortex Bessel beam in (5). The coefficient A given in (12) contains the material dependence. The torque relates to the dissipation through a linear proportionality (22) where the scale factor is a function of the phase difference and the 2920
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amplitude ratio of the two orthogonal waves. For a given dissipation, the torque is most efficient when the two waves are out of phase 90 and have identical amplitude. The viscous dissipation is independent on the phase difference. The analysis uses prior formulations9,15 for the firstorder vortex beams (m ¼ 1) where the viscous correction to the dipole scattering coefficient is introduced.16 Notice that because the absorption is taken to be dominated by the dipole field when ka is small [cf. (9)], the torque does not depend explicitly on the compressibility in this approximation. In the limit of a heavy solid sphere, the torque from the orthogonal waves agrees with prior analysis using boundary mean flow.2 Some generalizations from combining the approaches of viscous correction and boundary mean flow are examined for other axisymmetrical obstacles such as a heavy cylinder and disk where the coefficient A is replaced by (23). The results can be used for rotation rate prediction in acoustophoresis. The results are applicable to arbitrary standing waves or vortex beams with proper phase distribution because at small-kr all fields have the same form as first-order vortex wave fields,21 where the profile approaches to first-order Bessel function at small-kr region. ACKNOWLEDGMENTS
L.Z. acknowledges the support of the 2013–2014 F. V. Hunt Postdoctoral Research Fellowship in Acoustics. P.L.M. acknowledges support from ONR. 1
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using Bessel-function acoustic pressure fields,” Appl. Phys. Lett. 102, 123508 (2013). 15 L. K. Zhang and P. L. Marston, “Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres,” Phys. Rev. E 84, 035601 (2011). 16 M. Settnes and H. Bruus, “Forces acting on a small particle in an acoustical field in a viscous fluid,” Phys. Rev. E 85, 016327 (2012). 17 P. L. Marston, “Viscous contributions to low-frequency scattering, power absorption, radiation force, and radiation torque for spheres in acoustic beams,” Proc. Meet. Acoust. 19, 045005 (2013).
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L. P. Gorkov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl. 6, 773–775 (1962). 19 A. A. Doinikov, “Acoustic radiation force on a spherical particle in a viscous heat-conducting fluid. II. force on a rigid sphere,” J. Acoust. Soc. Am. 101, 722–730 (1997). 20 S. D. Danilov and M. A. Mironov, “Mean force on a small sphere in a sound field in a viscous fluid,” J. Acoust. Soc. Am. 107, 143–153 (2000). 21 J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
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