Acoustic Radiation Force and Torque on a Viscous Fluid Cylindrical Particle Nearby a Planar Rigid Wall

arXiv (2017)

Acoustic radiation force and torque on a viscous fluid cylindrical particle nearby a planar rigid wall

F.G. Mitria) Chevron, Area 52 Technology – ETC, Santa Fe, New Mexico 87508, USA

Abstract

This work presents a comprehensive analytical formalism for the modal series expansions of the acoustic radiation force and radiation torque experienced by a fluid viscous cylindrical object of arbitrary geometrical cross-section placed near a planar rigid wall. A plane progressive wave field with an arbitrary angle of incidence propagating in an inviscid fluid is assumed. The formalism developed here utilizes the features of the modal expansion method with boundary matching in cylindrical coordinates, the method of images and the translational addition theorem. Initially, an effective field incident on the particle (including the principal incident field, the reflected wave-field from the boundary and the scattered field from the image object) is defined. Subsequently, the incident effective field is utilized in conjunction with the scattered one from the object, to obtain closed-form expansions for the longitudinal and transversal force components, in addition to the axial torque component, based on the scattering in the far-field. The obtained solutions involve the angle of incidence, the modal coefficients of the scatterer and its image, and the particle-wall distance. Computations for the non-dimensional force and torque functions for a viscous fluid circular cylindrical cross-section are considered, and calculations exemplify the analysis with emphasis on changing the particle size, the particle-wall distance and the angle of incidence. It is found that the object experiences either an attractive or repulsive force directed toward or away from the boundary. Moreover, it reverses its rotation around its center of mass depending on its size, particle-wall distance and angle of incidence. Furthermore, radiation force and torque singularities occur (i.e., zero force and torque) such that the cylinder becomes irresponsive to the linear and angular momenta transfer yielded by the effective field. The results lead to an improved understanding of the force and radiation torque behaviors for a viscous particle in the field of plane progressive waves nearby a boundary, which is a commonly encountered situation in acoustofluidics, contrast agents in biomedicine, and fluid dynamics to name a few examples. The exact formalism presented here using the multipole expansion method, which is valid for any frequency range, could be used to validate other approaches using purely numerical methods, or frequency-limiting approximate models such as the Rayleigh and the Kirchhoff regimes.

Keywords: Radiation force; radiation torque; absorption; multiple-scattering; boundary; plane progressive waves.

1. Introduction which is yet to be developed in the scientific literature. Particularly, the rigorous formalism accounts for the multiple When evaluating the radiation force and radiation torque on reflections/scattering effects occurring between the particle a particle located nearby a boundary 1 or a chamber wall 2, it is and the rigid boundary by means of the modal expansion crucial to take into account the multiple interactions, i.e. method in cylindrical coordinates 19, the method of images 20,21 scatterings 3 and reflections. Therefore, a need to predict and the translational addition theorem 22. numerically the acoustic forces 1,2,4 and torque accurately near The procedure to obtain the acoustic radiation force and a chamber walls arises. This effort has been originally radiation torque consists on the integration of the time- presented in 1 for an elastic spherical contrast agent shell averaged Brillouin’s stress tensor 23,24 and its moment 25,26, nearby a porous flat boundary. respectively. The procedure uses an analysis of the scattering Several investigations were previously developed for a in the far-field, without any approximation in the evaluation of cylindrical particle in an unbounded fluid 5-14, where some the physical observables. An effective acoustic velocity recent analyses (not limited to a particular range of potential field incident on the particle is defined, which frequencies) considered the elliptical geometry 15-18. includes the primary incident field, the reflected waves from Nonetheless, those formalisms cannot be applied to a the boundary as well as the scattered field from the image cylindrical particle nearby a boundary, and it is important to object, is utilized with the scattered field from the object to develop an improved methodology taking the multiple obtain the radiation force and radiation torque functions. The reflections/scattering effects between the wall and the particle present investigation is substantiated by numerical into account. computations for a lossy fluid cylindrical particle of circular The aim of this investigation is therefore directed toward the cross-section illuminated by plane travelling waves with development of a novel analytical formalism for the acoustic arbitrary incidence. The numerical examples are chosen to radiation force and torque on a particle of arbitrary illustrate possible practical scenarios with emphases on the geometrical cross-section (in 2D) valid for any scattering distance of the particle from the boundary, the dimensionless regime (i.e., Rayleigh, Mie or geometrical/ray acoustics), size parameter, and the incidence angle of the primary waves.

a) Electronic mail: [email protected] 1

arXiv (2017)

wave-vector and r is the vector position. In the cylindrical system of coordinates (r,), Eq.(1) is expressed as

it ikr cos  incr,, t 0 e e  (2)   ei t i n e in  J kr e in  , 0  n   n

where k = /c is the wavenumber, c is the speed of sound in

the medium of wave propagation, and Jn  is the cylindrical Bessel function of the first kind. Due to the existence of the boundary, reflection of the incident (primary) field occurs, which is in turn incident on the

fluid object. The velocity potential field for the reflected FIG. 1. Sketch representing a cylindrical fluid object of arbitrary cross- waves, incident upon the fluid object, is expressed as section (in 2D) submerged in an unbounded non-viscous fluid. Its center of mass is located at a distance d from a rigid planar wall. Acoustic plane i tikr cos    2 ikd cos progressive wave illumination with an arbitrary incidence angle  is R0r,, t e e e assumed. The rotating arrow on the fluid object surface represents the induced spinning (counter-clockwise or clockwise) caused by the axial z-  (3) i t2 ikd cos  nin in  component of acoustic radiation torque vector.  0e e i e Jn  kr e ,  n The mathematical model and its related physical results may be important for applications involving the numerical where d is the distance from the center of mass of the particle predictions of the acoustic radiation force and radiation torque to the boundary along the direction  = 0 (Fig. 1). The scattered velocity potential field resulting from the induced by plane waves of arbitrary incidence on a particle interaction of the primary incident plane waves with the object located nearby a boundary, using exact partial-wave series expansions (PWSEs). The results can assist in validating is expressed as, numerical tools using the finite-element or boundary-element 27,28  methods (FEM/BEM) , and other techniques. Moreover, object i t1 in r,,, t e C H kr e (4) the 2D cylindrical particle geometry serves as a benchmark sca  0  nn  solution where the anticipated physical effects occurring in 2D n are also expected to exist for a 3D particle close to a boundary, and the present model should assist to further develop an where H 1  is the cylindrical Hankel function of the first improved formalism for the radiation force and radiation n 1 torque on a spherical particle located nearby a wall with kind of order n, and Cn is the modal coefficient to be arbitrary incidence. determined subsequently. Since the object is sound penetrable, the internal velocity potential field for the compressional waves propagating in its 2. Method core is written as

2.1. Plane waves scattering by a fluid cylindrical  particle of arbitrary geometry in 2D near a rigid flat objectr,,, t  ei t C int J  r e in (5) int  0  n n L  boundary n

Consider a particle with arbitrary geometrical cross-section, where  is the wavenumber corresponding to the subjected to a continuous-wave field of harmonic plane L travelling waves propagating in an inviscid fluid. The incident longitudinal/compressional waves propagating inside the core wave-field has an angle of incidence  with respect to the fluid material. For a viscous fluid medium, is a complex horizontal plane as shown in Fig. 1. The expression for the int incident velocity potential is given as number, accounting for sound absorption. Cn is the expansion coefficient for the internal waves.  eitkr , (1) The method of images is now applied so that the scattering inc 0 from the flat rigid boundary can be replaced by the scattering from the image object (mirrored by the wall). Therefore, the where 0 is the amplitude,  is the angular frequency, k is the scattered velocity potential contributed by the image object is 2

arXiv (2017) given as where,

 image imagei t1 in ' sca rt,,  r', ', t e D H kr ' e , (6) rd 2 sca  0  nn  n   (11)   ei t D H1 2. kd J kr e in 0 m n m  n   nm  where Dn is the modal coefficient of the image object. In addition, the internal velocity potential field internal to the image object is expressed as Following the same procedure, the expansion series for the total velocity potential field in the primed system of  coordinates is obtained as,

imager', ', t  ei t D int J  r ' e in ' , (7) int 0  n n L  rt', ', n tot   rd'2 int where Dn is the expansion coefficient for the internal waves  incr', ', t   R  r ', ', t (12) of the image object. Eqs.(4)-(7) are valid for any particle of arbitrary geometry object image scar', ', t   sca  r ', ', t , (in 2D) located nearby a flat rigid boundary and its image. The rd'2 expansion series (written in terms of cylindrical wave functions) were utilized earlier in investigations dealing with where, elastic wave 29-31 and acoustic scattering 15,17,32,33 using the T-  matrix 34. (Other discussions for non-circular cylinders can be r', ', t ei t i n ein J kr ' e in ' , (13) 15-18,35,36 inc 0  n found in ). n Before applying the appropriate boundary conditions to obtain the modal coefficients, the expressions for the velocity  potential fields must be modified such that they are all referred i t2 ikd cos  n in  in  ' r', ', t e e i e J kr ' e , (14) to the same coordinates system. The adequate boundary R0   n   conditions of continuity must be satisfied, either at the planar n rigid boundary surface (which were used previously in the and, context of the multiple acoustic scattering 37,38 from a rigid boundary 39,40), or at the surface of the particle and its   objecti t1 in ' 41 rt', ', e CH 2 kdJkre  ' . corresponding image in both systems of coordinates . Both scard'2 0 m m n n methods are commensurate with the same result. Using the nm  latter approach, the translational addition theorem is used to (15) express the corresponding PWSE in a particular system as a series in the other system. Since the particle is a sound penetrable fluid, the following Near the object and its image, the following expressions conditions of continuity should be imposed in both systems of hold 42, coordinates (i.e. object + image), such that,

 o r,,,,, t  object r t (16) 11 in'   im tot  rA int   H kr' e J kr H 2 kd e , r 2 d , (8)  rA  n m m n m  whereas, o 11object (17) totr,,,,, t nn   int  r t   rA  f rA    11 H   kr ein J kr' H   2 kd e im ' , r ' 2 d . (9) n m n m o image (18) m totr', ', t   int  r ', ', t , rA'  rA' 

Subsequently, in the fluid host medium surrounding the o 11 image (19) object, the total velocity potential is written as, totr', ', t nn   int  r ', ', t  ,  rA'   f rA' 

tot rt,,  rd 2 where  and f are the mass densities of the host fluid medium and fluid particle, respectively. The parameters n and A are  incr,,,, t   R  r t (10) the normal vector and the surface shape function of the 15-18 objectr,,,,, t   image  r t particle of arbitrary geometry , respectively. sca sca rd 2

arXiv (2017)

2.2. Radiation force and radiation torque expressions

The series expansions for the longitudinal and transversal vector force components are attained after defining an incident effective velocity potential field, which results from combining the principal incident field with the reflected waves from the flat wall as well as the scattered waves from the image object [see Eq.(20)]. This improved definition allows a simplification of the problem in hand, such that the determination of the components is equivalent to the sole object, illuminated by an effective field, which accounts for the multiple scatterings, re-scattered waves and particle-wall interactions. Recognizing that the (cylindrically-outgoing) scattered waves cannot be concurrently incident upon the object surface, FIG. 2. The same as in Fig. 1, but the cylinder has a circular cross-section of the incident effective velocity potential field in the system of radius a. coordinates (r,) is determined from Eq.(10) such that, 2 1 2 FSkd  is , (21) kr 2 0 eff object r,,,,,, t   r  t    r  t inc totrd2 sca image where  denotes the real part of a complex number,  incr,,,,,, t   R r  t   sca  r  t object* eff eff object  is  scaik  r  inc   inc   sca , and kr  i t n in in   2 ikd cos  in  0e i e e Jn  kr e , dSe dS , where dS r d is differential element of a  r n surface S of unit-length enclosing the particle. The normal unit    vector (pointing outwardly to the surface) is 1 in  D H2. kd J kr e  e= cos e sin e , where e and e are the unit m n m n r xy x y nm    vectors, the symbol  denotes time-averaging, and the (20) superscript * denotes the conjugate of a complex number. After determination of the effective velocity potential field, The force vector consists of a longitudinal (along the x- the formulation is extended to derive the analytical direction) and a transversal (along the y-direction) component. components (i.e., longitudinal and transversal) of the radiation Those are defined as force based on the far-field scattering approach 23,43-45. The mathematical functions are expressed, respectively, by the Fxx  e  following approximations;  F   F e 2 n yy    Jn  kr kr cos kr 24   and kr (22) i krn Yx 1 2  24  SE, Hn  kr  kr e , and the radiation (particle- c 0 kr Yy wall) force series expansions are obtained based on integrating 23,43,46 Brillouin’s stress over a surface with a large radius surrounding the object. An alternative method consists of where Sc is the cross-sectional surface of the particle, integrating Brillouin’s stress tensor over the surface of the 1 2 2 probed particle using the approach of the near-field scattering. Ek00 2  is a representative energy density Notice, however, that in an inviscid host fluid, these two parameter, and Y and Y are the non-dimensional approaches are equivalent 45,47. Nonetheless, the benefit of x y using the far-field scattering lies in achieving fewer algebraic longitudinal and transversal radiation force functions, manipulations. respectively. The expression for the force vector used previously 45 can be applied such that,

arXiv (2017)

Using the angular integrals, where ij is the Kronecker delta function, Eq.(21) gives the mathematical series for the longitudinal and transversal force components, respectively, as 2 i n' n cos n, n 1 n , n 1  ed   , (23) 0 sin i    n, n 1 n , n 1  (text continues on right column)

  2 n in in   2 ikd cos  ** Y  iee   CDHkdCC 2,  (24) x n m n m   n 11 n   kSc nm 

  2 n in in   2 ikd cos  ** Y   iee   CDHkdCC 2,  (25) y n m n m   n 11 n   kSc nm 

where  designates the imaginary part of a complex ˆ  Liz  . (28) number.  The closed-form partial-wave series expansion for the axial component (along the z-direction) of the radiation torque After some calculation using the property of the following vector is also obtained as follows. angular integral, Using the general expression for the radiation torque vector 25 (which was also used in the context of a cylindrically- 2 i n' n ed 2, nn,' (29) focused quasi-Gaussian beam incident on a viscoelastic 0 cylinder 14, plane waves on an elliptical cylinder 18 and a pair 35 of fluid cylinders ) given by, The axial radiation torque expression is obtained as,

N  v  r  v d S, (26) NVErad  , (30) kr  zz0 S

where V is the particle volume, and  is the non- where the symbol  denotes a tensor product, and z dimensional axial torque function, given by eff object the sole non-vanishing axial v =,inc   sca  radiation torque component [in the z-direction normal to the in e in  ein   2 ikd cos  C polar plane (r,)] can be rewritten as, 4  n *  zn 2  nC  . rad kV n  D H2 kd N z Nez ,  m n m   m (27)  eff* object*ˆ eff object   r inc   r  scaL z  inc   sca dS ,      (31) 2 S

Eqs.(24),(25) and (31) are generalized series expansions, ˆ where ez is the unit vector along the z-axis, and Lz is the suitable for a 2D object of arbitrary geometry located nearby a axial component of the angular momentum operator in polar flat boundary, and illuminated by a plane travelling wave-field coordinates given by, with an arbitrary angle of incidence.

arXiv (2017)

2.3. Circular fluid cylinder example 3. Numerical examples and discussions

The example of a liquid circular cylinder of radius a, as Numerical examples are considered for which a MATLAB shown in Fig. 2, is now considered. For the circular geometry, program is developed to compute the expansion coefficients the boundary conditions given by Eqs. (16) and (17) at r = a for the fluid viscous circular cylinder located nearby a rigid lead to, flat boundary, and subsequently compute the longitudinal and transversal force and the axial torque components (per-length). 2 i2 ikd cos i  1 For the circular cylinder case, Sc = 2a and V = a for a unit- i e e e J ka C H ka length cylinder. The parameters chosen for the viscous fluid circular  1 int cylinder mimic those of a red blood fluid material, pertinent to J ka  Dn H n2, kd C J L a some bio-acoustofluidics applications. The mass density of the n 3 fluid circular cylinder particles is f = 1099 kg/m , and the (32) speed of sound for the compressional waves in its core is cf = 1631 m/s. The host liquid is water ( = 1000 kg/m3 and speed i ei e2 ikd cos ei  J '  ka C H1'  ka of sound c = 1500 m/s). Inside the inner core cylinder   material, the (complex) wavenumber for the compressional  –3 1 waves is expressed as  ci1  , where  (= 10 ) J' ka D H  2, kdc C int J '  a Lf     n n    ffc   L  48 n is a dimensionless absorption coefficient . (33) Computations for the longitudinal and transversal force components, in addition to the axial torque component are where the primes in Eq.(33) [and Eq.(35), below] denote the performed where the angle of incidence of the plane progressive wave field and the dimensionless distance are derivatives of the wave functions, and cf is the speed of sound of the compressional waves in the fluid cylinder. varied, respectively, in the chosen ranges –90    90 and Correspondingly, the application of the boundary conditions ka < kd  15, for fixed values of the size parameter ka of the at the surface of the image cylinder leads to, circular cylinder. First, the example of a Rayleigh circular viscous fluid cylinder near a boundary having ka = 0.1 is

considered. Panel (a) of Fig. 3 displays the plot for the i e2ikd cos e i  ei   J ka D H1  ka longitudinal radiation force function, which exhibits periodic oscillations versus kd as  varies in the range –90    90.  These oscillations result from multiple reflections occurring 1 int J ka  Cn H n 2, kd D J L a between the circular cylinder and rigid wall, where they are n maximal at  = 0. As | | increases, the multiple reflections (34) become less pronounced due to geometrical consideration as the waves reflected from the boundary become less scattered 2ikd cos i i  ' 1' by the particle. The longitudinal force components varies i e e e J ka D H ka between positive and negative values depending on the  dimensionless distance kd and , suggesting a pushing or J' ka C H1 2. kdc D int J '  a pulling effects towards the acoustic source. At the crossing    n n     ffc   L  n point between positive and negative values, a longitudinal (35) radiation force function singularity occurs where Yx = 0. This effect occurs manifold as kd and  vary, suggesting that the The coupled system of Eqs.(32)-(35) can be calculated fluid viscous Rayleigh cylinder becomes unresponsive to the numerically by matrix inversion, so that the modal coefficients longitudinal force, and is only subjected to the transversal force component. Moreover, symmetry with respect to the axis C and D can be determined. Then, the force and torque n n  = 0 is noted such that Yx(–) = Yx(). Panel (b) shows the vector components can be computed after adequate truncation plot for the transversal radiation force function, which displays of the series, with a suitable limit that ensures convergence. asymmetric values such that Yy(–) = –Yy() as kd increases The truncation limit in the series has been set to Nmax = with respect to the axis  = 0, where Yy vanishes therein due to [(max(ka, kd)] + 35, which leads to adequate convergence and symmetry considerations. In contrast to what has been accuracy of the results with negligible truncation error in the observed for the longitudinal radiation force component in order of ~10-6. panel (a), as | | increases, the effect of the multiple reflections on Yy becomes more pronounced as its amplitude displays larger oscillations as | | approaches 90. Panel (c) shows the plot for the axial radiation torque function, which also displays asymmetry with respect to the axis  = 0, where it vanishes therein due to symmetry. Notice also that z(–) = –z(). In panel (c), it is shown that z varies between positive and 6

arXiv (2017)

FIG. 3. Panels (a) and (b) show the plots for the longitudinal and transversal FIG. 4. The same as in Fig. 3 but ka = 5. force functions, while panel (c) corresponds to the plot for the axial torque function, respectively, versus kd and  for a small (subwavelength) viscous circular cylinder located near a boundary at ka = 0.1. Fig. 3. The plot of Yx shows that both positive and negative values can be predicted, which anticipates the generation of negative values, suggesting that the viscous fluid circular repulsive and attractive longitudinal forces, contingent upon cylinder is subjected to a radiation torque causing its rotation the values of kd and , as ka increases. The symmetry with counter-clockwise or clockwise. Moreover, at the crossing respect to the axis  = 0 is also preserved such that Yx(–) = Yx(). The plot of Yy displayed in panel (b), is also about 250 point between positive and negative values of z, an axial times larger than the plot shown in panel (b) of Fig. 3 for the radiation torque function singularity occurs where z = 0 such that there is no spinning around the center of mass of the Rayleigh viscous fluid cylinder. Similarly to panel (b) of Fig. cylinder; this suggests that the viscous cylinder renders 3, Yy(–) = –Yy(), where Yy = 0 at  = 0 as required by “invisibility” to the transfer of angular momentum. symmetry. Panel (c) shows the plot for the axial radiation Increasing ka (= 5) is considered now, where the related torque function, which is also about 1000 times larger than its plots for the longitudinal and transversal components of the counterpart, as shown in panel (c) of Fig. 3. The oscillations of force functions are shown in panels (a) and (b) of Fig. 4, the axial radiation torque function are less emphasized, nonetheless, positive and negative values of  are predicted respectively. As ka increased, Yx is about 2.5 times larger than z its counterpart for the Rayleigh cylinder shown in panel (a) of numerically, which suggests that the viscous fluid cylinder

arXiv (2017)

FIG. 5. Panels (a) and (b) exhibit the plots for the longitudinal and transversal FIG. 6. The same as in Fig. 5 but kd = 15. force functions, while panel (c) shows to the plot for the axial torque function, respectively, versus ka and  for a viscous circular cylinder located near a boundary at kd = 5.5.

frequencies, the scattering and absorption increase, which both reverses its spinning around its center of mass depending on contribute to the force 23,43 and torque 14. Panels (a)-(c) also the values of kd and , as ka increases. Moreover, z(–) = – show that the symmetry of the function Yx(–) = Yx() as well z(), where z = 0 at  = 0 as required by symmetry. as the asymmetry of the functions Yy(–) = –Yy() and z(–) Another example of interest is to vary the dimensionless = –z() are preserved. size parameter and angle of incidence, respectively, in the As kd increases (= 15), the computed force and torque chosen ranges 0 < ka  5 and –90    90, for fixed values functions shown in panels (a)-(c) of Fig. 6 display noticeable of the non-dimensional particle-boundary distance kd. Panels undulations at a large but fixed ka while  varies. At a larger (a)-(c) of Fig. 5 exhibit the results for the longitudinal and particle-boundary dimensionless distance (i.e., kd = 15) the transversal force functions, in addition to the axial torque established (quasi)standing wave field generated between the function, for a lossy fluid cylinder located at kd = 5.5. In all object and the flat wall at a fixed ka value possesses more the plots, the amplitudes of the force and torque functions nodes and antinodes, which affects the amplitudes of the increase as ka increases, which is expected since at higher radiation force and torque functions.

arXiv (2017)

FIG. 7. Panels (a) and (b) show the results for the longitudinal and transversal FIG. 8. The same as in Fig. 7 but  = 45. force functions, while panel (c) corresponds to the plot for the axial torque function, respectively, versus ka and kd for a viscous circular cylinder located near a boundary at  = 0. positive and negative values depending on the choice of kd and ka, suggesting the possibility of pushing and pulling the The effect of varying the size parameter and the cylindrical particle away or toward the boundary. dimensionless distance, respectively, in the chosen ranges 0 < Increasing the angle of incidence to  = 45 is also ka  5 and ka < kd  15 is now considered for fixed values of considered, and the related computations for the radiation the angle of incidence . Practically, for any physical force and torque functions are displayed, respectively, in situation, the constraint kd > ka should be always satisfied, panels (a)-(c) of Fig. 8. As  deviates from zero, Yx displays indicating that the distance particle-wall should be larger than smaller amplitudes compared to those shown in panel (a) of the radius of the cylinder. After imposing this condition, the Fig. 7, while Yy is nonzero now. Panel (c) shows also that z  radiation force and torque functions are computed for  = 0, 0, and varies between + and – values as kd and ka vary. This and the corresponding results are displayed, respectively, in also suggests that the viscous fluid cylinder can spin around its Fig. 7. At this angle, only the longitudinal radiation force center of mass (counter-clockwise or clockwise) contingent function is nonzero, as Y and  vanish as required by y z upon the sign of z. At the crossing point where z vanishes, symmetry. Panel (a) of Fig. 7 shows that Yx oscillates between the particle becomes unresponsive to the transferal of the

arXiv (2017) angular momentum carried by the incident waves, yielding 4 Y. Qiao, X. Zhang, and G. Zhang, J. Acoust. Soc. Am. 141, 4633-4641 “rotational invisibility”. Notice that the angular velocity of the (2017). 5 J. Awatani, Memoirs of the Institute of Scientific and Industrial Research, cylinder can be determined (following the steps described in Osaka University 12, 95-102 (1955). 14 49 Section 4 of ) using the rotational friction coefficient for a 6 T. Hasegawa, K. Saka, N. Inoue, and K. Matsuzawa, J. Acoust. Soc. Am. permeable cylinder. 83, 1770–1775 (1988). 7 J. Wu, G. Du, S. S. Work, and D. M. Warshaw, J. Acoust. Soc. Am. 87, 581-586 (1990). 8 F. G. Mitri, J. Sound Vib. 284, 494-502 (2005). 4. Conclusion 9 F. G. Mitri, European Physical Journal B 44, 71-78 (2005). 10 F. G. Mitri and S. G. Chen, Physical Review E 71, 016306 (2005). 11 F. G. Mitri, Ultrasonics 50, 620-627 (2010). This work presented a comprehensive analysis for the 12 F. G. Mitri, Ultrasonics 62, 244–252 (2015). radiation force and radiation torque on a particle of arbitrary 13 F. G. Mitri, AIP Advances 5, 097205 (2015). geometrical cross-section located nearby a boundary in 2D. 14 F. G. Mitri, Wave Motion 66, 31-44 (2016). 15 Closed-form partial-wave expansions for the longitudinal and F. G. Mitri, J. Appl. Phys. 118, 184902 (2015). 16 F. G. Mitri, Journal of Applied Physics 118, 214903 (2015). transversal radiation force functions, in addition to the axial 17 F. G. Mitri, Ultrasonics 66, 27-33 (2016). radiation torque function (which remain suitable for any 18 F. G. Mitri, Physics of Fluids 28, 077104 (2016). particle of arbitrary geometry in 2D) are derived, stemming 19 P. M. Morse, Vibration and Sound, (Acoustical Society of America, 1981). 20 from the integration of the radiation stress tensor and its B. Van Der Pol, Physica 2, 843-853 (1935). 21 P. M. Morse and R. H. Bolt, Rev. Mod. Phys. 16, 69-150 (1944). moment. The analysis is also substantiated with computations 22 G. N. Watson, A Treatise on the Theory of Bessel Functions. (Cambridge for a circular viscous liquid cylinder immersed in a non- University Press, London, 1922). viscous fluid, and subjected to incident plane travelling waves 23 P. J. Westervelt, J. Acoust. Soc. Am. 23, 312-315 (1951). 24 with arbitrary incidence. The coupling effects of the scattered H. Olsen, W. Romberg, and H. Wergeland, J. Acoust. Soc. Am. 30, 69-76 (1958). waves from the particle and its image, and those reflected by 25 G. Maidanik, J. Acoust. Soc. Am. 30, 620-623 (1958). the flat rigid boundary are taken into account while deriving 26 W. E. Smith, Austr. J. Phys. 25, 275-282 (1972). the mathematical expansions for the force and torque 27 Y. Chai, W. Li, Z. Gong, and T. Li, Ocean Eng. 116, 129-141 (2016). 28 components. The multipole modal expansion method in Y. Chai, W. Li, T. Li, Z. Gong, and X. You, Eng. Anal. Bound. Elem. 72, 27-41 (2016). cylindrical coordinates, the method of images, in addition to 29 V. K. Varadan, V. V. Varadan, and Y. H. Pao, J. Acoust. Soc. Am. 63, the translational addition theorem of cylindrical wave 1310-1319 (1978). functions are used. Systematic examples are considered where 30 V. V. Varadan, J. Acoust. Soc. Am. 63, 1014-1024 (1978). 31 a series of undulatory/wiggly behaviors are manifested in the Acoustic, electromagnetic and elastic wave scattering - Focus on the T- matrix approach. (Pergamon, (V. K. Varadan and V. V. Varadan, Eds.) NY, plots of the radiation force and torque functions. Depending 1980). on the particle-boundary dimensionless distance kd, the 32 T. A. K. Pillai, V. V. Varadan, and V. K. Varadan, J. Acoust. Soc. Am. 72, 1032-1037 (1982). incidence angle  as well as the dimensionless size parameter 33 ka, situations have been predicted where the particle yields F. Léon, F. Chati, and J.-M. Conoir, J. Acoust. Soc. Am. 116, 686-692 (2004). total force and torque neutrality depending on the choice of 34 P. C. Waterman, J. Acoust. Soc. Am. 60, 567-580 (1976). the parameters , ka and kd. The present formalism finds 35 F. G. Mitri, J. Appl. Phys. 121, 144901 (2017). useful applications in bio-acoustofluidics, particle 36 F. G. Mitri, J. Appl. Phys. 121, 144904 (2017). 37 J. C. Bertrand and J. W. Young, J. Acoust. Soc. Am. 60, 1265-1269 (1976). manipulation, rotation and dynamics for a particle near a 38 S. M. Hasheminejad and M. A. Alibakhshi, Int. J. Mech. Sci. 49, 1-12 chamber wall. Other applications dealing with the interaction (2007). of acoustical waves on objects in underwater/ocean acoustics, 39 G. C. Gaunaurd and H. Huang, IEEE Trans. Ultrason. Ferroelectr. Freq. sonochemistry, reconfigurable metafluids and liquid crystals Control 43, 690-700 (1996). 40 W. K. Lui and K. M. Li, J. Acoust. Soc. Am. 127, 664-674 (2010). may benefit from the results of the present investigation. The 41 S. M. Hasheminejad and M. A. Alibakhshi, Forsch. Ing. 70, 179-186 expressions for the force and torque components can serve to (2006). predict similar behaviors for a 3D particle of arbitrary shape 42 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: (or spherical form) nearby a boundary, and this analysis with Formulas, Graphs, and Mathematical Tables. (Dover Publications, 1965). should assist in developing appropriate analytical modeling 43 P. J. Westervelt, J. Acoust. Soc. Am. 29, 26-29 (1957). using the multipole expansion method, the method of images 44 C. P. Lee and T. G. Wang, J. Acoust. Soc. Am. 93, 1637-1640 (1993). and the addition theorem in spherical coordinates. Notice that 45 F. G. Mitri, New J. Phys. 8, 138 (2006). 46 recent advances 50-56 have considered the cases of non- H. Olsen, H. Wergeland, and P. J. Westervelt, J. Acoust. Soc. Am. 30, 633- 634 (1958). spherical objects (i.e., spheroids) in Bessel beams, and the 47 F. G. Mitri and Z. E. A. Fellah, J. Phys. A 39, 6085-6096 (2006). scope of such analyses can be extended to consider the 48 F. G. Mitri and Z. E. A. Fellah, Ultrasonics 74, 233-240 (2017). interaction of acoustical waves with non-spherical particles 49 F. W. Wiegel, Phys. Lett. A 70, 112-113 (1979). nearby a boundary. 50 F. G. Mitri, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 62, 1809-1818 (2015). 51 F. G. Mitri, Ann. Phys. 363, 262-274 (2015). References 52 F. G. Mitri, Wave Motion 57, 231-238 (2015). 53 F. G. Mitri, EPL (Europhysics Letters) 112, 34002 (2015). 1 A. K. Miri and F. G. Mitri, Ultrasound Med. Biol. 37, 301-311 (2011). 54 Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, J. Sound Vib. 383, 233- 2 J. Wang and J. Dual, Ultrasonics 52, 325-332 (2012). 247 (2016). 3 E. A. Ivanov, Diffraction of electromagnetic waves on two bodies. (NASA 55 F. G. Mitri, Ultrasonics 74, 62-71 (2017). Technical Translation F-597, (Nauka i Tekhnika Press, Minsk, 1968), 56 Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, Ocean Eng. 129, 507- 1970). 519 (2017).