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Mean-Acoustic Fields Exerted on a Subwavelength Axisymmetric Particle

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Mean-Acoustic Fields Exerted on a Subwavelength Axisymmetric Particle Mean-acoustic fields exerted on a subwavelength axisymmetric particle Everton B. Lima1 and Glauber T. Silva1, a Physical Acoustics Group, Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o,AL 57072- 970, Brazil (Dated: 29 June 2021) The acoustic radiation force produced by ultrasonic waves is the \workhorse" of particle manipulation in acoustofluidics. Nonspherical particles are also subjected to a mean torque known as the acoustic radiation torque. Together they constitute the mean-acoustic fields exerted on the particle. Analytical methods alone cannot calculate these fields on arbitrar- ily shaped particles in actual fluids and no longer fit for purpose. Here, a semi-analytical approach is introduced for handling subwavelength axisymmetric particles immersed in an isotropic Newtonian fluid. The obtained mean-acoustic fields depend on the scattering co- efficients that reflect the monopole and dipole modes. These coefficients are determined by numerically solving the scattering problem. Our method is benchmarked by comparison with the exact result for a subwavelength rigid sphere in water. Besides, a more realistic case of a red blood cell immersed in blood plasma under a standing ultrasonic wave is investigated with our methodology. ©2021 Acoustical Society of America. [https://doi.org(DOI number)] [XYZ] Pages:1{11 I. INTRODUCTION acoustic fields have to be re-calculated at each time step in the interval of interest. As a consequence, particle Acoustofluidics is extensively based on the acoustic 1 dynamics analysis in acoustofluidics might be a very in- radiation force produced by ultrasonic waves to control tense computational task. We emphasize that such anal- and pattern micro/nanoparticles (cells, microorganisms, ysis is of fundamental importance for understanding and and viruses) in a liquid medium. Nonspherical particles 33 2 designing micro/nanorobots propelled by ultrasound. are also accompanied by the acoustic radiation torque. In acoustofluidics, subwavelength nonspherical par- The time-averaged radiation force and torque are referred ticles, which are much smaller than the wavelength, have to as the mean-acoustic fields that drive the particle dy- been conveniently modeled as small spheres mainly be- namics in most acoustofluidic settings. cause they fit the simple analytical expressions of the The analytical solutions for the mean-acoustic fields radiation force34 and torque.4 obtained in the scattering depend, apart from the incoming wave, on the geomet- dipole approximation. Recently, analytical results con- ric shape3{15 and material properties16{22 of the particle. 23{26 27 sidering a subwavelength spheroidal particle were also Moreover, the viscous and thermodynamic prop- derived in the dipole approximation.35 We stress that erties of the surrounding fluid can significantly change a complete theoretical description of the mean-acoustic the mean-acoustic fields on the particle. The solutions fields for subwavelength particles of arbitrary geometry mentioned above are found for particles with a simple in an actual fluids seems to be unattainable. geometric shape, such as spheres and spheroids. Gen- This article proposes a semi-analytical approach to erally, these solutions are derived with the multipole ex- describe the mean-acoustic fields interacting with an ar- pansion and angular spectrum methods. The equivalence bitrary axisymmetric particle in Newtonian thermovis- of these theoretical approaches have been demonstrated cous fluids. The obtained fields are related to the pres- in Ref. 28. sure and fluid velocity of the incident wave and the coeffi- The mean-acoustic fields exerted on particles with cients of the monopole and dipole modes of the scattered arXiv:2104.06090v2 [physics.flu-dyn] 26 Jun 2021 more elaborate geometries are usually obtained through waves. These coefficients are obtained from the projec- numerical techniques such as the finite element29{31 and 32 tion of the scattered pressure onto the angular part of boundary element method. Nevertheless, numerical the corresponding mode (monopole and dipole). In turn, methods compute the mean-acoustic fields for a given the scattered pressure is obtained via a finite element spatial configuration (position and orientation) of the solver. The numerical part of our method is verified particle relative to the incoming wave. To determine the against the exact solution for a rigid spherical particle particle position and orientation versus time, the mean- in a lossless fluid. Besides, we apply our method to com- pute the mean-acoustic fields exerted on a red blood cell by a standing plane wave in blood plasma. The results, agtomaz@fis.ufal.br such as trap location and spatial orientation of the RBC, J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics 1 are in good agreement with previously reported experi- ments. II. PHYSICAL MODEL A. General assumptions Consider a subwavelength axisymmetric particle, i.e., with rotational symmetry around its principal axis, in a fluid of density ρ0 and compressibility β0. The parti- cle scatters an incoming wave of pressure amplitude pin, angular frequency !, wavelength λ, and wavenumber k. In Fig.1, we sketch the the acoustic scattering prob- lem for an axisymmetric particle. The scattering is con- veniently described concerning the right-handed refer- ence frame Oxpypzp fixed in the geometric center of the particle (centroid), referred to as the p-frame. It is also useful to obtain the mean-acoustic fields in a fixed lab- oratory reference frame Oxyz (l-frame) attached to the particle's centroid. The cartesian unit vectors of both ref- FIG. 1. Problem sketch: an incoming wave (background im- erence systems are denoted by ei and eip , with i = x; y; z and ip = xp; yp; zp. Since the particle is invariant under age with a positive and negative crest in purple and red, re- infinitesimal rotations around the zp axis, we need two spectively) of wavelength λ is scattered by a small axisymmet- angles (α; β), known as the Euler angles, to transform ric particle (gray cylinder). Two Euler angles (α; β) describe a vector from the p-frame to the l-frame. The transfor- the particle orientation regarding a laboratory frame denoted mation structure is formed by two elementary rotations: by blue axes. The particle frame is represented by red axes. a counterclockwise rotation of an angle α around the z The dotted concentric circles and black arrows denote the out- axis, followed by a counterclockwise rotation of an angle going scattered wave. In the inset, we show the elementary 0 β around new y axis{see Fig.1. We thus introduce the rotations that map a vector from the particle frame (red axes) rotation tensor as into the laboratory frame (blue axes). R = cos α cos β exexp − sin α exeyp + cos α sin β exezp + sin α cos β eyexp + cos α eyeyp + sin α sin β eyezp distance to the particle surface are, respectively, − sin β ezexp + cos β ezezp ; (1) a = max Rp(θp); (3a) 0≤θp≤π where the product eieip is a dyadic, which forms the standard basis of second rank tensors in the Euclidean b = min Rp(θp): (3b) 0≤θ ≤π space. The Euler angles are given in the intervals 0 ≤ p α < 2π and 0 ≤ β ≤ π. We define the particle orientation In case of a spherical particle, a = b. The character- in both the p-frame and l-frame as istic size parameter of the subwavelength axisymmetric particle can be defined as ep = ezp ; (2a) 2πa el = R · ezp = cos α sin β ex + sin α sin β ey + cos β ez; ka = 1: (4) (2b) λ where the center dot represents the scalar product be- B. Scattering normal modes tween a tensor and vector or two vectors. Note The scattered pressure by the particle is obtained also in deriving Eq. (2b), we have used the expressions by solving the Helmholtz equation with boundary con- e e · e = e e · e = 0, and e e · e = e . The i xp zp i yp zp i zp zp i ditions on the particle surface and the Sommerfeld ra- inverse transformation from the l-frame to the p-frame is diation condition at the farfield (r ! 1). The series given in AppendixA. p solution represents the scattering pressure as an infinity We assume that the largest dimension of the parti- sum of partial waves. It is convenient to express the par- cle, denoted by a, is much smaller than the wavelength, tial waves in spherical coordinates (r ; θ ;' ). Hence, a λ, the so-called long-wavelength approximation. To p p p the nth-order partial wave is h (kr )Y m(θ ;' ), where determine a for an axisymmetric particle, we first notice n p n p p h is the spherical Hankel function of first type, Y m is that the particle occupies region, given in spherical co- n n the spherical harmonic, m is the number representing the ordinates, by Ω = f(r ; θ ;' ) j 0 ≤ r ≤ R (θ ); 0 ≤ p p p p p p p wave orbital angular momentum. In the case of subwave- θ ≤ π; 0 ≤ ' < 2πg, with R being the radial distance p p p length particles much smaller than the wavelength, the to the particle surface Sp. The largest and smallest radial 2 J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics partial wave expansion can be truncated at the dipole- coefficients and incident fields as35 moment term (n = 1). If we take a spherical particle rad as an example, we see that the truncation error is of the Fp = Re (exp exp + eyp eyp )Dxpyp + ezp ezp Dzpzp order of O[(ka)2].21 ∗ · vin,p ; (9a) In the dipole approximation, the partial wave expan- rp=0 2πi3ρ sion of the scattering pressure is D = − 0 s v · r + s v @ xpyp k2 k 11 in,p xpyp 10 in;zp zp 0 psc = p0 a00s00Y0 (θp;'p)h0(krp) ∗ pin −1 0 − is00 (1 + 2s11) ; (9b) + [a1;−1s1;−1Y1 (θp;'p) + a10s10Y1 (θp;'p) c0 + a s Y 1(θ ;' )] h (kr ); (5) 2πi3ρ 11 11 1 p p 1 p D = − 0 s v · r + s v @ zpzp k2 k 11 in,p xpyp 10 in;zp zp where p0 and anm are the pressure amplitude and beam- shape coefficient of the incoming wave, respectively.
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