Supplementary Material for ”Scattering properties of acoustic beams off spinning objects: Induced radiation force and torque”

This Supplementary Material gives details for both Mie theory and the finite-element modeling (implemented in COMSOL Multiphysics) of the spinning flow interacting with acoustic waves and compares their results. It also details the scattering and beam-shape coefficients of the spinning objects interacting with acoustic beams, as well as geometrical and material tunability.

II. Finite-element model for acoustics in spinning media 3 III. Beam-shape coefficients 7

References 13
I. MIE THEORY FOR ACOUSTICS IN SPINNING MEDIA Figure S.1 is a schematic of the problem that we plan to study in this Letter. We choose cylindrical coordinates (r, φ, z). The fields depend on the azimuthal angle φ via e ilφ . By using the convenient approximations [1][2][3][4] [and by ignoring the role of viscosity in this system, which is an acceptable approximation for water and taking into account the size of the cylinder (≈ 0.5 m)], and by denoting p, v r , v φ , the pressure field, the normal, and the azimuthal components of the velocity field, respectively, in the cylindrical coordinate system where e z is the unit-vector perpendicular to the plane of the figure (i.e., symmetry axis of the cylinder). The object is assumed to have the same physical properties as the surrounding medium, i.e., ρ/ρ 0 = β/β 0 = 1. The black arrows indicate acoustic scattering due to spinning.
(r, φ, z). It can be shown that the system obeys the linear system as function of these scalar fields. This system of coupled partially differential equations can be expressed as [2][3][4] .
In the standard framework of Mie theory, one expand the p in terms of Bessel and Hankel functions, and solving the expansion coefficients by taking boundary condition into account.

II. FINITE-ELEMENT MODEL FOR ACOUSTICS IN SPINNING MEDIA
In order to model the complex interaction of a flow (spinning fluid in our case) with acoustics, we make use of the finite element method (FEM) implemented in the commercial software COMSOL and its acoustic module of "linearized Navier-Stokes model" interface [5]. This is a robust model that describes the interaction of a stationary background flow with an acoustic field, as in our case. This interface is very general and takes into account several effects such as turbulent and non-isothermal flows. We implement an acoustic background (plane wave) field, use perfectly matched layers (PMLs) and proper boundary conditions (slip boundary conditions for the velocity, see Section the-linearized-navier-stokes-equations/ [5]. Moreover, in COMSOL, a material is defined by the following properties: density, dynamic viscosity, bulk viscosity, thermal conductivity, and heat capacity at constant pressure, denoted by ρ 0 , µ, µ B , k, and C P , respectively, showing the degree of generality of this model. We also add a background acoustic field to the domain, defined by pressure, velocity, and temperature p b , u b , and T b , respectively.
In addition to all these definitions, we implement the perfectly matched layers (PMLs) for the spinning flow and integrate the parameters of the study such as the scattering cross- Interestingly, we could use the Mie formalism in order to compute the scattering coefficients separately for both the object at rest (same as in Fig. S.2(a)) and the spinning one (of When spinning is induced, the situation dramatically changes and ±l multipoles are split owing to the acoustic analog of the Zeeman effect [6]. These figures undoubtedly show the effect on spinning on the interaction of sound with objects in motion.
Moreover, for the purely spinning-induced resonance, it can be seen in Fig These results serve as a numerical experiment that is closer to the reality compared with simple Mie theory models that help in getting a physical picture of the interaction between sound and spinning flows.

III. BEAM-SHAPE COEFFICIENTS
By making use of the orthogonality of the functions e ilφ , it can be shown that the expression of the beam-shape coefficients (BSC) is given by [7][8][9][10][11][12][13][14][15] In the case where the incident beam is simply a plane wave, Eq.
with r 0 the distance between the source and the cylinder's center and R 0 = (r 2 + r 2 0 + 2rr 0 cos φ) 1/2 the distance between the source's position and the observation point (r, φ). .

(S.5)
The second kind of acoustic source that we will analyze in our study is the quasi-Gaussian cylindrical beam [21,22] [see Fig. S.5(d)], whose BSC are given as [8,18,19] b qG l,s = i l+s I l+s (k 0 ξ) I 0 (k 0 ξ) , (S.6) where ξ is a parameter of the beam related to its waist (i.e., Rayleigh length). Now the field inside the cylinder shall be given by taking into account the modified Helmholtz equation, i.e., with k l the quasi-wavenumber in the spinning cylinder, b l the BSC, and a l the unknown coefficients of the internal pressure field.
The boundary conditions at the cylinder's outer radius r = a are (i) the continuity of total pressure and (ii) normal displacement, that is [1,2] p inc (a) + p scatt (a) = p (a) , By enforcing the boundary conditions (p and ζ r ) across the boundary r = a, we can get both the internal coefficients a l the scattering coefficients s l of order l, i.e., The field in the region a < r < r 0 is p 0 = p inc + p scatt , and hence the scattering coefficient is given by (S.10) with |M | the determinant of any matrix M and .   the SCS and radiation force for quasi-Gaussian beams of higher order and shows that when the order of the beam is 5 or higher, the force vanishes.
In order to analyze the robustness of our concept, we analyze the effect of changing the size of the scatterer on the radiation force in Fig. S.8 showing that even for smaller radii the effect is present. Last but not least, we characterize the effect of loss in density of the scatterer on the torque in Fig. S.9. The results show that the smaller the loss, the higher