Abstract
Research into the orbital angular momentum carried by helical wavefronts has been dominated by the fields of electromagnetism and acoustics, owing to its practical utility in sensing, communication, and tweezing. Despite the huge research effort across the wave community, only recently has elastic orbital angular momentum been theoretically shown to exhibit similar properties. Here we experimentally observe the transfer of elastic orbital angular momentum from a hollow elastic pipe to a fluid in which the pipe is partially submerged, in an elastic analogue of Durnin’s slitring experiment for optical beams. This transfer is achieved by coupling the dilatational component of guided flexural waves in the pipe with the pressure field in the fluid; the circumferential distribution of the normal stress in the pipe acts as a continuous phased pressure source in the fluid resulting in the generation of Bessellike acoustic beams. This demonstration has implications for future research into a new regime of orbital angular momentum for elastic waves, as well providing an alternative method to excite acoustic beams that carry orbital angular momentum that could create a paradigm shift for acoustic tweezing.
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Introduction
Over the past three decades, investigation into the orbital angular momentum (OAM) carried by helical waves has largely been reserved for light and sound^{1}. The realisation by Allen et al.^{2} that electromagnetic LaguerreGaussian (LG) beams, satisfying the paraxial wave equation^{3}, carry a welldefined OAM sparked a resurgence in interest in exploiting this mechanical property of light for optical tweezers^{4,5,6,7,8}. The celebrated success of utilising LG beams comes, in part, from the ease in which they can be generated; a variety of simple devices can form these modes, for example spiral phase plates, qplates, and spatial light modulators^{9,10}. The orbital angular momentum is associated with the spatial distribution of the beam, and not with the polarisation (that determines the intrinsic spin angular momentum)^{2}; the inclined phasefronts give rise to a welldefined OAM about the beam axis that is proportional to the azimuthal index, or topological charge, m. The azimuthal variation of phase forms a helical profile varying as e^{±imθ}, for azimuthal angle θ with the sign of m determining the handedness of the helix. In general any optical beam with inclined phase fronts can carry a welldefined OAM^{11}. Since this realisation other forms of optical beams that carry OAM have been investigated, by interference and superposition, such as higherorder Bessel beams^{12,13}, and more general vortex structures (knots) with noninteger OAM^{14,15,16,17}.
Translations of the phenomena surrounding OAM beams have been shown to hold for sound^{18}, with the transfer of torque for scalar waves within the nonparaxial regime being considered^{19,20}. Similar applications in acoustic tweezers exist for microscale biological manipulation and communication, to name a few^{21,22,23,24}. Conventionally the excitation of acoustic OAM modes rests on acoustic analogues of phase plates^{25,26}, resonant devices and antenna concepts^{27,28,29}, or by discrete phased arrays^{30,31,32}. Utilising discrete phased arrays is indeed a favoured method for the excitation of higherorder OAM acoustic modes, that carry topological charge m > 1^{33}. Importantly, such higherorder modes exhibit an instability phenomenon: a vortex of charge m > 1 is not stable and degenerates into ∣m∣ vortices (screw dislocations) of charge m/∣m∣^{6,34,35}.
Recently OAM has been extended further into the realm of elasticity, where both the spinphonon coupling in elastic media and the intrinsic spin of elastic waves have been investigated^{36,37}. The orbital angular momentum associated with only the compressional potential of flexural waves in elastic pipes has also been considered^{38}, with the complete theory shown by Bliokh^{39}, proving that the total angular momentum (the sum of spin and orbital contributions) is quantised, as in electromagnetism and acoustics. In this paper we experimentally verify the experiment proposed in^{38}, showing that the elastic OAMassociated flexural waves can be coupled to a fluid, thereby observing its transfer through the excitation of acoustic pressure fields that carry OAM. Guided ultrasonic modes in pipes are considered, specifically flexural modes with a natural circumferential variation of phase. The coupling to the acoustic pressure in a fluid is achieved by partially submerging one end of the pipe in the fluid, in a set up resembling an elastic analogue of Durnin’s slitring experiment^{13,40}. We show that the generation of Bessellike, higherorder acoustic OAM modes is possible, with clear agreement between simulation and experiments, highlighting specifically the features associated for an m = 3 topological charge. Before introducing the flexural modes and the pipe structure we recall the governing equations of elastic materials and the associated compressional OAM.
An isotropic, homogeneous linear elastic material supports waves governed by the NavierCauchy equations^{41}, following the Einstein summation convention:
and the constitutive law
with ξ_{i} the displacement and \(\ddot{{\xi }_{i}}\) its double time derivative; Lamé’s first and second parameters are denoted λ, μ respectively with ρ being the material density; σ_{ij} and C_{ijkl} are the stress and stiffness tensors respectively; and \({\varepsilon }_{ij}\equiv \frac{1}{2}({\xi }_{i,j}+{\xi }_{j,i})\) is the strain tensor (comma notation denotes partial differentiation, and \({\partial }_{i}\equiv \frac{\partial }{\partial {x}_{i}}\)).
The displacement comprises both shear and compressional motion, described by an equivoluminal vector potential Ψ_{i} and scalar dilatational potential Φ respectively such that, by Helmholtz decomposition, ξ_{i} = ∂_{i}Φ + ϵ_{ijk}∂_{j}Ψ_{k}. Elastic waves with inclined phasefronts naturally occur as flexural modes in pipe walls^{38}. Choosing cylindrical coordinates oriented with z along the pipe axis the potentials take the form
where α = r, θ, z. k_{z} is the wavenumber along the pipe axis and ω the radian frequency. Stressfree boundary conditions are imposed on the inner and outer radii r_{a}, r_{b} respectively, such that \({\sigma }_{rr}={\sigma }_{r\theta }={\sigma }_{rz}{ }_{{r}_{a,b}}=0\), along with the infinite cylinder gauge condition ∇⋅Ψ = 0. The resulting radial distribution of the dilatational and shear potentials (ϕ(r) and ψ_{α}(r)) are then described by a linear combination of Bessel functions and their modifications^{42}. We consider the coupling of elastic waves in a pipe with pressure fields in a fluid by partially submerging the pipe in the fluid. At the solidfluid boundary the acoustic pressure induces a fluid load on the solid structure, and the structural acceleration acts as a normal acceleration across the solidfluid boundary. The boundary conditions then manifest as the dynamic continuity of traction, and the kinematic continuity of the normal particle displacement at the interface between the solid and fluid, such that^{43}
where p is the acoustic pressure, I_{ij} is the unit tensor, u_{i} is the particle displacement in the fluid and n_{j} is the surface normal. In the fluid domain, the governing equations follow the standard linearised Euler equations. For harmonic motion these read
for fluid density ρ_{F} with the acoustic wavespeed in the fluid denoted c_{F}. The coupling boundary conditions of the elastoacoustic problem can then be expressed in the pure displacement formulation^{44}:
These equations are solved numerically throughout via the Finite Element Method (FEM)^{45}.
The vector system of elasticity supports two body waves, compression and shear, that travel with distinct wavespeeds given by \({c}_{p}=\sqrt{(\lambda +2\mu )/\rho }\) and \({c}_{s}=\sqrt{\mu /\rho }\) respectively. Clearly, in the fully coupled elastic system, longitudinal (compressional), transverse (shear) and hybrid components all contribute to the orbital angular momentum, and thus to the total angular momentum (orbital and spin)^{39}. The splitting of the displacement vector into the dilatational and shear potentials, as done in^{38}, is not required; however, the OAM associated with the compressional potential (that is proportional to the azimuthal index m) motivated this experimental work as fluids only support compressional waves and not shear.
Optical and acoustic tweezing are primary applications for the transfer of OAM, where particles can be trapped and manipulated at the vortex singularity at the beam centre^{21,46}. In the case of elastic OAM carried by flexural modes in pipes there is no such singularity as the pipe is hollow  a direct analogy cannot be drawn in this case as there is no elastic medium to suspend particles with at the pipe centre, along its axis. However, it is well known that vibrating solids radiate sound waves in fluids, and it is this coupling we leverage to excite an acoustic OAM mode via the flexural displacement field carried in the elastic pipe. We utilise the socalled elastic spiral phase pipe^{47} to efficiently generate flexural pipe modes and include qualitative comparisons with both Finite Element Method (FEM) simulations and Dynamic Mode Decomposition (DMD), and a comparison to classical discrete phased sources. We experimentally validate this OAM transfer, thereby developing a new continuousphased acoustic source in the form of flexural modes in pipes; Bessellike beams are generated by the radial distribution of the compressional potential.
Results and discussion
Guided flexural modes in pipes
Guided ultrasonic waves in pipes have long been studied, with the first analytical description being posed by Gazis in the late 1950s^{42,48}, for infinitely long pipes. They fall into three modal classes: longitudinal (L), torsional (T), and flexural (F), with a naming convention attributed to Silk and Bainton^{49} such that they are written L(m, n), T(m, n) and F(m, n). The integers m, n denote the circumferential and group order respectively; the circumferential order being analogous to the topological charge of optical vortex beams. The generation and inspection of these guided waves has found much success in nondestructive techniques and evaluation^{50,51}. The focus of this paper is to observe the coupling between nonaxisymmetric flexural modes F(m > 0, n) in pipes and acoustic waves in a fluid. As such we require a device to ensure their efficient generation.
The recent advent of the elastic spiral phase pipe (eSPP)^{47} achieves this. This structure removes the necessity to rely on conventional means of complex arrangements of transducers or phased arrays (e.g by comb arrays or nonaxisymmetric partial loading^{52,53}). Advantages of the eSPP include passively exciting arbitrary flexural modes with, crucially, singlehandedness (i.e., only one sign of m) by mode converting longitudinal modes (e.g., L(0, 2)) that can be easily to excited in isolation^{54}. This is a particularly attractive property as a candidate for using flexural modes that are sensitive to axial cracks, where the conventional longitudinal and torsional modes are weakly sensitive^{55,56}.
In Fig. 1 we detail the eSPP used in the experimental verification of the transfer of elastic OAM. In Fig. 1a we show the elastic spiral phase pipe that endows incoming axisymmetric waves with a helical phase profile, similar to OAM generation by optical and acoustic analogues^{25,26,57}. The spiral pipe used here has already been characterised^{47}, comprising an aluminium pipe of density ρ = 2710 kg m^{−3}, Young’s Modulus E = 70 GPa and Poisson’s ratio ν = 0.33. The inner and outer diameters of the pipe are d_{1} = 40 mm and d_{2} = 60 mm, respectively. One end of the pipe is open (to be submerged in fluid), while the other is capped with an aluminium disk of diameter d_{2} and thickness 10 mm, attached by six screws. The total length of the pipe is L = 900 mm. The spiral region of the pipe is specifically designed to convert L(0, 2) modes to F(3, 2) modes at 62 kHz, and is formed by CNC milling a thickness of 6 mm from the pipe into three spiral steps of length h_{s} = 63 mm. This step profile is determined via the method in ref. 47, where an effective refractive index relates the two speeds of the incoming and converted waves through
where m is the modal index of the desired flexural mode (m = 3, here) and \(\tilde{n}={c}_{f}/{c}_{i}\) is the ratio of the wavespeeds of the converted flexural and incident longitudinal waves respectively, with k_{i} the wavenumber of the incident mode along the pipe axis. To reduce the length of the step size the spiral is partitioned into three turns such that h_{s} = h/3. The wavespeeds of each mode are determined by calculating the modal dispersions supported by the pipe, evaluated by spectral collocation^{47,58}.
Transfer of elastic OAM
To experimentally confirm the transfer of elastic OAM, we consider the coupling of the compressional component of a guided flexural F(3, 2) mode in an elastic pipe, to the acoustic pressure field in a fluid (water) in which the pipe is partially submerged. Extensive timegated acoustic characterisation of the fluidfield pressure distributions were made using a scanning tank facility, shown in Fig. 1b–d. We show, in Fig. 2, the experimental observation of elastic orbital angular momentum transfer by an elastic spiral phase pipe. The timeseries data obtained (see Methods) is analysed by way of the Fast Fourier Transform (FFT), giving the spatialfrequency components comprising the acoustic signal in the fluid. Figure 2a, b show the real pressure field and the phase, respectively, of the FFT of FEM simulations (see Methods) at 58 kHz, 20 mm below the submerged end of the pipe. The discrepancy between the design frequency and the shown frequency arise due to imperfections in the eSPP milling procedure, as a consequence of the finite size of the drill head, as well as the effect of real damping and viscous properties of the materials at high frequencies. This results in a slight rounding of the spiral tips causing a step profile that causes a small change from the design frequency; the device is not purely monochromatic and works over a range of frequencies near the design frequency, with varying efficiency^{47}. The corresponding experimental pressure fields are shown in Fig. 2c, d; there is clear qualitative agreement between simulation and experiment: we observe the predicted triplehelix phase profile with three phase singularities in the form of acoustic vortices. The splitting of the central vortex into three firstorder charges results from the instability of higherorder OAM modes^{34}. Similar to optical beams, these modes are vulnerable to perturbation by any coherent background^{59}, that itself does not require any dislocation lines (vortices)^{60}; unconverted compressional waves form such a background resulting in the observed decay of the highorder screw dislocations on a sum of dislocations of charge one. These are clearly visible in the simulations and experimental data. Additionally, in the experiment, there is an amplitude modulation due to the physical eSPP only approximating the exact circularhelicoid structure.
Figure 1b shows a zoom of the measurement area, with example experimental result superimposed, highlighting that due to both the parallax associated with refraction in the alignment, and the physical dimensions of the hydrophone, only an approximate depth of the relative position of the acoustic centre of the hydrophone (i.e., the plane where the pressure is accurately mapped) can be determined. The matching of the field profiles is observed at a depth of 20 mm below the pipe. This is, approximately, the closest possible approach of the acoustic centre of the hydrophone. The implications of this are highlighted in Fig. 3. For an acoustic Bessellike beam carrying OAM, one expects a zero in acoustic intensity due to the phase singularity at the centre of the beam, here the intensity in given by
where, in the frequency domain, the velocity, v is related to the pressure p through \({{{{{{{\boldsymbol{v}}}}}}}}=\frac{1}{i\omega \rho }\nabla p\) and * denotes the complex conjugate. At the observable depth this is obscured due to the background field excited from the compressional mode that is unconverted by the eSPP, since it is not perfectly efficient^{47}. As such at the measurement plane, marked by the dashdotted line in Fig. 3b, there is an amplitude modulation of the Bessellike nature resulting from modal interference. However, close to the end of the pipe, e.g., at the plane marked by the dashed line in Fig. 3b, the doughnutlike profile of the beam is unperturbed; this is seen in Fig. 3a that shows the Fourieranalysed FEM acoustic intensity and phase as a hued colourmap.
To confirm that the observed pressure field in the fluid is the dominant mode within the system, despite the pipe being modally rich, we perform Dynamic Mode Decomposition (DMD) on the experimental data set (see Methods). This is a technique popularised by Schmid^{61} that extracts the singular values of a matrix representing the timeevolution of the complete data set, and thus determines the dominant dynamics of the system. In Fig. 2e, f we show the results of the DMD on the experimental data, corroborating the assertion that the helical pressure field propagating through the fluid is dominant, as a direct result of the coupling from the compressional component of the incident F(3,2) mode designed to be excited by the eSPP.
We further explain the amplitude variation of the pressure field in the fluid by considering the superposition of the OAM beam with background sources. We do so by an analogy to amplitudemodulateddiscretephased acoustic sources that are conventionally used for exciting acoustic beam shapes that carry OAM.
Analogy to discrete phased arrays
Conventional methods for exciting acoustic beams that carry orbital angular momentum rely on discrete phased sources, such as circular arrays of loudspeakers. The interference of the monopolarlike sources then approximates a beam with a helical phasefront. Often acoustic waveguides are used to enable the beam waist to be formed a desired distance away from the sources^{30,32}. In Fig. 4 we show the discrete phased analogy via a frequency domain FEM simulation for two cases; (I) a ring of 12 phased point acoustic sources with equal amplitude, and (II) the same ring but with additional central source and amplitude variation. We include this model purely as a qualitative analogue to the acoustic OAM mode excited by the pipe; the vector elastic system cannot be reduced to a purely scalar (monopole) source as higher order vector components (e.g., dipole) contribute^{62,63}. The geometry considered is such that the point sources lie on a ring of diameter 50 mm, as if placed at the midpoint of the pipe thickness, lying atop a cylindrical volume of water 0.1 m deep and 0.15 m in diameter, akin to the FEM simulations of the main experiment (see Methods). Each source is coloured to represent the relative phase shift (of π/4 radians) to the adjacent sources, and has an amplitude represented by their relative size. The array is chosen so that a topological charge of m = 3 is achieved. Figure 4c, d show, for case (I), the real pressure field, phase, and acoustic intensity respectively for an excitation frequency of 58 kHz at a distance 20 mm from the source plane. As there is no other sources present this well approximates an acoustic beam carrying OAM. For case (II), an additional source is present at the centre of the ring, and the amplitude of three sources is also modified.
The eSPP considered throughout can be seen to act as a continuous phased acoustic source, with the phase profile determined by the circumferential order of the flexural mode, as shown in Fig. 3a. The analogy to the additional source in case (II) represents the amplitude modulation in the experiment due to the background provided by the unconverted compressional wave, and other modes present in the pipe. This intuitive analogy gives qualitative agreement to the experimental fields. As such we pose that the pipe acts as an amplitudemodulatedcontinuousphased source for acoustic OAM beams.
We additionally show in Fig. 5 a comparison between FEM modelling and experimental results through the absolute pressure field at 63 kHz, and in Fig. 6 show (at 53 kHz) similar comparisons at different depths, demonstrating the divergence of the acoustic beam with propagation depth (shown in Methods).
Conclusions
Generating acoustic vortex beams that carry OAM has been instrumental to the development of optical and acoustic tweezers. Specifically in acoustics, the generation of these modes conventionally relies on discrete phased arrays. By considering the elastic orbital angular momentum associated with compressional motion we have demonstrated the experimental observation of elastic orbital angular momentum transfer from guided flexural modes in a pipe to acoustic waves in a fluid, verifying the experiment proposed in ref. 38, and thus providing a new avenue to generate acoustic OAM beams.
The applications of this phenomena are therefore aligned with those of acoustic tweezers, including sensing, communication and microfluidic control. Extensions of this new methodology are anticipated to fluidfilled pipes, where there exist attractive applications in, for example, nondestructive testing in pipenetworks.
Methods
Experimental setup
Measurements were performed in a water tank without wall or surface treatments, with dimensions 3.0 × 1.8 × 1.2 m (L × W × D). The pipe was suspended vertically above the tank using nylon fishing line attached to a mount so that the end of the pipe was submerged approximately 20 mm into the fluid (Fig. 1d); the partial seal on the capped end of the pipe ensures fluid is present within the pipe, up to the same depth of submersion. A piezoelectric PZT8 disc of thickness 12 mm and diameter 35 mm was glued to the centre of the cap to provide excitation with a 5cycle pulse centred on 60 kHz. The piezoelectric excites the L(0, 2) mode which then efficiently excites the F(3, 2) mode via mode conversion in the spiral region as outlined in ref. 47.
To obtain pressure field maps of sound radiated from the submerged end of the pipe, the signal at the detection hydrophone (Brüel & Kjær 8103 hydrophone) was scanned in space using an xyz scanning stage (inhouse built with Aerotech controllers). The hydrophone was vertically mounted to a perforated perspex arm, to match the propagation direction of the acoustic field. The acoustic propagation was then spatially mapped in 2.5 mm steps across a 75 × 75 mm^{2} area centred beneath the pipe; the voltage, V, from the detector was recorded as a function of time, t, at each position in the scan. At each spatial point, the acoustic pressure field is averaged over the detecting area of the hydrophone head and the signals were averaged in time over 20 repeat pulses to improve the signaltonoise ratio. The detector was sampled with sample rate f_{s} = 9.62 MHz to record the signal for 5.2 ms at each point. The resulting usable frequency range for this sourcedetector response function was between 26−90 kHz. The resulting signals are timewindowed corresponding to the timeofflight of the flexural wave packet so that the pressure field excited by the F(3, 2) pulse is isolated in the fluid. We then confirm this is a dominant mode of the system through Dynamic Mode Decomposition.
In Figs. 5 and 6, we provide two further qualitative comparisons between FEM simulations and experiment, at 63 kHz and 53 kHz, respectively, showing the variation in acoustic mode shape in the fluid that is dictated by the dominant mode in the pipe at these frequencies. Additionally, in Fig. 6 we show the effect of propagation distance by presenting the results at deeper measurement planes. As expected the dominant mode propagates in the fluid and the acoustic beam diverges with propagation depth. Experimental data set available at ref. 64.
Dynamic Mode Decomposition
Dynamic Mode Decomposition is a technique developed by Schmid^{61} that enables a data set, be it numerical or experimental, to be analysed so that the dominant dynamics can be observed. This is a particularly attractive method here given the large number of modes excited within the pipe. Here we briefly outline the methodology following^{61}.
DMD rests on representing an original timeseries data set D as a sum of n mode shapes associated with the radian frequency ω_{n} such that
where ζ_{n} is the n^{th} mode shape. The data we analyse is the temporal evolution of the acoustic pressure at a series of gridpoints in space. The data is rearranged into a single matrix such that each column represents on frame of the data:
Where X is the complete data set and x_{i} is the data at times i = 1,…,N. As the governing equations for the acoustic propagation are linear, the data at each time step can be related by a matrix A such that
and thus
The dynamics of the system are then governed by the eigenvalues and eigenvectors of A, which can be approximated by several numerical methods. Here, as in Schmid’s original paper^{61}, we use singular value decomposition (SVD). We shall also consider the shifted matrix
for sufficiently large N (i.e., a long time signal) X and \(\bar{X}\) will have a near identical structure. By SVD, we write
where U and V contain the left and rightsingular vectors respectively, with the singular values along the diagonal of S. If the relative size of successive singular values to the first few is small, then the size of the matrices can be reduced, with the reduced forms subsequently written as e.g., \(\hat{U}\). The matrix U contains the socalled principal directions, that are used to rewrite the data in a new basis and define
Using the reduced forms, (13) then becomes
This approximation of A then contains all the information needed to take one frame of the data to the next. The eigenvalues and eigenvectors of \(\tilde{A}\) then are obtained by converting back to the original basis such that
where ζ_{n} is the n^{th} mode for the nth eigenvector η_{n}. The results of this decomposition on the timeseries data obtained in the experiments is shown in Fig. 2e, f, showing that this is a dominant mode shape.
Finite Element Modelling
The commercial FEM software COMSOL Multiphysics^{®} was used to perform timedomain simulations of the suspended pipe geometry. The acoustics and structural mechanics module were used with acousticsolid interaction to couple the displacement field in the pipe with the acoustic pressure fields in the air and water. A schematic of the simulation domain is shown in Fig. 7, with cylindrical wave radiation conditions on the dashed boundaries. The same 5cycle tone burst, centred on 60 kHz excitation was used and applied as a boundary load to the top cap of the pipe (area marked with magenta circle in Fig. 7) to simulate the effect of the piezoelectic disc source (not actually modelled in the geometry). The numerical pressure field was then extracted in the fluid, with the same spatial resolution as used in the experiment. Fourier analysis was then performed via the Fast Fourier Transform to obtain the spatialfrequency spectra, as done in the experiment. The results were analysed at several planes beneath the pipe to determine the position of the acoustic centre of the hydrophone and used to show the excitation of a LGlike acoustic beam near the submerged surface of the pipe (Fig. 3).
For the comparison with a discrete phased acoustic array, only the acoustics module was used, with the simulation domain shown in Fig. 4, using monopolarlike point acoustic sources. The simulation domain here matches the region of water in the main simulation.
Data availability
The experimental data that support the findings of this study are available at https://doi.org/10.24378/exe.4205. All other data are available from the corresponding author upon reasonable request.
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Acknowledgements
G.J.C gratefully acknowledges financial support from the Royal Commission for the Exhibition of 1851 in the form of a Research Fellowship. J.M.D.P. acknowledges the financial support from the H2020 FETproactive project MetaVEH under grant agreement No. 952039. T.A.S. gratefully acknowledges financial support from DSTL. The authors thank Prof. R.V. Craster, Prof. A.P. Hibbins and Dr. S.A.R. Horsely for useful conversations, and to G.T. Starkey for assistance in mount construction.
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G.J.C, J.M.D.P. and T.A.S. performed the experiment, subsequent analysis, interpretation of the results and contributed to writing the manuscript.
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Chaplain, G.J., De Ponti, J.M. & Starkey, T.A. Elastic orbital angular momentum transfer from an elastic pipe to a fluid. Commun Phys 5, 279 (2022). https://doi.org/10.1038/s42005022010570
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DOI: https://doi.org/10.1038/s42005022010570
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