Acoustofluidic technology combines acoustic and microfluidic technologies to realize particle manipulation in microchannels driven by acoustic waves, and the acoustic radiation force (ARF) with boundaries is important for particle manipulation in an acoustofluidic device. In the work reported here, the ARF on a free cylinder immersed in a viscous fluid with an incident plane wave between two impedance boundaries is derived analytically and calculated numerically. The influence of multiple scattering between the particle and the impedance boundaries is described by means of image theory, the finite-series method, and the translational addition theorem, and multiple scattering is included partly in image theory. The ARF on a free rigid cylinder in a viscous fluid is analyzed by numerical calculation, with consideration given to the effects of the distances from cylinder edge to boundaries, fluid viscosity, cylinder size, and boundary reflectivity. The results show that the interaction between the two boundaries and the cylinder makes the ARF change more violently with different frequencies, while increasing the viscosity can reduce the amplitude of the ARF in boundary space. This study provides a theoretical basis for particle manipulation by the ARF in acoustofluidics.
•The acoustic radiation force due to plane-wave incidence is derived for a free cylinder immersed in a viscous fluid between two boundaries.
•A larger negative acoustic radiation force can be obtained when two impedance boundaries exist.
•Theoretical guidance is provided for applying the acoustic radiation force in particle manipulation.
Acoustofluidic technology combines acoustics with microfluidic technology and uses acoustic waves as a driving force to realize contactless particle manipulation of fluids or particles in acoustofluidic devices.1,2 The acoustic radiation force (ARF) and acoustic streaming are influential for particle manipulation in a fluid. When acoustic waves encounter an obstacle in the medium, refraction, diffraction, and scattering occur, thereby changing the energy and momentum of the acoustic waves. The ARF arises from the nonlinearity of the acoustic field and refers to the nonzero average pressure on the material surface caused by the incident waves.3 Contactless particle manipulation, acoustic suspension, cell screening, and control of nanomaterials can be achieved by using the ARF and acoustic radiation moment,4–11 and this approach has been used widely in biomedicine, materials science, and ultrasound devices.
Since King proposed the theory of the ARF in 1934 and calculated it on a rigid sphere for the first time,12 there have been many theoretical and experimental studies of the ARF on spherical and cylindrical particles under various conditions.13–18 In practical applications, particles are normally kept in a limited space with a dissipative medium, and boundaries, viscosity, and heat conduction of the medium can affect the ARF. In 1994, Gaunaurd and Huang studied the acoustic scattering of a sphere near a planar boundary with plane waves incident at an arbitrary angle by means of image theory and the additive theorem in spherical coordinates.19 With the image method, the real sphere is on one side of the boundary and there is an image sphere on the opposite side, and the whole acoustic field has four parts: (i) the incident acoustic field, (ii) the acoustic field reflected from the boundary, and the acoustic fields scattered from (iii) the sphere and (iv) the image sphere. In 2004, Hasheminejad and Azarpeyvand studied the ARF of an infinite cylinder vibrating near an impedance boundary based on a simple local reaction surface model.20 In 2011, Miri and Mitri considered the ARF of a spherical shell near a vessel wall and discussed the effects of the vessel wall on the force using a porous impedance boundary.21 In 2012, Wang and Dual calculated the ARF of a rigid cylinder near a boundary under a standing wave and verified the ARF with finite-element simulation.22 In 2017, Qiao analyzed the ARF of a fluid cylinder near an impedance boundary.23 In 2019, Zang calculated the ARF of a liquid sphere between two impedance boundaries under Gaussian wave incidence, finding different behavior to that with a single boundary.24 In 2020, Qiao deduced the ARF of a free cylinder in a viscous fluid near a rigid boundary and studied the particle trajectories under different conditions.25 In 2021, Zang calculated the ARF of an elastic spherical shell near an impedance boundary in an ideal fluid under zero-order quasi-Bessel–Gauss waves.26
However, most of the aforementioned studies accounted for the ARF with only a single boundary or an ideal fluid, and they did not deal with the ARF of a particle moving freely between two impedance boundaries in a viscous fluid, as in the transport and movement of particles in blood vessels and acoustofluidic devices. Herein, based on previous research, the acoustic radiation on a free cylinder between two impedance boundaries in a viscous fluid is analyzed by means of image theory and the translational addition theorem in cylindrical coordinates in two dimensions. In many cases, analyzing acoustic scattering and the ARF in two dimensions is a quick and easy way to reveal physical phenomena. It is faster than doing so in 3D spherical coordinates and aids development of that analytical solution. Numerical simulation is reported to facilitate our discussion of the ARF, and the effects of boundaries, fluid properties, cylinder parameters, and particle–boundary distance on the ARF for a free rigid cylinder are discussed.
Consider a free rigid cylinder of radius R immersed in a viscous fluid between two impedance boundaries, with the fluid assumed to be infinite, homogeneous, and isotropic. As shown in Fig. 1, the distances between the cylinder and the two boundaries are d1 and d2, and an infinite plane wave is incident along the Ox axis perpendicular to the two parallel boundaries.
According to image theory, infinitely many image cylinders can substitute for the multiple scattering between the cylindrical particle and the boundaries.27,28 However, the effects of other image cylinders decay rapidly with distance and can be ignored in the calculation of the ARF. Instead, two image cylinders can be used to replace the effects of multiple scattering between the particle and the boundaries, with the image and original cylinders arranged symmetrically about the boundaries. If higher precision is required, then the effects of other image cylinders farther away must be considered. In Fig. 1, the distances between the two image cylinders and the two boundaries are d1 and d2, respectively, and three cylindrical coordinate systems (r, φ), (r1, φ1), and (r2, φ2) are established, centered on the original cylinder and the two image cylinders, respectively. Also, because the two boundaries are on opposite sides of the cylinder, the distances d1 and d2 have opposite signs.
A. Scattering of plane wave on free cylinder immersed in viscous fluid between two impedance boundaries
Consider a plane wave of frequency ω/2π incident vertically on the boundary in a viscous fluid. In the system (r, φ), the velocity potential is expressed as
where is the amplitude of the incident plane wave whose intensity is I, c0 and ρ0 are the acoustic velocity and density of the viscous fluid, respectively, k = ω/c0 is the wave number, ω is the angular frequency, and t is time. Also, is the complex wave number of the longitudinal wave, which can be obtained from the acoustic equations for a barotropic viscous fluid;29 here, k = Re(k1), where μ and λ are the dynamic viscosity coefficient and the second viscosity coefficient, respectively. The incident velocity potential is expanded as a series by cylindrical function as
where Jn is the Bessel function of the first kind of order n.
In a viscous fluid, the scattering velocity potential of a free cylinder under an incident plane wave satisfies the following:29,30
where Φs and Ψs are the scattering longitudinal velocity potential and shear velocity potential, respectively, of a free cylinder in a viscous fluid, An and Bn are the scattering coefficients determined by the boundary conditions, and k1 and k2 are the complex wave numbers of the longitudinal and shear waves, respectively, where k2 = (1 + i)/L and is the penetration depth of the vicious wave, which is the thickness of the viscous boundary layer.
Based on the image method, we obtain the reflected waves of the plane wave at the two boundaries and the scattering waves of the two images. The reflected waves at the two boundaries are expressed as
where Rs1 and Rs2 are the surface reflection coefficients of the two impedance boundaries, which when the particle is not too close to either boundary can be approximated as20,31
where βi(ω) = ρ0c0/Zi(ω) is the relative admittance of impedance boundary i, and Zi(ω) is its surface normal impedance.20,31
Using image theory, cylindrical coordinate systems are established with the geometric centers of the two image cylinders as their origins. In their respective coordinate systems, the velocity potentials of the scattered longitudinal and shear waves of the two image cylinders in the viscous fluid can be expressed as
where Φs1 and Ψs1 are the velocity potentials of the scattered longitudinal and shear waves, respectively, of image cylinder 1 in the viscous fluid, Φs2 and Ψs2 are those of image cylinder 2, and An and Bn are the scattering coefficients determined by the boundary conditions.
To solve for the scattering coefficients, the velocity potentials expressed in different coordinate systems must be transformed into the same coordinate system. Using the translation addition theorem in cylindrical coordinates, all velocity potentials can be expressed in the same coordinate system,32 and Eqs. (10)–(13) become
The total acoustic field includes the incident wave, the reflection waves of the two boundaries, and the scattering waves of the cylinder and two image cylinders. According to Eqs. (2), (5)–(8), and (14)–(17), the total acoustic field can be represented as longitudinal waves Φ and shear waves Ψ, i.e.,
To facilitate the calculation, the total velocity potentials are abbreviated as
where Re and Im are the calculations representing the real and imaginary parts, respectively.
For a free rigid cylinder immersed in a viscous fluid, there are no transmitted acoustic waves inside the cylinder because the rigid boundary reflects the waves completely. At the boundary between the surface of a cylinder and a viscous fluid, the boundary condition
is satisfied, where v = ∇Φ + ∇ ×Ψ is the velocity vector of the fluid, and U is the velocity vector of the particle, which can be obtained from the kinetic equation33
where m = πR2ρ is the mass of the cylinder per unit length, ρ is the density of the cylinder, and σ is the stress tensor expressed as34
where E is a unit tensor, e = ((∇v) + (∇v)T)/2 is the deformation velocity tensor, and the superscript T indicates matrix transposition. If the particle can move freely under the acoustic field, then a dynamic equation of motion is generated. For a free cylinder, the partial derivative with respect to time should be calculated in a fixed coordinate system for determining the velocity potential,33 and it is expressed as
For a rigid cylinder, to simplify the subsequent calculations, the boundary conditions of the particle surface in Eq. (31) can be expressed in the radial and tangential directions as
and the scattering coefficients An and Bn can be obtained by solving the matrix equation, whose specific expression and coefficients are
where the submatrices Cr, Cφ, Dr, Dφ, Er, Eφ, Gr, and Gφ are diagonal matrices with elements Cr,n, Cφ,n, Dr,n, Dφ,n, Er,n, Eφ,n, Gr,n, and Gφ,n, respectively. The matrices I1n and I2n are square matrices of order m with elements I1,mn and I2,mn, respectively, and O is a zero matrix with the same dimensions as those of I1,mn and I2,mn. The relevant elements are given specifically as
and when |n| ≠ 1, we have
when n = 1, we have
and when n = −1, we have
B. Acoustic radiation force
The ARF of a particle in a viscous fluid can be expressed as35
where means the time average, S is the surface area of the particle, and σ is the stress tensor as in Eq. (33). The plane wave is incident along the Ox axis perpendicular to the two impedance boundaries, and because of the symmetry of the sound field, the component force along the Ox axis is the resultant force on the cylinder. The component of Eq. (39) on the x-axis is expressed as
where η = ρ0/ρ. For a free cylinder, an additional term involving η is added to the scattering coefficients, and the density ratio also affects the results of the ARF. When η → 0, the obtained ARF corresponds to that on a fixed cylinder with no additional terms in a viscous fluid.
When the viscosity coefficients μ → 0 and λ → 0, the cylindrical particle becomes localized in an ideal fluid, and there is no shear wave for the scattering of the cylinder. It is found that the axial ARF applied to a free cylinder between two boundaries in an ideal fluid is
The ARF can also be expressed by the dimensionless ARF function Yp, which represents the radiation force per unit energy density and unit cross-section surface. The relationship between Fx and Yp is
where S = 2R is the cross-sectional area of the cylinder per unit length, i.e., the area of the cylinder per unit length affected by the incident plane wave. There is a fixed multiple relationship between Fx and Yp, and their trends are completely consistent.
III. SIMULATION AND DISCUSSION
This section reports a numerical simulation conducted for the axial ARF on a free cylindrical particle located between two impedance boundaries. The numerical model was built under a plane-wave acoustic field incident perpendicular to the boundaries in a viscous fluid, the main purpose being to explain more intuitively how various factors (viscosity, impedance boundaries, and the freedom of the cylindrical particle) impact the ARF. To simplify the situation, the reflection coefficients of the two boundaries were assumed to be the same but with opposite signs.24 In the numerical simulation, the acoustic energy of the incident plane wave was I = 175.5 W/m2,25,36 and the radius of the rigid cylindrical particle was R = 1 × 10−4 m. The involved parameters of the fluid medium are given in Table I, and to ensure accurate simulation results, the truncation constants in the calculation were Nmax = Mmax = 40.
|.||Density .||Acoustic .||Dynamic .|
|Liquid .||ρ0 (kg/m3) .||velocity c0 (m/s) .||viscosity μ (Pa⋅s) .|
|.||Density .||Acoustic .||Dynamic .|
|Liquid .||ρ0 (kg/m3) .||velocity c0 (m/s) .||viscosity μ (Pa⋅s) .|
A. Initial test
To assess the effectiveness of the method, the cases of no reflecting boundaries and only one impedance boundary in water were considered using Eqs. (45) and (46). Specifically, the ARFs on the rigid cylinder in the infinite and half-infinite areas were calculated with variation of kR, and the calculation results are shown in Figs. 2 and 3. Combining Eqs. (45) and (46), the ARFs on the fixed (η = 0) and free (η = 1/2) rigid cylinders are investigated. For the fixed cylinder, the results of Eqs. (45) and (46) are identical and agree with those in Ref. 37; in Fig. 2, the ARF calculated by Eq. (25) in Ref. 37 is shown by the black squares. For the free cylindrical particle, the ARF is less than that for the fixed cylinder when kR is relatively small, but with increasing kR, the ARFs of the two situations become more identical, which was also described in Ref. 25.
Figure 3 shows how the axial ARF varies with kd when there is only one rigid boundary, where d is the distance between the center of the cylinder and the boundary. Using Eq. (45), the calculation was conducted for kR = 0.126 and 0.063, and −Fx was calculated with Eq. (45) and standardized with ρ0A2R. The obtained results agree with those in Ref. 22.
B. Effects of impedance boundaries
In this subsection, we consider how impedance boundaries in a viscous fluid influence a free cylindrical particle. First, the ARF on a fixed cylinder in water was calculated with only one impedance boundary, leading to the variation of the ARF with kR under d = 2R and 4R. As shown in Fig. 4, the ARF has a large range of variation when kR is relatively small, this being due to the impact of the impedance boundary. With the boundary, the ARF is negative at specific frequencies, which can drag the particle toward the acoustic source. In addition, the variation of the ARF becomes more pronounced with increase of the boundary reflection coefficient, which was also described in Ref. 21. The boundary has no significant impact on the locations of the peaks and troughs of Yp, with those of the five curves being in roughly the same locations. The results in Fig. 4 agree with those in Ref. 38, thereby verifying the feasibility of the theoretical equations.
Figure 5 shows the results of simulating the ARF on the free cylindrical particle (η = 1/2) in glycerol with one impedance boundary, leading to the variation of the ARF with kR for d = 2R and 4R with different values of the reflection coefficient Rs. As with the case in water, the ARF is again negative at some specific locations. The variation of the ARF becomes more pronounced with increasing boundary reflectance, and its oscillation becomes more rapid with increasing distance between the boundary and the cylinder’s center.
Figure 6 shows how having two impedance boundaries in a viscous fluid affects the ARF. A free rigid cylindrical particle (η = 1/2) is immersed in the viscous fluid, and the distances between the particle and boundaries 1 and 2 are d1 = 2R and d2 = 4R, respectively. According to the results, compared with having one boundary, the amplitude of the ARF is increased significantly in the high-frequency region and a larger negative ARF is acquired. In addition, the added boundary brings more-complex multi-scattering, and the locations of the peaks and troughs of the ARFs are different from those in the former situation. Also, with increase of the boundary reflection coefficient, the amplitude of oscillation becomes more pronounced. According to the simulation results in Figs. 4–6, having one or more impedance boundaries leads to a series of peaks and troughs, and at some frequencies the ARF is negative, causing the cylindrical particle to be pulled toward the acoustic source.
Next, we consider how the distance between the free cylinder and one of the boundaries affects the ARF in the viscous fluid. The distance between the free cylinder and boundary 1 is d1 = 2R, with kR = 0.5 and η = 1/2, and Fig. 7 shows how Yp varies with d2/R. According to the results, the variation of Yp is periodic and is analogous to a sinusoidal curve. Curves with different values of the reflection coefficient have the same period because of the complex exponential function in the derivation process.24 Moreover, the amplitude of the ARF increases with increase of the boundary reflection coefficient.
Figure 8 shows how the ARF on a free rigid cylinder in the viscous fluid between two impedance boundaries varies with both d2/R and kR for η = 1/2 and Rs = 0.5. According to the calculation results, the peaks and troughs of the ARF become more concentrated with increase of d2/R because of the interaction between the actual cylinder and the image cylinder. More concretely, it is caused by the faster oscillation of the phase difference between the actual cylinder and the image cylinder with the increasing boundary distance. In specific frequency bands, negative ARF can be acquired to realize the capture of particles.39,40 Besides, the influence of d2/R on the ARF is negligible when kR is relatively small, which is because the particle is too small to be detected by the incident wave. When kR is larger, the ARF shows periodic variation with increasing d2/R.
C. Effects of viscosity
Figure 9 shows how the thickness L of the viscous boundary layer affects the ARF on a free cylinder between two impedance boundaries for η = 1/2, d1 = d2 = 2R, and Rs = 0.5. For a smaller cylindrical particle (smaller kR), Yp decreases with increasing L. For a larger cylindrical particle, Yp increases with increasing L, and the viscosity of the boundary layer enhances the ARF on the particle. The results differ from those with an infinite viscous fluid,30 which indicates that the viscous effect in the bounded viscous fluid is weaker than that in the unbounded fluid.25
D. Effects of cylinder size
Figure 10 shows the ARF on a free rigid cylindrical particle with four different values of its radius in the viscous fluid between two impedance boundaries for Rs = 0.5, d1 = 2R, d2 = 4R, and η = 1/2. The frequency range for the incident acoustic wave is 0–5 MHz. From the simulation results, the corresponding frequency for the peak of Yp decreases with increasing particle radius. The amplitude increases as well, and a similar phenomenon also appears in an ideal fluid.38 In addition, we used the present formula to calculate the ARF of a rigid cylinder near one impedance boundary in an ideal fluid. Figure 11 was produced using the same parameters as those in Ref. 38 with d = 1.5R and Rs = 0.5, and Eq. (46) was used for double boundaries to calculate the variation of the ARF on a rigid cylinder near one impedance boundary with frequency for different values of the radius. The results obtained are consistent with those in Ref. 38.
E. Effects of density ratio η
In this subsection, we consider how η affects the ARF on a free rigid cylinder. Figure 12(a) shows how the ARF varies under different values of η and the distances between the particle and the boundaries for d1 = 2R, 0 ≤ η ≤ 6, 2 ≤ d2/R ≤ 6, and L = 0.1R. According to the results, different values of η lead to different values of the ARF. When d2/R is relatively small, Yp decreases initially and then increases with increasing η, and when d2/R is sufficiently large, Yp increases initially and then decreases with increasing η. Additionally, when η is sufficiently large, the ARF varies less. Figure 12(b) shows the ARF under different values of η and kR. As the results show, when η is relatively small, its impact is remarkable, and for larger η, the variations of the ARF with kR for different values of η have similar tendencies.
Finally, we investigate the ARF on a cylindrical particle between two impedance boundaries under four conditions: (i) a viscous fluid, (ii) an ideal fluid, (iii) a free rigid cylinder, and (iv) a fixed rigid cylinder for d1 = 2R, d2 = 3R, and Rs = 0.5, and the calculation results are shown in Fig. 13. According to the results, for a rigid cylinder, the ARF in a viscous fluid is less than that in an ideal fluid as a whole, and this is the result of the decreased acoustic pressure due to the impact of viscosity.25 For a rigid cylinder in the same fluid, when kR is relatively small, the ARF on a fixed cylinder is much larger than that on a free cylinder. With increasing kR, the ARF curves for the free and fixed cylinders increasingly overlap. Unlike the fixed rigid cylinder, the free cylindrical particle in the acoustic field generates a modified term in the dynamic equation that causes the deviation of the amplitude of the ARF.
In this paper, the ARF on a free rigid cylinder between two impedance boundaries in a viscous fluid was calculated. The solution for scattering waves under normal plane-wave incidence was given, and the ARF on a free rigid cylinder in the viscous fluid between two impedance boundaries was deduced by means of image theory and the translation addition theorem in cylindrical coordinates. Also, how the ARF on a free cylinder varies with boundary parameters, fluid viscosity, particle-to-boundary distance, and fluid–particle density ratio was simulated numerically. The results showed that having one or more boundaries increases the oscillation amplitude of the ARF, and a larger negative ARF can be obtained with two impedance boundaries, with the particle pulled toward the acoustic source. In the case of one or more boundaries, the amplitude of the ARF in a viscous fluid is generally less than that in an ideal fluid. The fluid–particle density ratio of a free cylinder also affects the force.
In conclusion, this work offers some theoretical guidance for applying the ARF in particle manipulation and acoustofluidics. Although we considered the influence of the two boundaries through images, there will still be multiple scattering between the boundaries and particles, and so it is necessary to consider the influence of higher-order scattering if higher accuracy is needed. Furthermore, the ARF for multiple particles near two boundaries in a viscous fluid can be considered; in that case, in addition to the multiple scattering between particles and boundaries, there is also the multiple scattering between particles. Also, analyzing 2D scattering and ARF problems can help in developing analytical solutions in 3D cases. Our next step is to study further the ARF and motion trajectories of particles between impedance boundaries in a viscous fluid by experiments and finite-element simulations.
This work was supported by the National Key R&D Program of China (Grant No. 2020YFA0211400), the State Key Program of the National Natural Science Foundation of China (Grant No. 11834008), the National Natural Science Foundation of China (Grant Nos. 12174192 and 11774167), the State Key Laboratory of Acoustics, Chinese Academy of Science (Grant No. SKLA202210), and the Key Laboratory of Underwater Acoustic Environment, Chinese Academy of Sciences (Grant No. SSHJ-KFKT-1701).
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Xinlei Liu is currently studying for a master’s degree as a member of the Key Laboratory of Modern Acoustics, Institute of Acoustics and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, China. Her research is focused on nonlinear acoustics and acoustic radiation force.
Zhaoyu Deng is currently studying for a doctoral degree as a member of the Key Laboratory of Modern Acoustics, Institute of Acoustics and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, China. His research interests include artificial bubble arrays and nonlinear acoustics.
Li Ma graduated from Nanjing University, China in 1989, majoring in acoustics, and he received master’s and doctoral degrees from Harbin Engineering University and the Institute of Acoustics of the Chinese Academy of Sciences, respectively. He is now engaged mainly in research into physical acoustics and has presided over more than 60 research projects.
Xiaozhou Liu received a Ph.D. in acoustics from Nanjing University, China in 1999 and has been a Professor at Nanjing University since 2007. He was a Visiting Scholar at Pennsylvania State University, USA in 2009. Over the past 30 years, he has conducted both theoretical and experimental research on acoustics and has authored over 150 research papers. His current research interests include nonlinear acoustics, medical ultrasound, and ultrasonic nondestructive testing.