Acoustic radiation force on a rigid cylinder between two impedance boundaries in a viscous fluid

Liu, Xinlei; Deng, Zhaoyu; Ma, Li; Liu, Xiaozhou

doi:10.1063/10.0013562

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.

HIGHLIGHTS

•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.

I. INTRODUCTION

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.

II. THEORY

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.

FIG. 1.

View large Download slide

Schematic of acoustic scattering for a cylindrical particle between two impedance 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

Φ_{i} = A e^{i k_{1} r \cos φ} e^{- i ω t},

(1)

where $A = \sqrt{2 I / ρ_{0} c_{0} k^{2}}$ 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, $k_{1} = (ω / c_{0}) / \sqrt{1 - i ω (λ + μ) / ρ_{0} c_{0}^{2}}$ 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

Φ_{i} = A \sum_{n = - \infty}^{\infty} i^{n} J_{n} (k_{1} r) e^{i n φ} e^{- i ω t},

(2)

where J_n 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

[(1 + \frac{λ + 2 μ}{c_{0}^{2} ρ_{0}} \frac{\partial}{\partial t}) Δ - \frac{1}{c_{0}^{2}} \frac{\partial^{2}}{\partial t^{2}}] Φ_{s} = 0,

(3)

(ν Δ - \frac{\partial}{\partial t}) Ψ_{s} = 0,

(4)

where ν = μ/ρ₀ is the kinematic viscosity and depends on the temperature, and pressure of the viscous fluid. In the system (r, φ), the solutions of Eqs. (3) and (4) can be expressed as

Φ_{s} = \sum_{n = - \infty}^{\infty} A_{n} H_{n}^{(1)} (k_{1} r) e^{i n φ} e^{- i ω t},

(5)

Ψ_{s} = \sum_{n = - \infty}^{\infty} B_{n} H_{n}^{(1)} (k_{2} r) e^{i n φ} e^{- i ω t},

(6)

where Φ_s and Ψ_s are the scattering longitudinal velocity potential and shear velocity potential, respectively, of a free cylinder in a viscous fluid, A_n and B_n 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 $L = \sqrt{2 ν / ω}$ 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

Φ_{r 1} = A \sum_{n = - \infty}^{\infty} R_{s 1} {(- 1)}^{n} i^{n} J_{n} (k_{1} r) e^{i n φ} e^{2 i k_{1} d_{1}} e^{- i ω t},

(7)

Φ_{r 2} = A \sum_{n = - \infty}^{\infty} R_{s 2} {(- 1)}^{n} i^{n} J_{n} (k_{1} r) e^{i n φ} e^{2 i k_{1} d_{2}} e^{- i ω t},

(8)

where R_s1 and R_s2 are the surface reflection coefficients of the two impedance boundaries, which when the particle is not too close to either boundary can be approximated as^20,31

R_{s_{i}} \approx \frac{1 - β_{i} (ω)}{1 + β_{i} (ω)}, (i = 1, 2),

(9)

where β_i(ω) = ρ₀c₀/Z_i(ω) is the relative admittance of impedance boundary i, and Z_i(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

Φ_{s 1} = \sum_{n = - \infty}^{\infty} {(- 1)}^{n} R_{s 1} A_{n} H_{n}^{(1)} (k_{1} r_{1}) e^{i n φ_{1}} e^{- i ω t},

(10)

Ψ_{s 1} = \sum_{n = - \infty}^{\infty} {(- 1)}^{n} R_{s 1} B_{n} H_{n}^{(1)} (k_{2} r_{1}) e^{i n φ_{1}} e^{- i ω t},

(11)

Φ_{s 2} = \sum_{n = - \infty}^{\infty} {(- 1)}^{n} R_{s 2} A_{n} H_{n}^{(1)} (k_{1} r_{2}) e^{i n φ_{2}} e^{- i ω t},

(12)

Ψ_{s 2} = \sum_{n = - \infty}^{\infty} {(- 1)}^{n} R_{s 2} B_{n} {H_{n}}^{(1)} (k_{2} r_{2}) e^{i n φ_{2}} e^{- i ω t},

(13)

where Φ_s1 and Ψ_s1 are the velocity potentials of the scattered longitudinal and shear waves, respectively, of image cylinder 1 in the viscous fluid, Φ_s2 and Ψ_s2 are those of image cylinder 2, and A_n and B_n 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

Φ_{s 1}^{'} = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} {(- 1)}^{m} R_{s 1} A_{m} H_{n - m}^{(1)} (2 k_{1} d_{1}) J_{n} (k_{1} r) e^{i n φ} e^{- i ω t},

(14)

Ψ_{s 1}^{'} = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} {(- 1)}^{m} R_{s 1} B_{m} H_{n - m}^{(1)} (2 k_{2} d_{1}) J_{n} (k_{2} r) e^{i n φ} e^{- i ω t},

(15)

Φ_{s 2}^{'} = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} {(- 1)}^{m} R_{s 2} A_{m} H_{n - m}^{(1)} (2 k_{1} d_{2}) J_{n} (k_{1} r) e^{i n φ} e^{- i ω t},

(16)

Ψ_{s 2}^{'} = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} {(- 1)}^{m} R_{s 2} B_{m} H_{n - m}^{(1)} (2 k_{2} d_{2}) J_{n} (k_{2} r) e^{i n φ} e^{- i ω t} .

(17)

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.,

\begin{align} Φ & = \sum_{n = - \infty}^{\infty} [F_{n} J_{n} (k_{1} r) + A_{n} H_{n}^{(1)} (k_{1} r) + R_{s 1} S_{1 n} J_{n} (k_{1} r)) \\ + (R_{s 2} S_{2 n} J_{n} (k_{1} r)] e^{i n φ} e^{- i ω t}, \end{align}

(18)

Ψ = \sum_{n = - \infty}^{\infty} [B_{n} {H_{n}}^{(1)} (k_{2} r) + R_{s 1} Q_{1 n} J_{n} (k_{2} r) + R_{s 2} Q_{2 n} J_{n} (k_{2} r)] e^{i n φ} e^{- i ω t},

(19)

where

\begin{align} F_{n} & = A [\exp (\frac{i n π}{2}) + R_{s 1} \exp (- \frac{i n π}{2} + 2 i k_{1} d_{1})) \\ + (R_{s 2} \exp (- \frac{i n π}{2} + 2 i k_{1} d_{2})], \end{align}

(20)

S_{1 n} = \sum_{m = - \infty}^{\infty} {(- 1)}^{m} A_{m} H_{n - m}^{(1)} (2 k_{1} d_{1}),

(21)

S_{2 n} = \sum_{m = - \infty}^{\infty} {(- 1)}^{m} A_{m} H_{n - m}^{(1)} (2 k_{1} d_{2}),

(22)

Q_{1 n} = \sum_{m = - \infty}^{\infty} {(- 1)}^{m} B_{m} H_{n - m}^{(1)} (2 k_{2} d_{1}),

(23)

Q_{2 n} = \sum_{m = - \infty}^{\infty} {(- 1)}^{m} B_{m} H_{n - m}^{(1)} (2 k_{2} d_{2}) .

(24)

To facilitate the calculation, the total velocity potentials are abbreviated as

Φ = \sum_{n = - \infty}^{\infty} (K_{n} + i L_{n}) e^{i n φ} e^{- i ω t},

(25)

Ψ = \sum_{n = - \infty}^{\infty} (M_{n} + i N_{n}) e^{i n φ} e^{- i ω t},

(26)

where

K_{n} = R e [F_{n} J_{n} (k_{1} r) + A_{n} H_{n}^{(1)} (k_{1} r) + R_{s 1} S_{1 n} J_{n} (k_{1} r) + R_{s 2} S_{2 n} J_{n} (k_{1} r)],

(27)

L_{n} = I m [F_{n} J_{n} (k_{1} r) + A_{n} H_{n}^{(1)} (k_{1} r) + R_{s 1} S_{1 n} J_{n} (k_{1} r) + R_{s 2} S_{2 n} J_{n} (k_{1} r)],

(28)

M_{n} = R e [B_{n} {H_{n}}^{(1)} (k_{2} r) + R_{s 1} Q_{1 n} J_{n} (k_{2} r) + R_{s 2} Q_{2 n} J_{n} (k_{2} r)],

(29)

N_{n} = I m [B_{n} {H_{n}}^{(1)} (k_{2} r) + R_{s 1} Q_{1 n} J_{n} (k_{2} r) + R_{s 2} Q_{2 n} J_{n} (k_{2} r)],

(30)

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

v = U,

(31)

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

m \dot{U} = \iint_{s_{0}} σ d S,

(32)

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

σ = (- p + λ \nabla \cdot v) E + 2 μ e,

(33)

\begin{align} p & = ρ_{0} (\frac{λ + 2 μ}{ρ_{0}} Δ - \frac{\partial}{\partial t}) Φ + \frac{ρ_{0}}{2 c_{0}} {(\frac{\partial Φ}{\partial t})}^{2} \\ - \frac{ρ_{0}}{2} {(\vec{\nabla Φ})}^{2} - \frac{λ + 2 μ}{c_{0}^{2}} \frac{\partial Φ}{\partial t} Δ Φ, \end{align}

(34)

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

\frac{\partial Φ}{\partial t} = \frac{d Φ}{d t} - U \cdot \nabla Φ .

(35)

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

{(v_{r}|}_{r = R} = U_{x} \cos φ + U_{y} \sin φ,

(36)

{(v_{φ}|}_{r = R} = - U_{x} \sin φ + U_{y} \cos φ .

(37)

A matrix equation for the sum of the scattering coefficients can be obtained by incorporating Eqs. (19) and (20) for the velocity potentials of the total sound field into Eqs. (32)–(37), i.e.,

a X = b,

(38)

and the scattering coefficients A_n and B_n can be obtained by solving the matrix equation, whose specific expression and coefficients are

a = [\begin{matrix} C_{r} & D_{r} \\ C_{φ} & D_{φ} \end{matrix}] - [\begin{matrix} E_{r} & G_{r} \\ E_{φ} & G_{φ} \end{matrix}] [\begin{matrix} I_{1 n} & O \\ O & I_{2 n} \end{matrix}],

\begin{array}{l} X & = [A_{- n} A_{- n + 1} \cdot \cdot \cdot A_{- 1} A_{0} A_{1} \cdot \cdot \cdot A_{n - 1} A_{n};) \\ {(B_{- n} B_{- n + 1} \cdot \cdot \cdot B_{- 1} B_{0} B_{1} \cdot \cdot \cdot B_{n - 1} B_{n}]}^{T}, \end{array}

\begin{array}{l} b & = [P_{r, - n} P_{r, - n + 1} \cdot \cdot \cdot P_{r, 0} \cdot \cdot \cdot P_{r, n - 1} P_{r, n};) \\ {(P_{φ, - n} P_{φ, - n + 1} \cdot \cdot \cdot P_{φ, 0} \cdot \cdot \cdot P_{φ, n - 1} P_{φ, n}]}^{T}, \end{array}

where the submatrices C_r, C_φ, D_r, D_φ, E_r, E_φ, G_r, and G_φ are diagonal matrices with elements C_r,n, C_φ,n, D_r,n, D_φ,n, E_r,n, E_φ,n, G_r,n, and G_φ,n, respectively. The matrices I_1n and I_2n are square matrices of order m with elements I_1,mn and I_2,mn, respectively, and O is a zero matrix with the same dimensions as those of I_1,mn and I_2,mn. The relevant elements are given specifically as

I_{1, m n} = {(- 1)}^{m} [R_{s 1} H_{n - m}^{(1)} (2 k_{1} d_{1}) + R_{s 2} H_{n - m}^{(1)} (2 k_{1} d_{2})],

I_{2, m n} = {(- 1)}^{m} [R_{s 1} H_{n - m}^{(1)} (2 k_{2} d_{1}) + R_{s 2} H_{n - m}^{(1)} (2 k_{2} d_{2})],

and when |n| ≠ 1, we have

C_{r, n} = n H_{n}^{(1)} (k_{1} r) - k_{1} r H_{n + 1}^{(1)} (k_{1} r),

C_{φ, n} = i n {H_{n}}^{(1)} (k_{1} r),

D_{r, n} = i n {H_{n}}^{(1)} (k_{2} r),

D_{φ, n} = - n H_{n}^{(1)} (k_{2} r) + k_{2} r H_{n + 1}^{(1)} (k_{2} r),

E_{r, n} = - n J_{n} (k_{1} r) + k_{1} r J_{n + 1} (k_{1} r),

E_{φ, n} = - i n J_{n} (k_{1} r),

G_{r, n} = - i n J_{n} (k_{2} r),

G_{φ, n} = n J_{n} (k_{2} r) - k_{2} r J_{n + 1} (k_{2} r),

P_{r, n} = - F_{n} [n J_{n} (k_{1} r) - k_{1} r J_{n + 1} (k_{1} r)],

P_{φ, n} = - n F_{n} J_{n} (k_{1} r),

when n = 1, we have

C_{r, n} = n H_{n}^{(1)} (k_{1} r) - k_{1} r H_{n + 1}^{(1)} (k_{1} r) - n η {H_{n}}^{(1)} (k_{1} r),

D_{r, n} = i (1 - η) n {H_{n}}^{(1)} (k_{2} r),

C_{φ, n} = i n {H_{n}}^{(1)} (k_{1} r) - i η [n H_{n}^{(1)} (k_{1} r) - k_{1} r H_{n + 1}^{(1)} (k_{1} r)],

D_{φ, n} = - n H_{n}^{(1)} (k_{2} r) + k_{2} r H_{n + 1}^{(1)} (k_{2} r) + n η {H_{n}}^{(1)} (k_{2} r),

E_{r, n} = - n J_{n} (k_{1} r) + k_{1} r J_{n + 1} (k_{1} r) + n η J_{n} (k_{1} r),

E_{φ, n} = i (η - 1) n J_{n} (k_{1} r),

G_{r, n} = i (η - 1) n J_{n} (k_{2} r),

G_{φ, n} = n J_{n} (k_{2} r) - k_{2} r J_{n + 1} (k_{2} r) - n η J_{n} (k_{2} r),

P_{r, n} = - F_{n} {[n J_{n} (k_{1} r) - k_{1} r J_{n + 1} (k_{1} r)] - n η J_{n} (k_{1} r)},

P_{φ, n} = - n (1 - η) F_{n} J_{n} (k_{1} r),

and when n = −1, we have

C_{r, n} = n H_{n}^{(1)} (k_{1} r) - k_{1} r H_{n + 1}^{(1)} (k_{1} r) + n η {H_{n}}^{(1)} (k_{1} r),

D_{r, n} = i (1 - η) n {H_{n}}^{(1)} (k_{2} r),

C_{φ, n} = i n {H_{n}}^{(1)} (k_{1} r) - i η [n H_{n}^{(1)} (k_{1} r) - k_{1} r H_{n + 1}^{(1)} (k_{1} r)],

D_{φ, n} = - n H_{n}^{(1)} (k_{2} r) + k_{2} r H_{n + 1}^{(1)} (k_{2} r) - n η {H_{n}}^{(1)} (k_{2} r),

E_{r, n} = - n J_{n} (k_{1} r) + k_{1} r J_{n + 1} (k_{1} r) - n η J_{n} (k_{1} r),

E_{φ, n} = i (η - 1) n J_{n} (k_{1} r),

G_{r, n} = i (η - 1) n J_{n} (k_{2} r),

G_{φ, n} = n J_{n} (k_{2} r) - k_{2} r J_{n + 1} (k_{2} r) + n η J_{n} (k_{2} r),

P_{r, n} = F_{n} {[n J_{n} (k_{1} r) - k_{1} r J_{n + 1} (k_{1} r)] + n η J_{n} (k_{1} r)},

P_{φ, n} = n (1 - η) F_{n} J_{n} (k_{1} r) .

B. Acoustic radiation force

The ARF of a particle in a viscous fluid can be expressed as³⁵

F = ⟨\iint_{s_{0}} σ d S⟩,

(39)

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

F_{x} = ⟨\int_{0}^{2 π} (σ_{r r} \cos φ - σ_{r φ} \sin φ) R d φ⟩,

(40)

where

σ_{r r} = - p + 2 μ \frac{\partial v_{r}}{\partial r} + λ (\frac{\partial v_{r}}{\partial r} + \frac{1}{r} \frac{\partial v_{φ}}{\partial_{φ}} + \frac{v_{r}}{r}),

(41)

σ_{r φ} = μ (\frac{1}{r} \frac{\partial v_{r}}{\partial φ} + \frac{\partial v_{φ}}{\partial r} - \frac{v_{φ}}{r}),

(42)

v_{r} = \frac{\partial Φ}{\partial r} + \frac{1}{r} \frac{\partial Ψ}{\partial φ},

(43)

v_{φ} = \frac{1}{r} \frac{\partial Φ}{\partial φ} - \frac{\partial Ψ}{\partial r} .

(44)

Substituting Eqs. (18), (19), (33)–(35), and (41)–(44) into Eq. (39) gives the axial ARF as

\begin{align} F_{x} & = \frac{η ρ_{0} π}{R} [- (K_{1} + N_{1}) K_{2} - (L_{1} - M_{1}) L_{2} + 2 M_{- 1} M_{- 2} + (N_{- 1} - K_{- 1}) K_{- 2} - (L_{- 1} + M_{- 1}) L_{- 2} + L_{- 1} M_{- 2} - K_{- 1} N_{- 2}] \\ - \frac{ρ_{0} π R ω^{2}}{4 c_{0}^{2}} \sum_{n = - \infty}^{\infty} (K_{n} K_{n + 1} + L_{n} L_{n + 1} + K_{n} K_{n - 1} + L_{n} L_{n - 1}) + \frac{ρ_{0} π}{4 R} \sum_{n = - \infty}^{\infty} [n (n + 1) (K_{n} K_{n + 1} + L_{n} L_{n + 1})) \\ + (n (n - 1) (K_{n} K_{n - 1} + L_{n} L_{n - 1})] + \frac{ρ_{0} π}{4 R} \sum_{n = - \infty}^{\infty} [n (n + 1) (M_{n} M_{n + 1} + N_{n} N_{n + 1}) + n (n - 1) (M_{n} M_{n - 1} + N_{n} N_{n - 1})] \\ - \frac{ρ_{0}^{2} π R (λ + 2 μ) ω^{3}}{2 [ρ_{0}^{2} c_{0}^{2} + ω^{2} {(λ + 2 μ)}^{2}]} \sum_{n = - \infty}^{\infty} (L_{n} K_{n + 1} + L_{n} K_{n - 1} - K_{n} L_{n + 1} - K_{n} L_{n - 1}) + \frac{ρ_{0} π R {(λ + 2 μ)}^{2} ω^{4}}{2 c_{0}^{2} [ρ_{0}^{2} c_{0}^{2} + ω^{2} {(λ + 2 μ)}^{2}]} \\ \times \sum_{n = - \infty}^{\infty} (K_{n} K_{n + 1} + L_{n} L_{n + 1} + K_{n} K_{n - 1} + L_{n} L_{n - 1}), \end{align}

(45)

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

\begin{align} F_{x} & = - \frac{η ρ_{0} π}{R} (K_{1} K_{2} + L_{1} L_{2} + K_{- 1} K_{- 2} + L_{- 1} L_{- 2}) \\ - \frac{ρ_{0} π R ω^{2}}{4 c_{0}^{2}} \sum_{n = - \infty}^{\infty} (K_{n} K_{n + 1} + L_{n} L_{n + 1} + K_{n} K_{n - 1} + L_{n} L_{n - 1}) \\ + \frac{ρ_{0} π}{4 R} \sum_{n = - \infty}^{\infty} [n (n + 1) (K_{n} K_{n + 1} + L_{n} L_{n + 1}) + n (n - 1)) \\ (\times (K_{n} K_{n - 1} + L_{n} L_{n - 1})] . \end{align}

(46)

The ARF can also be expressed by the dimensionless ARF function Y_p, which represents the radiation force per unit energy density and unit cross-section surface. The relationship between F_x and Y_p is

F_{x} = Y_{p} S ⟨E⟩,

(47)

⟨E⟩ = \frac{1}{2} ρ_{0} k^{2} A^{2} = \frac{I}{c_{0}}, S = 2 R,

(48)

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 F_x and Y_p, 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 N_max = M_max = 40.

TABLE I.

Acoustic parameters of fluids in simulation.

	Density	Acoustic	Dynamic
Liquid	ρ₀ (kg/m³)	velocity c₀ (m/s)	viscosity μ (Pa⋅s)
Water	1000	1480	0.001
Glycerol	1260	1920	1.499

	Density	Acoustic	Dynamic
Liquid	ρ₀ (kg/m³)	velocity c₀ (m/s)	viscosity μ (Pa⋅s)
Water	1000	1480	0.001
Glycerol	1260	1920	1.499

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.

FIG. 2.

View large Download slide

Y_p versus kR for a fixed (η = 0) and free (η = 1/2) rigid cylinder in water with no boundaries.

FIG. 3.

View large Download slide

Normalized acoustic radiation force (ARF) versus kd for a fixed rigid cylinder near one rigid boundary in an ideal fluid: kR = (a) 0.126 and (b) 0.063.

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 −F_x 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 Y_p, 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.

FIG. 4.

View large Download slide

Y_p versus kR for a fixed rigid cylinder near one impedance boundary in water with different values of the reflection coefficient: d = (a) 2R and (b) 4R.

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 R_s. 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.

FIG. 5.

View large Download slide

Y_p versus kR for a free rigid cylinder (η = 1/2) near one impedance boundary in a viscous fluid with different values of the reflection coefficient: d = (a) 2R and (b) 4R.

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.

FIG. 6.

View large Download slide

Y_p versus kR for a free rigid cylinder (η = 1/2) between two impedance boundaries in a viscous fluid with different values of the reflection coefficient for d₁ = 2R and d₂ = 4R.

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 Y_p varies with d₂/R. According to the results, the variation of Y_p 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.

FIG. 7.

View large Download slide

Y_p versus d₂/R for a free rigid cylinder (η = 1/2) in a viscous fluid between two impedance boundaries with different values of the reflection coefficient for kR = 0.5.

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 R_s = 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.

FIG. 8.

View large Download slide

Y_p versus kR for a free rigid cylinder (η = 1/2) between two impedance boundaries in a viscous fluid at different values of d₂/R: (a) 3D color map; (b) 2D contours.

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 R_s = 0.5. For a smaller cylindrical particle (smaller kR), Y_p decreases with increasing L. For a larger cylindrical particle, Y_p 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

FIG. 9.

View large Download slide

Y_p versus kR for a free rigid cylinder between two impedance boundaries in a viscous fluid with different values of viscosity.

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 R_s = 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 Y_p 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 R_s = 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.

FIG. 10.

View large Download slide

Y_p versus frequency for a free rigid cylinder (η = 1/2) between two impedance boundaries in a viscous fluid with different values of cylinder radius.

FIG. 11.

View large Download slide

Y_p versus frequency for a rigid cylinder (η = 0) near one impedance boundary in water with different values of cylinder radius.

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, Y_p decreases initially and then increases with increasing η, and when d₂/R is sufficiently large, Y_p 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.

FIG. 12.

View large Download slide

Y_p versus η for a free cylinder between two impedance boundaries in a viscous fluid at different values of (a) d₂/R and (b) kR.

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 R_s = 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.

FIG. 13.

View large Download slide

Y_p versus kR for a free rigid cylinder between two impedance boundaries in four cases: ideal fluid, viscous fluid, fixed rigid cylinder, and free rigid cylinder.

IV. CONCLUSIONS

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.

ACKNOWLEDGMENTS

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).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

REFERENCES

1.

Rufo

J

,

Cai

F

,

Friend

J

,

Wiklund

M

,

Huang

TJ

.

Acoustofluidics for biomedical applications

.

Nat Rev Methods Primers

2022

;

2

:

30

.

https://doi.org/10.1038/s43586-022-00109-7

Google Scholar

Crossref

2.

Wu

H

,

Tang

Z

,

You

R

,

Pan

S

,

Liu

W

,

Zhang

H

,

Li

T

,

Yang

Y

,

Sun

C

,

Pang

W

,

Duan

X

.

Manipulations of micro/nanoparticles using gigahertz acoustic streaming tweezers

.

Nanotechnol Precis Eng

2022

;

5

:

023001

.

https://doi.org/10.1063/10.0009954

Google Scholar

Crossref

3.

Goldberg

ZA

, Acoustic radiation pressure in high-intensity ultrasonic fields.

New York

:

Plenum

.

1971

.

4.

Baudoin

M

,

Thomas

J-L

.

Acoustic tweezers for particle and fluid micromanipulation

.

Annu Rev Fluid Mech

2020

;

52

:

205

–

234

.

https://doi.org/10.1146/annurev-fluid-010719-060154

Google Scholar

Crossref

5.

Ozcelik

A

,

Rufo

J

,

Guo

F

,

Gu

Y

,

Li

P

,

Lata

J

,

Huang

TJ

.

Acoustic tweezers for the life sciences

.

Nat Methods

2018

;

15

:

1021

–

1028

.

https://doi.org/10.1038/s41592-018-0222-9

Google Scholar

Crossref

PubMed

6.

Andrade

MAB

,

Pérez

N

,

Adamowski

JC

.

Review of progress in acoustic levitation

.

Braz J Phys

2018

;

48

:

190

–

213

.

https://doi.org/10.1007/s13538-017-0552-6

Google Scholar

Crossref

7.

Tang

T

,

Huang

L

.

Acoustic radiation force for multiple particles over a wide size-scale by multiple ultrasound sources

.

J Sound Vib

2021

;

509

:

116256

.

https://doi.org/10.1016/j.jsv.2021.116256

Google Scholar

Crossref

8.

Yang

S

,

Tian

Z

,

Wang

Z

,

Rufo

J

,

Li

P

,

Mai

J

,

Xia

J

,

Bachman

H

,

Huang

PH

,

Wu

M

.

Harmonic acoustics for dynamic and selective particle manipulation

.

Nat Mater

2022

;

21

:

540

–

546

.

https://doi.org/10.1038/s41563-022-01210-8

Google Scholar

Crossref

PubMed

9.

Tian

Z

,

Yang

S

,

Huang

PH

,

Wang

Z

,

Zhang

P

,

Gu

Y

,

Bachman

H

,

Chen

C

,

Wu

M

,

Xie

Y

,

Huang

TJ

.

Wave number–spiral acoustic tweezers for dynamic and reconfigurable manipulation of particles and cells

.

Sci Adv

2019

;

5

:

eaau6062

.

https://doi.org/10.1126/sciadv.aau6062

Google Scholar

Crossref

PubMed

10.

Zehnter

S

,

Andrade

MAB

,

Ament

C

.

Acoustic levitation of a Mie sphere using a 2D transducer array

.

J Appl Phys

2021

;

129

:

134901

.

https://doi.org/10.1063/5.0037344

Google Scholar

Crossref

11.

Zhang

P

,

Rufo

J

,

Chen

C

, et al.

Acoustoelectronic nanotweezers enable dynamic and large-scale control of nanomaterials

.

Nat Commun

2021

;

12

:

3844

.

https://doi.org/10.1038/s41467-021-24101-z

Google Scholar

Crossref

PubMed

12.

King

LV

.

On the acoustic radiation pressure on spheres

.

Proc R Soc London, Ser A

1934

;

147

:

212

–

240

.

https://doi.org/10.1098/rspa.1934.0215

Google Scholar

Crossref

13.

Yosioka

K

,

Kawasima

Y

.

Acoustic radiation pressure on a compressible sphere

.

Acta Acust Acust

1955

;

5

:

167

–

173

.

Google Scholar

14.

Wu

J

,

Du

G

.

Acoustic radiation force on a small compressible sphere in a focused beam

.

J Acoust Soc Am

1990

;

87

:

997

–

1003

.

https://doi.org/10.1121/1.399435

Google Scholar

Crossref

15.

Hasegawa

T

,

Hino

Y

,

Annou

A

,

Noda

H

,

Kato

M

,

Inoue

N

.

Acoustic radiation pressure acting on spherical and cylindrical shells

.

J Acoust Soc Am

1993

;

93

:

154

.

https://doi.org/10.1121/1.405653

Google Scholar

Crossref

16.

Mitri

FG

.

Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves

.

J Appl Phys

2015

;

118

:

214903

.

https://doi.org/10.1063/1.4936617

Google Scholar

Crossref

17.

Zhang

X

,

Yun

Q

,

Zhang

G

,

Sun

X

.

Computation of the acoustic radiation force on a rigid cylinder in off-axial Gaussian beam using the translational addition theorem

.

Acta Acust Acust

2016

;

102

:

334

–

340

.

https://doi.org/10.3813/aaa.918949

Google Scholar

Crossref

18.

Marston

PL

.

Quasi-Gaussian Bessel-beam superposition: Application to the scattering of focused waves by spheres

.

J Acoust Soc Am

2011

;

129

:

1773

–

1782

.

https://doi.org/10.1121/1.3559704

Google Scholar

Crossref

PubMed

19.

Gaunaurd

GC

,

Huang

H

.

Acoustic scattering by a spherical body near a plane boundary

.

J Acoust Soc Am

1994

;

96

:

2526

–

2536

.

https://doi.org/10.1121/1.410126

Google Scholar

Crossref

20.

Hasheminejad

SM

,

Azarpeyvand

M

.

Modal vibrations of a cylindrical radiator over an impedance plane

.

J Sound Vib

2004

;

278

:

461

–

477

.

https://doi.org/10.1016/j.jsv.2003.10.039

Google Scholar

Crossref

21.

Miri

AK

,

Mitri

FG

.

Acoustic radiation force on a spherical contrast agent shell near a vessel porous wall–theory

.

Ultrasound Med Biol

2011

;

37

:

301

–

311

.

https://doi.org/10.1016/j.ultrasmedbio.2010.11.006

Google Scholar

Crossref

PubMed

22.

Wang

J

,

Dual

J

.

Theoretical and numerical calculation of the acoustic radiation force acting on a circular rigid cylinder near a flat wall in a standing wave excitation in an ideal fluid

.

Ultrasonics

2012

;

52

:

325

–

332

.

https://doi.org/10.1016/j.ultras.2011.09.002

Google Scholar

Crossref

PubMed

23.

Qiao

Y

,

Zhang

X

,

Zhang

G

.

Acoustic radiation force on a fluid cylindrical particle immersed in water near an impedance boundary

.

J Acoust Soc Am

2017

;

141

:

4633

–

4641

.

https://doi.org/10.1121/1.4986624

Google Scholar

Crossref

PubMed

24.

Zang

Y

,

Qiao

Y

,

Liu

J

,

Liu

X

.

Axial acoustic radiation force on a fluid sphere between two impedance boundaries for Gaussian beam

.

Chin Phys B

2019

;

28

:

034301

.

https://doi.org/10.1088/1674-1056/28/3/034301

Google Scholar

Crossref

25.

Qiao

Y

,

Zhang

X

,

Gong

M

,

Wang

H

,

Liu

X

.

Acoustic radiation force and motion of a free cylinder in a viscous fluid with a boundary defined by a plane wave incident at an arbitrary angle

.

J Appl Phys

2020

;

128

:

044902

.

https://doi.org/10.1063/5.0005866

Google Scholar

Crossref

26.

Zang

YC

,

Lin

WJ

,

Su

C

,

Wu

PF

.

Axial acoustic radiation force on an elastic spherical shell near an impedance boundary for zero-order quasi-Bessel–Gauss beam

.

Chin Phys B

2021

;

30

:

044301

.

https://doi.org/10.1088/1674-1056/abca27

Google Scholar

Crossref

27.

Van Der Pol

B

.

Theory of the reflection of the light from a point source by a finitely conducting flat mirror, with an application to radiotelegraphy

.

Physica

1935

;

2

:

843

–

853

.

https://doi.org/10.1016/s0031-8914(35)90168-9

Google Scholar

Crossref

28.

Morse

PM

,

Bolt

RH

.

Sound waves in rooms

.

Rev Mod Phys

1944

;

16

:

69

.

https://doi.org/10.1103/revmodphys.16.69

Google Scholar

Crossref

29.

Lin

WH

,

Raptis

AC

.

Acoustic scattering by elastic solid cylinders and spheres in viscous fluids

.

J Acoust Soc Am

1983

;

73

:

736

–

748

.

https://doi.org/10.1121/1.389039

Google Scholar

Crossref

30.

Guz

N

.

On the representation of solutions of linearized Navier-Stokes equations

.

Dokl Akad Nauk SSSR

1980

;

253

:

825

–

827

.

Google Scholar

31.

Hasheminejad

SM

.

Modal acoustic force on a spherical radiator in an acoustic halfspace with locally reacting boundary

.

Acta Acust Acust

2001

;

87

:

443

–

453

.

Google Scholar

32.

Abramowitz

M

,

Stegun

IA

. Handbook of mathematical functions: With formulas, graphs, and mathematical tables.

US Government Printing Office

.

1965

.

33.

Embleton

TFW

.

Mean force on a sphere in a spherical sound field. I. (Theoretical)

.

J Acoust Soc Am

1954

;

26

:

40

–

45

.

https://doi.org/10.1121/1.1907286

Google Scholar

Crossref

34.

Guz

AN

,

Zhuk

AP

,

Hydrodynamic forces acting in an acoustic field in a viscous fluid

.

Dokl Akad Nauk SSSR

1982

;

266

:

32

–

35

.

Google Scholar

35.

Guz

AN

,

Zhuk

AP

.

Dynamics of a rigid cylinder near a plane boundary in the radiation field of an acoustic wave

.

J Fluid Struct

2009

;

25

:

1206

–

1212

.

https://doi.org/10.1016/j.jfluidstructs.2009.06.003

Google Scholar

Crossref

36.

Guz

AN

,

Zhuk

AP

.

Motion of solid particles in a liquid under the action of an acoustic field: The mechanism of radiation pressure

.

Int Appl Mech

2004

;

40

:

246

–

265

.

https://doi.org/10.1023/b:inam.0000031907.28520.7b

Google Scholar

Crossref

37.

Azarpeyvand

M

,

Azarpeyvand

M

.

Acoustic radiation force on a rigid cylinder in a focused Gaussian beam

.

J Sound Vib

2013

;

332

:

2338

–

2349

.

https://doi.org/10.1016/j.jsv.2012.11.002

Google Scholar

Crossref

38.

Qiao

YP

,

Zhang

XF

,

Acoustic radiation force on a rigid cylindrical particle in water near an impedance interface

.

J Nanjing Univ

2017

;

53

:

19

(in Chinese).

Google Scholar

39.

Hasheminejad

SM

,

Badsar

SA

.

Acoustic scattering by a pair of poroelastic spheres

.

Q J Mech Appl Math

2004

;

57

:

95

–

113

.

https://doi.org/10.1093/qjmam/57.1.95

Google Scholar

Crossref

40.

Hasheminejad

SM

,

Alibakhshi

MA

.

Diffraction of sound by a poroelastic cylindrical absorber near an impedance plane

.

Int J Mech Sci

2007

;

49

:

1

–

12

.

https://doi.org/10.1016/j.ijmecsci.2006.08.011

Google Scholar

Crossref

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.

2022

Author(s)

Acoustic radiation force on a rigid cylinder between two impedance boundaries in a viscous fluid

HIGHLIGHTS

I. INTRODUCTION

II. THEORY

A. Scattering of plane wave on free cylinder immersed in viscous fluid between two impedance boundaries

B. Acoustic radiation force

III. SIMULATION AND DISCUSSION

A. Initial test

B. Effects of impedance boundaries

C. Effects of viscosity

D. Effects of cylinder size

E. Effects of density ratio η

IV. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Acoustic radiation force on a rigid cylinder between two impedance boundaries in a viscous fluid

HIGHLIGHTS

I. INTRODUCTION

II. THEORY

A. Scattering of plane wave on free cylinder immersed in viscous fluid between two impedance boundaries

B. Acoustic radiation force

III. SIMULATION AND DISCUSSION

A. Initial test

B. Effects of impedance boundaries

C. Effects of viscosity

D. Effects of cylinder size

E. Effects of density ratio η

IV. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only