The Acoustophoretic Radiation Force node defines a force on particles in an acoustic pressure field. For any force to be applied to the particles, the acoustic pressure field and acoustic velocity field must be solved for in a Frequency Domain study, prior to solving for the particle trajectories in the time domain. If the force on the particles appears to be zero, check that a Frequency Domain study has been selected in the Values of variables not solved for in the settings for the Time Dependent study for the particle trajectories.

The acoustophoretic radiation force is applicable for small particles in the Rayleigh limit, where the product of particle radius and wave number is much smaller than unity. In this limit, scattering theory applies and the scattering coefficients can be computed analytically for drops and solid particles. The first classical expression below is the Ideal model given by Gor’kov (Ref. 22). If the particle radius becomes comparable to the viscous or thermal boundary layer thicknesses of the fluid, viscous and thermal effects need to be taken into account (Ref. 23, Ref. 24). The Viscous and Thermoviscous models include these additional effects.

The acoustic radiation force Frad is a function of the acoustic pressure field p (SI unit: Pa) and the acoustic velocity field uin (SI unit: m/s),

where rp (SI unit: m) is the particle radius and κs (SI unit: 1/m) is the isentropic compressibility of the fluid surrounding the particle,

where ρ (SI unit: kg/m3) is the fluid density and c (SI unit: m/s) is its adiabatic sound speed.

The dimensionless quantities f0 and f1 are respectively the monopole and dipole scattering coefficients. Their expressions change depending on whether the particles are solids or liquid, and on whether the effects of viscosity and thermal conductivity are considered in the derivation of the force.

The Acoustophoretic Radiation Force supports three different types of Thermodynamic loss model: Ideal, Viscous, and Thermoviscous. In addition, the model particles may be solid particles or liquid droplets. In total there are six different expressions for the monopole and dipole scattering coefficients. In the following sections, the monopole and dipole scattering coefficients will be indicated with a superscript sl if the particles are solid, or a superscript fl if the particles are liquid droplets.

If Solid particle is selected from the Particle type list, and Ideal is selected from the Thermodynamic loss model list, then the monopole and dipole scattering coefficients are

If Liquid droplet is selected from the Particle type list, and Ideal is selected from the Thermodynamic loss model list, then the monopole and dipole scattering coefficients are

If Solid particle is selected from the Particle type list, and Viscous is selected from the Thermodynamic loss model list, then the monopole and dipole scattering coefficients are

The function G(xs) is a dimensionless help function,

where xs is the dimensionless shear wave number of the fluid,

and in turn rp (SI unit: m) is the particle radius and ks (SI unit: 1/m) is the shear (or viscous) wave number of the fluid,

where δs (SI unit: m) is the shear (or viscous) boundary layer thickness,

where μ (SI unit: Pa·s) is the fluid dynamic viscosity, and ω (SI unit: rad/m) is the angular frequency of the acoustic pressure field.

If Liquid droplet is selected from the Particle type list, and Viscous is selected from the Thermodynamic loss model list, then the monopole and dipole scattering coefficients are

The functions F(xs, xs,p) and G(xs) are dimensionless help functions,

where xs and xs,p are respectively the dimensionless shear wave numbers of the surrounding fluid and the droplet,

and in turn rp (SI unit: m) is the particle radius, and ks and ks,p (SI unit: 1/m) are respectively the shear (or viscous) wave number of the surrounding fluid and the droplet,

where δs and δs,p (SI unit: m) are respectively the shear (or viscous) boundary layer thickness of the surrounding fluid and the droplet,

where μ (SI unit: Pa·s) is the dynamic viscosity of the surrounding fluid, μp (SI unit: Pa·s) is the dynamic viscosity of the liquid droplet, and ω (SI unit: rad/m) is the angular frequency of the acoustic pressure field.

If Solid particle is selected from the Particle type list, and Thermoviscous is selected from the Thermodynamic loss model list, then the monopole and dipole scattering coefficients are

The functions G(xs) and H(xth, xth,p) are dimensionless help functions,

both the particle thermal conductivity kp and the fluid thermal conductivity k (SI unit: W/(m K)) are assumed isotropic, hence they are treated as scalar quantities. In addition, xth is the dimensionless thermal wave number of the fluid, xth,p is the dimensionless thermal wave number of the particle, and xs is the dimensionless shear wave number of the fluid,

and in turn rp (SI unit: m) is the particle radius and ks (SI unit: 1/m) is the shear (or viscous) wave number of the fluid,

where δs (SI unit: m) is the shear (or viscous) boundary layer thickness,

where μ (SI unit: Pa s) is the fluid dynamic viscosity, and ω (SI unit: rad/m) is the angular frequency of the acoustic pressure field.

Furthermore, kth and kth,p (SI unit: 1/m) are, respectively, the thermal wave numbers of the surrounding fluid and the solid particle,

In the surrounding fluid, δth (SI unit: m) is the thermal boundary layer thickness,

γ (dimensionless) is the ratio of specific heats, Γs (dimensionless) is the viscous bulk damping factor, and Γth (dimensionless) is the thermal bulk damping factor. The bulk damping factors are defined as

where μB (SI unit: Pa·s) is the fluid bulk viscosity and k0 (SI unit: rad/m) is the lossless wave number,

For the solid particle, δth,p (SI unit: m) is the thermal boundary layer thickness,

γp (dimensionless) is the ratio of specific heats, and Γth (dimensionless) is the thermal bulk damping factor,

Xp is a dimensionless help variable,

The lossless wave number of the particle, k0,p (SI unit: rad/m) is based on the compressional speed of sound,

If Liquid droplet is selected from the Particle type list, and Thermoviscous is selected from the Thermodynamic loss model list, then the monopole and dipole scattering coefficients are

The functions F(xs, xs,p), G(xs), and H(xth, xth,p) are dimensionless help functions,

both the droplet thermal conductivity kp and the surrounding fluid thermal conductivity k (SI unit: W/(m K)) are assumed isotropic, hence they are treated as scalar quantities. In addition, xth is the dimensionless thermal wave number of the surrounding fluid, xth,p is the dimensionless thermal wave number of the droplet, xs is the dimensionless shear wave number of the surrounding fluid, and xth,p is the dimensionless shear wave number of the droplet,

and in turn rp (SI unit: m) is the droplet radius, ks (SI unit: 1/m) is the shear (or viscous) wave number of the fluid, and ks,p (SI unit: 1/m) is the shear (or viscous) wave number of the droplet,

δs and δs,p (SI unit: m) are respectively the shear (or viscous) boundary layer thickness of the surrounding fluid and the droplet,

μ and μp (SI unit: Pa s) are respectively the dynamic viscosity of the surrounding fluid and the droplet, and ω (SI unit: rad/m) is the angular frequency of the acoustic pressure field.

kth and kth,p (SI unit: 1/m) are, respectively, the thermal wave numbers of the surrounding fluid and the droplet,

where μB and μB,p (SI unit: Pa·s) are respectively the bulk viscosity of the surrounding fluid and the droplet. The lossless wave numbers are defined in terms of the speed of sound,