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 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/m
3) 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
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ρp (SI unit: kg/m 3) is the particle density,
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cp,p (SI unit: m/s) is the pressure-wave speed of the solid particle,
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cs,p (SI unit: m/s) is the shear-wave speed of the solid particle,
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ρ (SI unit: kg/m 3) is the fluid density, and
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c (SI unit: m/s) is the fluid speed of sound.
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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
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ρp (SI unit: kg/m 3) is the droplet density,
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cp (SI unit: m/s) is the droplet speed of sound,
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ρ (SI unit: kg/m 3) is the fluid density, and
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c (SI unit: m/s) is the fluid speed of sound.
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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
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ρp (SI unit: kg/m 3) is the particle density,
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cp,p (SI unit: m/s) is the pressure-wave speed of the solid particle,
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cs,p (SI unit: m/s) is the shear-wave speed of the solid particle,
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ρ (SI unit: kg/m 3) is the fluid density, and
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c (SI unit: m/s) is the fluid speed of sound.
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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
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ρp (SI unit: kg/m 3) is the droplet density,
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cp (SI unit: m/s) is the droplet speed of sound,
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ρ (SI unit: kg/m 3) is the fluid density, and
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c (SI unit: m/s) is the fluid speed of sound.
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The functions F(xs, xs,p) and
G(xs) are dimensionless help functions,
where xs and x
s,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
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ρp (SI unit: kg/m 3) is the particle density,
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cp,p (SI unit: m/s) is the pressure-wave speed of the solid particle,
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cs,p (SI unit: m/s) is the shear-wave speed of the solid particle,
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αp,p (SI unit: 1/K) is the isobaric coefficient of thermal expansion of the particle,
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Cp,p (SI unit: J/(kg K)) is the particle thermal conductivity,
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ρ (SI unit: kg/m 3) is the fluid density,
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c (SI unit: m/s) is the fluid speed of sound,
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αp (SI unit: 1/K) is the isobaric coefficient of thermal expansion of the fluid, and
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Cp (SI unit: J/(kg K)) is the fluid thermal conductivity.
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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
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ρp (SI unit: kg/m 3) is the droplet density,
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cp (SI unit: m/s) is the adiabatic sound speed of the droplet,
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αp,p (SI unit: 1/K) is the isobaric coefficient of thermal expansion of the droplet,
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Cp,p (SI unit: J/(kg K)) is the droplet thermal conductivity,
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ρ (SI unit: kg/m 3) is the fluid density,
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c (SI unit: m/s) is the fluid speed of sound,
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αp (SI unit: 1/K) is the isobaric coefficient of thermal expansion of the fluid, and
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Cp (SI unit: J/(kg K)) is the fluid thermal conductivity.
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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,
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γ (dimensionless) is the ratio of specific heats of the surrounding fluid,
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γp (dimensionless) is the ratio of specific heats of the droplet,
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δth (SI unit: m) is the thermal boundary layer thickness of the surrounding fluid,
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δt h,p (SI unit: m) is the thermal boundary layer thickness of the droplet,
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Γs (dimensionless) is the shear bulk damping factor of the fluid,
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Γs,p (dimensionless) is the shear bulk damping factor of the droplet,
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Γth (dimensionless) is the thermal bulk damping factor of the fluid, and
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Γth (dimensionless) is the thermal bulk damping factor of the droplet.
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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,