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The acoustic radiation force on a spherical thermoviscous particle in a thermoviscous fluid including scattering and microstreaming

Bjørn G. Winckelmann, Henrik Bruus

TL;DR

This study extends the analytical theory of the acoustic radiation force on a single spherical particle by incorporating elastic solids, temperature- and density-dependent viscosities, tangential Stokes-drift boundary conditions, and inner streaming inside droplets. Using unified potential theory and partial-wave expansions, the authors derive general first- and second-order solutions for both fluids and solids and obtain closed-form long-wavelength expressions for F^rad = F^rad_{11} + F^rad_{2,\text{in}}. They show that microstreaming can dominate F^rad for small particles and provide detailed comparisons with Gor'kov, Settnes–Bruus, Karlsen–Bruus, and Doinikov in various limiting cases, highlighting the importance of boundary-layer effects and thermal coupling. The results refine predictions for acoustofluidic manipulation and aerosol dynamics, particularly in regimes where boundary layers are non-negligible or where viscosity depends on temperature and density.

Abstract

We derive general analytical expressions for the time-averaged acoustic radiation force on a small spherical particle suspended in a fluid and located in an axisymmetric incident acoustic wave. We treat the cases of the particle being either an elastic solid or a fluid particle. The effects of particle vibrations, acoustic scattering, acoustic microstreaming, heat conduction, and temperature-dependent fluid viscosity are all included in the theory. Acoustic streaming inside the particle is also taken into account for the case of a fluid particle. No restrictions are placed on the widths of the viscous and thermal boundary layers relative to the particle radius. We compare the resulting acoustic radiation force with that obtained from previous theories in the literature, and we identify limits, where the theories agree, and specific cases of particle and fluid materials, where qualitative or significant quantitative deviations between the theories arise.

The acoustic radiation force on a spherical thermoviscous particle in a thermoviscous fluid including scattering and microstreaming

TL;DR

This study extends the analytical theory of the acoustic radiation force on a single spherical particle by incorporating elastic solids, temperature- and density-dependent viscosities, tangential Stokes-drift boundary conditions, and inner streaming inside droplets. Using unified potential theory and partial-wave expansions, the authors derive general first- and second-order solutions for both fluids and solids and obtain closed-form long-wavelength expressions for F^rad = F^rad_{11} + F^rad_{2,\text{in}}. They show that microstreaming can dominate F^rad for small particles and provide detailed comparisons with Gor'kov, Settnes–Bruus, Karlsen–Bruus, and Doinikov in various limiting cases, highlighting the importance of boundary-layer effects and thermal coupling. The results refine predictions for acoustofluidic manipulation and aerosol dynamics, particularly in regimes where boundary layers are non-negligible or where viscosity depends on temperature and density.

Abstract

We derive general analytical expressions for the time-averaged acoustic radiation force on a small spherical particle suspended in a fluid and located in an axisymmetric incident acoustic wave. We treat the cases of the particle being either an elastic solid or a fluid particle. The effects of particle vibrations, acoustic scattering, acoustic microstreaming, heat conduction, and temperature-dependent fluid viscosity are all included in the theory. Acoustic streaming inside the particle is also taken into account for the case of a fluid particle. No restrictions are placed on the widths of the viscous and thermal boundary layers relative to the particle radius. We compare the resulting acoustic radiation force with that obtained from previous theories in the literature, and we identify limits, where the theories agree, and specific cases of particle and fluid materials, where qualitative or significant quantitative deviations between the theories arise.
Paper Structure (30 sections, 37 equations, 4 figures, 3 tables)

This paper contains 30 sections, 37 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Thermodynamic identities
  4. Thermoviscous fluids
  5. Thermoelastic solids
  6. The first-order problem
  7. Unified potential theory for fluids and solids
  8. Partial-wave expansions and scattering coefficients
  9. The second-order problem
  10. The general solution
  11. Fluid-solid boundary conditions
  12. Fluid-fluid boundary conditions
  13. The acoustic radiation force
  14. Solid particle
  15. Fluid particle
  16. ...and 15 more sections

Figures (4)

  • Figure 1: (a) The spherical particle with radius $a$, and the referenced coordinate systems used in the calculation. (b) The incident pressure wave, described by the potential $\phi_\mathrm{1c}^\mathrm{in}$ and the corresponding scattered response, described by $\phi_\mathrm{1c}^\mathrm{sc}$, $\phi_\mathrm{1t}^\mathrm{sc}$, $\bm{\psi}^\mathrm{sc}_1$, $\phi_\mathrm{1c}'$, $\phi_\mathrm{1t}'$, and $\bm{\psi}^\mathrm{\prime}_1$. (c) the second order time-averaged streaming rolls generated by the first-order wave scattering.
  • Figure 2: The acoustic contrast factor $\Phi_\mathrm{ac}$ plotted versus normalized boundary-layer width $\delta_s/a$. Brown lines are the ideal-fluid theory following Gor'kov Gorkov1962. Green dashed lines are from Settnes and Bruus Settnes2012, magenta lines are from Karlsen and Bruus Karlsen2015, red lines are from Doinikov 1997 Doinikov1997a, and blue lines are the present theory by Winckelmann and Bruus. (a) A polystyrene sphere in water at frequency $f=1$ MHz. (b) A polystyrene sphere in oil at $f=1$ MHz. (c) A polystyrene sphere in air at $f=1$ kHz. The inset shows the large-particle behavior. (d) A copper sphere in oil at $f=1$ MHz.
  • Figure 3: The acoustic contrast factor $\Phi_\mathrm{ac}$ plotted versus normalized boundary-layer thickness $\delta_s/a$. Brown lines are the ideal-fluid theory following Gor'kov Gorkov1962. Green dashed lines are from Settnes and Bruus Settnes2012, magenta lines are from Karlsen and Bruus Karlsen2015, red lines are from Doinikov 1997 Doinikov1997b, and blue lines are the present theory by Winckelmann and Bruus. (a) An oil droplet in water at frequency $f=1$ MHz. (b) A water droplet in oil at $f=1$ MHz. (c) A water droplet in air at $f=1$ kHz. (d) An oil droplet in air at $f=1$ kHz. The insets in (c) and (d) show the large-particle behavior.
  • Figure 4: The acoustic contrast factor $\Phi_\mathrm{ac}$ from Eq. (\ref{['eq:Phiac']}) plotted versus $\delta_s/a$ computed for a fluid particle in a fluid using the fluid-fluid boundary condition (\ref{['eq:Second_order_no_slip_fl']}) (full curve) and the fluid-solid boundary condition (\ref{['eq:Second_order_no_slip_sl']}) (dashed curve). (a) An oil droplet in water at frequency $f=1$ MHz. (b) A water droplet in oil at $f=1$ MHz.