Forces acting on a small particle in an acoustical field in a viscous fluid

We calculate the acoustic radiation force from an ultrasound wave on a compressible, spherical particle suspended in a viscous fluid. Using Prandtl-Schlichting boundary-layer theory, we include the kinematic viscosity of the solvent and derive an analytical expression for the resulting radiation force, which is valid for any particle radius and boundary-layer thickness provided that both of these length scales are much smaller than the wavelength of the ultrasound wave (millimeters in water at megahertz frequencies). The acoustophoretic response of suspended microparticles is predicted and analyzed using parameter values typically employed in microchannel acoustophoresis.

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Figure 1

(a) Sketch of the far-field region $r\gg \lambda$ of an incoming acoustic wave ${\phi }_{\mathrm{in}}^{}$ (vertical lines) of wavelength $\lambda$ scattering off a small particle (black dot) with radius $a\ll \lambda$, leading to the outgoing scattered wave ${\phi }_{\mathrm{sc}}^{}$ (circles and arrows). The resulting first-order wave is ${\phi }_1^{}={\phi }_{\mathrm{in}}^{}+{\phi }_{\mathrm{sc}}^{}$. (b) Sketch of a compressible spherical particle of radius $a$, compressibility ${\kappa }_{\mathrm{p}}^{{}_{}}$, and density ${\rho }_{\mathrm{p}}^{{}_{}}$, surrounded by the incompressible, viscous acoustic boundary layer of width $\sim$$5\delta$ (dark shade) with density ${\rho }_0^{{}_{}}$ and kinematic viscosity $\nu$. Outside is the compressible inviscid bulk (light shade) of compressibility ${\kappa }_0^{}$ and density ${\rho }_0^{{}_{}}$. The bulk liquid is divided into the near-field region for $r\ll \lambda$, with the instantaneous scattered field ${\phi }_{\mathrm{sc}}^{}\left(t\right)$, and the far-field region with time-retarded scattered field ${\phi }_{\mathrm{sc}}^{}(t-r/c_0^{})$.

Figure 2

(a) The real part of the viscosity-dependent dipole scattering coefficient relative to its inviscid counterpart $f_2^{{\mathrm{r}}_{}}(\tilde{\rho },\tilde{\delta })/f_2^{{\mathrm{r}}_{}}(\tilde{\rho },0)$ plotted vs the nondimensionalized thickness $\tilde{\delta }$ of the viscous, acoustic boundary layer. (b) The imaginary part $f_2^{{\mathrm{i}}_{}}(\tilde{\rho },\tilde{\delta })$ vs $\tilde{\delta }$ for $\tilde{\rho }=1.25$, 1.50, 1.75, and 2.00. For the same parameters $0.17<f_2^{{\mathrm{r}}_{}}(\rho ,\infty )<0.67$.

Figure 3

The detectability $\mathcal{D}$ for solid microspheres of radius $a$ ranging from 0.1 to $10\mu \mathrm{m}$ in a sodium chloride solution as a function of NaCl concentration (mass fraction $S$), see Appendix . (a) Polystyrene spheres for which $\mathcal{D}$ is typically a few percent or lower. (b) Pyrex spheres for which $\mathcal{D}$ easily can be larger than 10$\%$.

Figure 4

Contour plots of the acoustic contrast factor $\Phi (\tilde{\kappa },\tilde{\rho },\tilde{\delta })$ as function of $\tilde{\kappa }$ and $\tilde{\rho }$ for fixed values of $\tilde{\delta }$, from $\Phi <-0.5$ (black) to $\Phi >0.5$ (white) in steps of 0.1. The position in the $(\tilde{\rho },\tilde{\kappa })$ plane for various material parameters are marked by crosses and the following labels: polystyrene (PS), polymethacrylate (PM), melamine resin (MR), a typical biological cell (Cell), water (wa), glycerol (gl), and percoll (pc). (a) The inviscid case $\tilde{\delta }=0$. (b) The viscous case $\tilde{\delta }=1$.