L4 · Chapter 8

Reading the 2012 paper by Barnkob and co-workers that separated radiation force from streaming

Sandbox Physics · 2026-10-29

The formulas of the last two chapters have a neat consequence: radiation drift grows with the square of particle size, while streaming drift does not depend on size. In 2012 Barnkob, Augustsson, Laurell and Bruus tested this directly with micro particle image velocimetry. This chapter reads the paper in order and asks at each step what they measured, which number a conclusion rests on, and what we can recompute.

What they measured

The device is a straight channel in a silicon–glass chip, 377 µm wide and 157 µm high, with a piezo exciting a standing wave across the width. They filmed polystyrene particles 0.6 µm to 10 µm across in the horizontal symmetry plane at mid-height and used micro particle image velocimetry (micro-PIV) to get the velocity at each position.

There are four series: two in pure water at 1.940 MHz with the channel half a wavelength wide (the second repeats the first), one in pure water at 3.900 MHz with the channel one wavelength wide, and one in a 3 : 1 water–glycerol mixture at 2.027 MHz. Velocities taken at different voltages are rescaled to 1 V with the rule that velocity grows as voltage squared.

Source: R. Barnkob · M. Settnes and H. Bruus

The model they test

In the symmetry plane the horizontal particle velocity is the sum of the radiation drift and the streaming drag, and both are proportional to $\sin(2ky)$ . So each particle size needs only one fitted amplitude $u_a$ . Theory says $u_a$ is a straight line in $a^2$ , with the intercept from streaming and the slope from radiation.

The paper corrects Rayleigh’s parallel-plate coefficient twice: by 0.82 for the rectangular cross-section and by 1.26 for thermoviscous effects, giving $s\approx0.194$ . The radiation term is also divided by a wall-drag factor $\chi$ , because particles near a wall feel extra drag.

u_a=u_0\frac{a^2}{a_0^2}+u_0s,\qquad \frac{u^{\mathrm{rad}}}{u^{\mathrm{str}}}=\frac{1}{6s}\frac{\Phi}{\nu}\,\omega a^2

Eqs. 12 and 13 of the paper. The right expression shows that the field strength cancels: the ratio depends only on frequency, particle size and the combination

\nu/\Phi

Source: R. Barnkob · M. Settnes and H. Bruus

How the numbers support the conclusion

First, each of the four series shows a clear $a^2$ dependence (Fig. 8). The fitted energy density per volt differs by more than a factor of 700 between series, meaning the resonances were excited with very different strength, while the streaming coefficient falls between 0.18 and 0.25, with an uncertainty-weighted mean of $0.192\pm0.010$ , in line with the theoretical 0.194 (Table IV).

Second, once velocities are divided by $u_0$ and squared sizes by $a_0^2$ , 22 measurements from three series fall on one line (Fig. 9).

Third, the ratios of Table V. Treating 0.6 µm particles as moved by streaming alone and 10 µm particles by radiation alone, their velocity ratio is 49.6 at 1.940 MHz and 88.4 at 3.900 MHz: 1.8 times larger for 2.0 times the frequency. Water against water–glycerol gives about 3.0 (the text writes 48.6/16.4, the table 49.6), against 3.2 predicted from $\nu/\Phi$ . And $(u_5-u_{0.6})/(u_{10}-u_{0.6})$ should be 0.25 and is measured at 0.27 to 0.29.

The paper lists its own sources of error: particle density and compressibility vary between makers and batches, size distributions are not exactly Gaussian, local concentration changes cause particle interactions, and the particles change the liquid viscosity. It also notes that real channel fields can be far less uniform than the one-dimensional standing wave used here, and that the streaming field still needs mapping over the whole cross-section and length.

Source: R. Barnkob

Worked example: recompute Table II and the critical diameter

From Table II: polystyrene $\rho_p=1050\,\mathrm{kg/m^3}$ and $\kappa_p=249\,\mathrm{TPa^{-1}}$ ; water $\rho_0=997\,\mathrm{kg/m^3}$ and $\kappa_0=448\,\mathrm{TPa^{-1}}$ . Then $f_1=1-249/448=0.444$ , $f_2=2(1.053-1)/(2\times1.053+1)=0.034$ and $\Phi=0.444/3+0.034/2=0.165$ , rounded to 0.17 in the table.

Table II gives $\nu/\Phi$ as 5.25 mm²/s in water and 16.8 mm²/s in water–glycerol. With $s=0.194$ and $2a_c=\sqrt{24\nu s/(\omega\Phi)}$ : water at 1.940 MHz gives 1.42 µm, at 3.900 MHz 1.00 µm, and water–glycerol at 2.027 MHz 2.48 µm.

Now predict the first ratio of Table V. In water at 1.940 MHz, $a_0=1.61\,\mu\mathrm{m}$ . With the measured diameters of 10.16 µm and 0.59 µm, Eq. 13 gives radiation-to-streaming ratios $r_{10}=51.5$ and $r_{0.6}=0.17$ . Under the paper’s assumption (large particles pure radiation, small ones pure streaming) the velocity ratio is 51.5. Keeping the other part of each particle’s motion, Eq. 12 gives $(1+r_{10})/(1+r_{0.6})=44.7$ . The measured 49.6 lies between the two. Neither prediction includes wall drag; this is our recomputation, not a number from the paper.

The acoustic trap experiment of this course recomputes the contrast factor and penetration depths of Table II, the numbers from Eqs. 5 and 14, and the wall factors of Table I. Its particle tracks are model output computed from the closed-form fields.

Source: R. Barnkob · M. Settnes and H. Bruus

Try it in the Lab

Repeat the paper’s size scan in the experiment

Set the particle diameter to 0.6, 1, 2, 5 and 10 µm in turn and note the “Radiation drift over streaming drift” readout.
Plot the readout against $(2a)^2$ and check that it is a straight line through the origin with slope $1/(4sa_0^2)$ .
Set the half-wavelengths across the channel to 2 and compare the change for a 5 µm particle with the paper’s change from 1.940 MHz to 3.900 MHz.

Exercises with explained answers

Why does the paper conclude that a higher piezo voltage cannot focus 0.6 µm particles?
Show the answer
Radiation drift and streaming drift are both proportional to $E_{\mathrm{ac}}$ , which grows as the voltage squared, so their ratio does not depend on the voltage. It depends only on $\omega a^2\Phi/\nu$ , and improves only with higher frequency, a less viscous liquid or a larger contrast factor.
In Table V the ratio grows only 1.8 times when the frequency doubles. Does that refute the theory?
Show the answer
That does not follow. The pure ratio should grow $3.900/1.940=2.01$ times, and the paper judges the measured 1.8 to agree well. Moreover, 0.6 µm particles still carry radiation drift: by Eq. 13 about 0.17 of their streaming drift at 1.940 MHz and about 0.35 at 3.900 MHz. Keeping these terms (without wall drag), Eq. 12 predicts the measured velocity ratio to grow only about 1.73 times, closer to the measurement. This is our recomputation.
Having read only the abstract, could you state that the critical diameter is about 1.4 µm?
Show the answer
No. The abstract says only that the ratio scales with size squared, frequency and contrast factor and inversely with the kinematic viscosity. The 1.4 µm comes from Eq. 14 in the main text, for polystyrene in water at 2 MHz with the corrected streaming coefficient. A different liquid, frequency or particle material changes it.

Sources and what we recomputed

R. Barnkob, P. Augustsson, T. Laurell and H. Bruus, Acoustic radiation- and streaming-induced microparticle velocities determined by micro-PIV in an ultrasound symmetry plane, arXiv:1208.6534v1; Physical Review E 86, 056307 (2012)
Version 1, 31 August 2012, read in full. Eq. 1 (standing-wave field), Eq. 2 (radiation force), Eqs. 5–7 (streaming scale and Rayleigh streaming), Eqs. 9–10 (geometry and thermoviscous corrections), Eqs. 12–14 (velocity amplitude, velocity ratio, critical diameter), Eqs. 16–18 and Table I (wall drag), Table II (material data), Tables IV–V and Figs. 8–9 (measured energy densities, streaming coefficients and velocity ratios). We recompute the theory values; the measured data are quoted, not reproduced.
M. Settnes and H. Bruus, Forces acting on a small particle in an acoustical field in a viscous fluid, arXiv:1110.6037v1; Physical Review E 85, 016327 (2012)
Version 1, 27 October 2011, read in full. Eq. 1 (viscous penetration depth), Eqs. 4–5 (first-order pressure and compressibility), Eqs. 48a–e (Gor’kov potential with the viscous dipole coefficient) and Table I (Pyrex and cell parameters).