Bulk-driven acoustic streaming at resonance in closed microcavities

Bulk-driven acoustic (Eckart) streaming is the steady flow resulting from the time-averaged acoustic energy flux density in the bulk of a viscous fluid. In simple cases, like the one-dimensional single standing-wave resonance, this energy flux is negligible, and therefore the bulk-driven streaming is often ignored relative to the boundary-driven (Rayleigh) streaming in the analysis of resonating acoustofluidic devices with length scales comparable to the acoustic wavelength. However, in closed acoustic microcavities with viscous dissipation, two overlapping resonances may be excited at the same frequency as a double mode. In contrast to single modes, the double modes can support a steady rotating acoustic energy flux density and thus a corresponding rotating bulk-driven acoustic streaming. We derive analytical solutions for the double modes in a rectangular-box-shaped cavity including the viscous boundary layers, and use them to map out possible rotating patterns of bulk-driven acoustic streaming. Remarkably, the rotating bulk-driven streaming may be excited by a nonrotating actuation, and we determine the optimal geometry that maximizes this excitation. In the optimal geometry, we finally simulate a horizontal $2\times 2$, $4\times 4$, and $6\times 6$ streaming-roll pattern in a shallow square cavity. We find that the high-frequency $6\times 6$ streaming-roll pattern is dominated by the bulk-driven streaming as opposed to the low-frequency $2\times 2$ streaming pattern, which is dominated by the boundary-driven streaming.

Figure 1

The modulus $|F^{lmn}|$ (left axis) and phase ${\varphi }_F^{lmn}$ (right axis) of the frequency-dependent factor $F^{lmn}$ in Eq. (12c). The double arrow marks the full-width-at-half-maximum linewidth $\sqrt{3}{k}_0^{lmn}{\mathrm{\Gamma }}^{lmn}$ of $|F^{lmn}|$.

Figure 2

Rotating acoustics in the $xy$ plane ($K_z^n=0$) of a square cavity with $L_x=L_y$. (a) $K_x-K_y$ space diagram showing the allowed values of $K_x^l$ and $K_y^m$ (black dots). Each quarter circle (b)–(j) corresponds to the angular frequency $c_{\mathrm{fl}}\sqrt{{\left(K_x^l\right)}^2+{\left(K_y^m\right)}^2}$, where the double mode $lm0+ml0$ is excited. (b)–(j) The corresponding acoustic rotation of the double mode $lm0+ml0$, where the arrows represent the body force ${\mathit{f}}_{ml0}^{lm0}$ and the colors show the $z$ component of the spatial dependency $\mathbf{\nabla }\times {\mathcal{K}}_{ml0}^{lm0}$ in Eq. (18b) of the acoustic angular momentum density from $-2k_0^2$ (light cyan) to $2k_0^2$ (dark magenta).

Figure 3

Plots of $|F^{ml0}|$, $|F^{lm0}|$, ${\varphi }_F^{ml0}$, ${\varphi }_F^{lm0}$, and ${\mathcal{F}}_{lm}$ as a function of ${\tilde{k}}_0-{\tilde{k}}_{lm}^{★}$, each for the four values 3.00, 1.50, 0.58, and 0.10 of the mode separation ${\tilde{\mathrm{\Delta }}}_{lm}$. (a) The modulus $\left|F\right|$ and (b) the phase ${\varphi }_F$ of $F^{lm0}$ (right, solid) and $F^{ml0}$ (left, dashed) given in Eq. (28). At the optimal mode separation ${\tilde{\mathrm{\Delta }}}_{lm}^{\mathrm{opt}}=\frac{1}{\sqrt{3}}\approx 0.58$ (orange), the phase difference at the center frequency is $\frac{\pi }{3}$. (c) The frequency-dependent factor ${\mathcal{F}}_{lm}\left(k_0\right)$, from Eq. (24), which describes the strength of the acoustic body force ${\mathit{f}}_{\mathrm{ac}}$, when the modes of same color in (a) and (b) are combined.

Figure 4

The acoustic pressure $p_1$ for water in the nearly square cavity with parameters given in Tables 2 and 3, where the double mode $l00+0l0$ is excited by the nonrotating actuation $G_1$ [Eqs. (30) and (34)]. (a) Pressure contour plot for $l=6$ of the analytic solution $\mathrm{Im}\left(p_1^{\mathrm{ana}}\right)$ [Eq. (35)] (for $x<\frac{1}{2}{L}_x$), and the numerical solution $\mathrm{Im}\left(p_1^{\mathrm{num}}\right)$ (for $x>\frac{1}{2}{L}_x$). (b) Line plots of the pressure $\mathrm{Im}\left(p_1^{\mathrm{ana}}\right)$ (solid) and $\mathrm{Im}\left(p_1^{\mathrm{num}}\right)$ (dots) along the green line in (a) at $y=\frac{1}{2}{L}_y$ and $z=L_z$ for the modes $l=2$ (blue), $l=4$ (orange), and $l=6$ (yellow). Animations of the rotating acoustic fields are shown in the Supplemental Material [41].

Figure 5

Simulation of the rotating acoustics in the $x\text{−}y$ plane for the double mode $200+020$ in water for three aspect ratios $A\approx 1.002$, 1, 0.998 each for the nonrotating ($G_1$) and rotating ($G_{16}$) actuation perpendicular to the plane. Color plots between $\pm {10}^{-4}\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ (light cyan to dark magenta) of the $z$ component ${\mathcal{L}}_{\mathrm{ac},z}$ of the angular momentum density ${\mathcal{L}}_{\mathrm{ac}}$, and vector plots of the acoustic body force ${\mathit{f}}_{\mathrm{ac}}$ (max 2.00 $\mathrm{N}{\mathrm{m}}^{-3}$). In (a)–(c) we use the single nonrotating Gaussian actuation $G_1$ (large red circles) [Eqs. (30) and (34)]. In (d)–(f) we use the rotating actuation profile $G_{16}$ of four quadruples of phase shifted Gaussian profiles given in Eq. (C1) of Appendix pp3 [light red (up), black (0), dark blue (down), and black (0) circles, respectively] to generate an actuation rotating in the direction of the thick brown arrows. $A\approx 1.002$ and 0.998 are chosen to optimize the acoustic rotation for the nonrotating actuation [see Eq. (29)].

Figure 6

The largest possible value (positive and negative) of ${\mathcal{F}}_{ml0}^{lm0}\propto {\mathit{f}}_{\mathrm{ac}}$ versus varying mode separation ${\tilde{\mathrm{\Delta }}}_{lm}$ (lower $x$ axis) or aspect ratio $A$ (upper $x$ axis), shown for nonrotating actuations (blue) and rotating actuations (light cyan). The numerical values (dots) are found by evaluating ${\left({\mathcal{F}}_{020}^{200}\right)}_{\mathrm{num}}$ from Eq. (36) in the simulation of the double mode $200+020$ shown in Figs. 5 for the nonrotating and Figs. 5–5 for the rotating actuation. The analytical curves (solid) are from the expressions derived in Eq. (B4a) (nonrotating act.), Eq. (B7a) (rotating act., corotating), and Eq. (B8a) (rotating act., counter-rotating) of Appendix pp2. The black-edged circles correspond to those of the same color in Fig. 3.

Figure 7

Numerical simulation of the acoustic streaming velocity ${\mathit{v}}_2$ (black arrows) and its magnitude from 0 (black) to 0.17 mm/s (white) induced by the nonrotating actuation $G_1$. The setup and the parameters are the same as in Fig. 4 with the parameters listed in Tables 2 and 3. The horizontal plane at $z=\frac{1}{2}{L}_z$ (dark blue edge) and the vertical plane at $y=\frac{1}{2}{L}_y$ (light green edge) are displayed in Fig. 8.

Figure 8

Simulation of the acoustic streaming velocity ${\mathit{v}}_2$ in water from 0 (black) to 0.17 mm/s (white) in the horizontal ($x\text{−}y$) center plane at $z=\frac{1}{2}{L}_z$ (dark blue edge as in Fig. 7) and the vertical ($x\text{−}z$) center plane at $y=\frac{1}{2}{L}_y$ (light green edge as in Fig. 7). The length of the longest arrow represents $v_2^{\max }$ listed in each plot. The used parameters are listed in Tables 2 and 3. In each column the double mode $l00+0l0$ is excited for $l=2,4,$ and 6, respectively. The first row (a)–(c) shows the bulk-driven streaming, the second row (d)–(f) shows the boundary-driven streaming, and the third row (g)–(i) shows the total streaming. 3D results corresponding to (i) are shown in Fig. 7.

Figure 9

Simulation of the acoustic streaming ${\mathit{v}}_2$ from 0 mm/s (black) to $v_2^{\max }$ (white) for the exact same geometry as in Figs. 8, but with the water replaced by pyridine, having the material parameters listed in Table 2 and actuation amplitude $u_{1z}^0$ and actuation frequency $f_{l0}^{★}$ given in Table 3. The maximum values in (a)–(c) are 2.8, 8.2, and 13.9 times the corresponding values in Figs. 8.