Hybrid finite-difference/lattice Boltzmann simulations of microchannel and nanochannel acoustic streaming driven by surface acoustic waves

A two-dimensional hybrid numerical method that allows full coupling of the elastic motion in a piezoelectric solid (modeled using a finite-difference time-domain technique) with the resultant compressional flow in a fluid (simulated using a lattice Boltzmann scheme) is developed to study the acoustic streaming that arises in both microchannels and nanochannels under surface acoustic wave (SAW) excitation. In addition to verifying the model through a comparison of the simulations with results from experimental and numerical studies of microchannel and nanochannel flows driven by both standing and traveling SAWs in the literature, we highlight salient features of the flow field that arise and discuss the underlying mechanisms responsible for the flow. In microchannels, boundary layer streaming is the dominant mechanism when the channel height is below the sound wavelength in the liquid, whereas Eckart streaming—arising as a consequence of the attenuation of the sound wave in the liquid—dominates in the form of periodic vortices for larger channel heights. The absence of Eckart streaming and the overlapping of boundary layers in nanochannels with heights below the boundary layer thickness, on the other hand, give rise to a time-averaged dynamic acoustic pressure that results in an inertial-dominant flow, which paradoxically possesses a parabolic-like velocity profile resembling pressure-driven laminar flow. In contrast, if the nanochannel were to be filled instead with air, the significantly lower fluid density leads to a considerable reduction in the dynamic acoustic pressure and hence inertial forcing such that boundary layer streaming once again dominates, asymptotically imposing a slip condition along the channel surface that results in a negative pluglike velocity profile.

Figure 1

Schematic depiction of the computational domains employed to model acoustic streaming in micro/nanochannels under excitation by (a) traveling or (b) standing SAWs. In either case, we model a section of the fluid domain in the channel with height $H_{\mathrm{g}}$ and length $L_{\mathrm{g}}$. In panels (a), two cases are examined wherein the top superstrate is either (i) passive (piezoelectric solid without IDTs) or (ii) active (piezoelectric solid with IDTs along which a SAW is activated together with that on the bottom substrate). In panel (b), the top superstrate is deliberately replaced with a rigid wall boundary condition (depicted by the brown line) in order to compare our simulation results with that reported in the literature (Ref. [68]). To minimize wave reflections from the boundaries in the solid domains, perfectly matched layers (absorbers) are adopted. The bold solid red lines show the locations where the solid and fluid domains are coupled, whereas the red boxes, with lateral dimensions of $2.2H_{\mathrm{g}}$, indicate the approximate locations for which the results are shown.

Figure 2

Contour plots of the computed normalized velocity magnitude of the instantaneous acoustic particle velocity $U_1/U_{1,\max }$ (at $t=250f_{\mathrm{SAW}}^{-1})$ and time-averaged acoustic streaming velocity $U_{\mathrm{dc}}/U_{\mathrm{dc},\max }$ fields $(t=100f_{\mathrm{SAW}}^{-1}$ to $t=250f_{\mathrm{SAW}}^{-1})$ for channel heights of (a) $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 0.2$ and (b) $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 1.43$, for the case where the top plate comprises (i) a passive piezoelectric (LN) superstrate without any IDTs or (ii) a piezoelectric (LN) superstrate with IDTs that are activated such that a 30-MHz SAW is generated on it together with that on the bottom substrate, as illustrated in Figs. 1 and 1, respectively. The results shown are for the domain indicated by the red box of lateral dimension $2.2H_{\mathrm{g}}$ shown in Fig. 1. In the simulations, the viscosity of the fluid is $\mu =0.05$ Pa s and thus $H_{\mathrm{g}}/{\alpha }_{\mathrm{f}}^{-1}\approx 0.01$ and 0.08 for $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 0.2$ and 1.43, respectively. The computed streamlines are overlaid on the streaming velocity contours. We note that the contour plots in panel (a) only display the flow in a very small section of the channel length $\left(\approx 2.2H_{\mathrm{g}}\right)$; the averaged velocity across the entire channel length is instead shown in Fig. 3.

Figure 3

Normalized acoustic streaming velocity along the $x$ direction in the microchannel $\left(H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 0.2\right)$ across section ${\mathrm{AA}}^{\prime }$ [see Fig. 2] for three different liquid viscosities: $\circ \mu =0.025$ Pa s $\left(H_{\mathrm{g}}/{\alpha }_{\mathrm{f}}^{-1}\approx 0.006\right)$; $\diamond \mu =0.05$ Pa s $\left(H_{\mathrm{g}}/{\alpha }_{\mathrm{f}}^{-1}\approx 0.01\right)$; $\times \mu =0.1$ Pa s $\left(H_{\mathrm{g}}/{\alpha }_{\mathrm{f}}^{-1}\approx 0.02\right)$.

Figure 4

Sketches (not to scale) illustrating the behavior of a fluid element fixed in space induced by a traveling surface wave along a solid surface adjacent to it. In panel (a), the traveling surface wave consists of the SAW that is generated by the active IDTs on the bottom piezoelectric (LN) substrate, and which propagates along it. In panel (b), the traveling surface wave is induced on and propagates along the top passive piezoelectric superstrate (no IDTs) as a consequence of the radiation pressure transmitted to it from the bottom piezoelectric substrate.

Figure 5

(a) Sketch (not to scale) illustrating the retrograde (anticlockwise) motion of the solid elements during propagation of the SAW excited on the bottom piezoelectric (LN) substrate, and the prograde (clockwise) motion of the solid elements during propagation of the surface wave induced on the top passive piezoelectric (LN) superstrate due to the transmission of the acoustic radiation pressure through the liquid. The phase difference $\varphi$ between the out-of-plane displacement (in the $y$ direction) of the top LN superstrate and the bottom LN substrate is a function of the channel height $H_{\mathrm{g}}$. [(b), (c)] Contour plots of the computed magnitude of the instantaneous acoustic particle velocity for the following combinations: (b) LN (top superstrate; passive) and LN (bottom substrate) and (c) glass (top superstrate) and LN (bottom substrate).

Figure 6

Experimental observations of SAW-driven open microchannel flow reproduced from Ref. [34]. (a) Schematic of the setup in which a $100\text{−}\mu \mathrm{m}$-deep and 10-mm-long microchannel is etched into the LN substrate; 20-MHz SAWs $\left({\lambda }_{\mathrm{f}}\approx 73\mu \mathrm{m}\right)$ are launched from the IDT on the left and travel along both sides of the microchannel walls, concurrently leaking their energy into the fluid (water) from both walls to give rise to (b) a standing wave transverse across the channel, and consequently acoustic streaming within the channel. The streaming comprises (c) positive unidirectional flow when the channel width $W$ is smaller than the sound wavelength in the fluid $\left(W/{\lambda }_{\mathrm{f}}\approx 0.41\right)$ that transitions into [(d), (e)] net positive periodic vortical flows when the channel is widened beyond the sound wavelength in the fluid: (d) $W/{\lambda }_{\mathrm{f}}\approx 2.05$ and (e) $W/{\lambda }_{\mathrm{f}}\approx 2.74$.

Figure 7

Contour plots of the computed time-averaged acoustic streaming velocity field $U_{\mathrm{dc}}/U_{\mathrm{dc},\max }(t=100f_{\mathrm{SAW}}^{-1}$ to $t=250f_{\mathrm{SAW}}^{-1})$ for a channel of height (a) $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 0.2$ and (b) $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 1.43$ driven by a 30-MHz standing SAW excited from beneath along the LN substrate. The computed streamlines are overlaid on the streaming velocity contours. (c) Sketch (not to scale) showing the displacement of the solid elements under standing SAW excitation and the corresponding motion of a fluid element adjacent to the solid substrate. Note that the motion of these elements alternates between clockwise and counterclockwise rotation every quarter SAW wavelength, thus giving rise to recirculation cells with a characteristic dimension of ${\lambda }_{\mathrm{SAW}}/4$.

Figure 8

Contour plot of the computed time-averaged acoustic streaming velocity field $U_{\mathrm{dc}}/U_{\mathrm{dc},\max }(t=100f_{\mathrm{SAW}}^{-1}$ to $t=250f_{\mathrm{SAW}}^{-1})$ for a channel of height (a) $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 0.2$ and (b) $H_{\mathrm{g}}/{\lambda }_{\mathrm{f}}\approx 1.43$ driven by a 30-MHz standing SAW excited (i) from beneath along the bottom LN substrate (the top superstrate remaining passive) and (ii) along both top and bottom LN solids. We note that the conditions for this case are exactly the same as that yielding the results in Fig. 7, other than that the rigid top wall boundary along $y=H_{\mathrm{g}}$ has now been replaced with either (i) a passive or (ii) active piezoelectric superstrate. The computed streamlines are overlaid over the streaming velocity contours.

Figure 9

Normalized average streaming velocity ${\overline{u}}_{\mathrm{dc}}={(L_{\mathrm{g}}-2L_{\mathrm{s}})}^{-1}{\int }_{L_{\mathrm{s}}}^{L_{\mathrm{g}}-L_{\mathrm{s}}}{u}_{\mathrm{dc}}dx$ of (a) water and (b) air in channels of height $H_{\mathrm{g}}=200\mathrm{nm}(\diamond )$, 400 nm $(\circ )$, and 600 nm $(\times )$ driven by (i) a 30-MHz traveling SAW along the bottom LN substrate (the top LN superstrate is passive) and (ii) 30-MHz traveling SAWs along both top and bottom LN solids. The viscous boundary layer thicknesses ${\delta }_v$ for water and air are approximately 150 and 460 nm, respectively.

Figure 10

Comparison between the velocity profile calculated from the numerical simulation $(\circ )$ with the approximate theoretical solution in Eq. (A13) for a channel of height $H_{\mathrm{g}}/\left(2{\delta }_v\right)\approx 0.7$ along which a traveling SAW is excited at the bottom substrate (the top LN superstrate remains passive). Three cases for the theoretical prediction are shown: $\left(▵\right){\delta }^{-1}\approx 0$ and $R<1$, which corresponds to the case in which the viscous dissipation within the boundary layer in Eqs. (A8)–(A11) is neglected and wherein the top substrate is not a perfect reflector, $(\diamond ){\delta }_v^{-1}\ne 0$ and $R<1$, and $(\times ){\delta }_v^{-1}\ne 0$ and $R=1$.

Figure 11

Sketches (not to scale) illustrating the motion of the solid elements along the substrate and superstrate when a traveling SAW generated by the IDT propagates (a) along the bottom LN substrate and (b) along both the top superstrate and the bottom substrate. Note that in panel (a), the motion of the solid elements along the top passive LN superstrate arises due to the transmitted acoustic radiation from the bottom substrate. By following a pair of arbitrary points in Lagrangian space over time, it can be seen that these points move in the same direction and at the same velocity as the SAW propagation.