Theory and modeling of nonperturbative effects in thermoviscous acoustofluidics

A theoretical model of thermal boundary layers and acoustic heating in microscale acoustofluidic devices is presented. Based on it, an iterative numerical model is developed that enables numerical simulation of nonlinear thermoviscous effects due to acoustic heating and thermal advection. Effective boundary conditions are derived and used to enable simulations in three dimensions. The theory shows how friction in the viscous boundary layers causes local heating of the acoustofluidic device. The resulting temperature field spawns thermoacoustic bulk streaming that dominates the traditional boundary-driven Rayleigh streaming at relatively high acoustic energy densities. The model enables simulations of microscale acoustofluidics with high acoustic energy densities and streaming velocities in a range beyond the reach of perturbation theory, and is relevant for design and fabrication of high-throughput acoustofluidic devices.

Figure 1

(a) The temperature $T_1=T_1^{\mathrm{xl},d}+T_1^{\mathrm{xl},\delta }$ and velocity ${\mathit{v}}_1={\mathit{v}}_1^{\mathrm{xl},d}+{\mathit{v}}_1^{\mathrm{xl},\delta }$ across a fluid (xl = fl)–solid (xl = sl) interface with the local normal vector ${\mathit{e}}_z$ and the local parallel vector ${\mathit{e}}_{\parallel }$ (either ${\mathit{e}}_x$ or ${\mathit{e}}_y$). On the interface, the bulk fields $T_1^d$ and ${\mathit{v}}_1^d$ have a discontinuity $\mathrm{\Delta }{T}_1$ and $\mathrm{\Delta }{\mathit{v}}_1$, respectively, due to the thin thermal ($T_1^{\mathrm{xl},\delta }$) and viscous (${\mathit{v}}_1^{\mathrm{xl},\delta }$) boundary layers. (b) Zoom-in on the thermal boundary layers $T_1^{\mathrm{fl},\delta }$ and $T_1^{\mathrm{sl},\delta }$ with respective thicknesses ${\delta }_t^{\mathrm{sl}}$ and ${\delta }_t^{\mathrm{sl}}$. (c) Zoom-in on the viscous boundary layer ${\mathit{v}}_1^{\mathrm{fl},\delta }$ in the fluid with thickness ${\delta }_s$.

Figure 2

Simulation of the effective (left, “Eff”) and the full model (right, “Full”) at the horizontal half-wave resonance $f=1.929$ MHz. (a) The two-dimensional (2D) silicon-glass model with a water channel and the bottom-edge actuation ${\mathit{u}}_{\mathrm{exc}}$. (b) Color plot at the energy density $E_{\mathrm{ac}}=28\text{J}/{\text{m}}^3$ of the displacement $|{\mathit{u}}_1|$ and pressure $p_1$ in the fluid. (c) Color plot at $E_{\mathrm{ac}}=28\text{J}/{\text{m}}^3$ of the steady temperature field $\mathrm{\Delta }{T}_0=T_0-T_0^{\mathrm{bot}}$ from 0 (black) to $8.7\text{mK}$ (yellow). (d) Color plot at $E_{\mathrm{ac}}=2680\text{J}/{\text{m}}^3$ of $\mathrm{\Delta }{T}_0$ from 0 (black) to $230\text{mK}$ (yellow). (e) Vector plot at $E_{\mathrm{ac}}=28\text{J}/{\text{m}}^3$ of the streaming ${\mathit{v}}_0$ and color plot of its magnitude $v_0$ from 0 (blue) to $34\text{µm}/\text{s}$ (yellow). (f) Same as (e) but at $E_{\mathrm{ac}}=2680\text{J}/{\text{m}}^3$ and with $v_0$ from 0 (blue) to $4.0\text{mm}/\text{s}$ (yellow).

Figure 3

Line plot at $E_{\mathrm{ac}}=28\text{J}/{\text{m}}^3$ and $f=1.929$ MHz of the simulated steady temperature $\mathrm{\Delta }{T}_0$ in the effective and full model along the vertical red line $y=\frac{1}{4}{W}_{\mathrm{fl}}$ shown in the inset. The corresponding color plot of $\mathrm{\Delta }{T}_0$ is shown in Fig. 2.

Figure 4

(a) Color plot from 0 (black) to $8.7\text{mK}$ (yellow) at $E_{\mathrm{ac}}=28\text{J}/{\text{m}}^3$ and $f=1.929$ MHz of $\mathrm{\Delta }{T}_0$ from Fig. 2 zoomed in on the fluid domain. (b) Same as (a) but for $E_{\mathrm{ac}}=9000\text{J}/{\text{m}}^3$ and a color scale from 0 (black) to $2375\text{mK}$ (yellow). (c) Line plot of the normalized temperature rise $\mathrm{\Delta }{T}_0/\max \left(\mathrm{\Delta }{T}_0\right)$ along the vertical line at $y=0$ shown in the inset. (d) Same as (c) but along the horizontal line through the center of the microchannel shown in the inset.

Figure 5

Study at frequency $f=1.893$ MHz. (a) Sketch of the 2D model of the glass chip with a fluid channel. (b) Color plot at the energy density $E_{\mathrm{ac}}=25\text{J}/{\text{m}}^3$ of the displacement $|{\mathit{u}}_1|$ and pressure $p_1$ in the fluid. (c) Color plot at $E_{\mathrm{ac}}=25\text{J}/{\text{m}}^3$ of the steady temperature field $\mathrm{\Delta }{T}_0=T_0-T_0^{\mathrm{bot}}$ from black (0) to yellow ($16\text{mK}$). (d) Color plot at $E_{\mathrm{ac}}=2000\text{J}/{\text{m}}^3$ of $\mathrm{\Delta }{T}_0$ from black (0) to yellow ($1.25\text{K}$). (e) Vector plot at $E_{\mathrm{ac}}=25\text{J}/{\text{m}}^3$ of the streaming ${\mathit{v}}_0$ and color plot of its magnitude $v_0$ from blue (0) to yellow ($30\text{µm}/\text{s}$). (f) Same as (e) but at $E_{\mathrm{ac}}=2000\text{J}/{\text{m}}^3$ and with the color scale of $v_0$ from blue (0) to yellow ($2.4\text{mm}/\text{s}$).

Figure 6

Study at frequency $f=1.860$ MHz. (a) Sketch of the 2D model of the glass-silicon chip with a oil channel. (b) Color plot at the energy density $E_{\mathrm{ac}}=25\text{J}/{\text{m}}^3$ of the displacement $|{\mathit{u}}_1|$ and pressure $p_1$ in the fluid. (c) Color plot at $E_{\mathrm{ac}}=25\text{J}/{\text{m}}^3$ of the steady temperature field $\mathrm{\Delta }{T}_0=T_0-T_0^{\mathrm{bot}}$ from black (0) to yellow ($95\text{mK}$). (d) Color plot at $E_{\mathrm{ac}}=2855\text{J}/{\text{m}}^3$ of $\mathrm{\Delta }{T}_0$ from black (0) to yellow ($10\text{K}$). (e) Vector plot at $E_{\mathrm{ac}}=25\text{J}/{\text{m}}^3$ of the streaming ${\mathit{v}}_0$ and color plot of its magnitude $v_0$ from blue (0) to yellow ($30\text{µm}/\text{s}$). (f) Same as (e) but at $E_{\mathrm{ac}}=2855\text{J}/{\text{m}}^3$ and with the color scale of $v_0$ from blue (0) to yellow ($3.5\text{mm}/\text{s}$).

Figure 7

(a) Domain for simulating $T_0, p_1$, and $|{\mathit{u}}_1|$ in a quarter of the glass-Si-glass system, actuated at the top (green) by ${\mathit{u}}_{\mathrm{exc}}={\mathit{u}}_1^{\mathrm{top}}=d_0{\mathit{e}}_z$ for $0<x<\frac{1}{2}{L}_{\mathrm{PZT}}, \frac{1}{2}W<y<\frac{1}{2}{W}_{\mathrm{sl}}$, and $z=z_{\mathrm{top}}$. The system is studied at the horizontal half-wave resonance in the channel at $f=0.957\text{MHz}$. (b) Color plot of $T_0$ from $20{}^{\circ }\text{C}$ (black) to $20.8{}^{\circ }\text{C}$ (yellow) due to the absorption of light from an LED with $P=5$ mW. (c) Color plot of the acoustic displacement $|{\mathit{u}}_1|$ from $0\text{nm}$ (blue) to $18\text{nm}$ (yellow) in the solid, and the acoustic pressure $p_1$ in the water-filled $0.76\times 0.36{\text{mm}}^2$ microchannel from $0\text{MPa}$ (gray) to $1.2\text{MPa}$ (red).

Figure 8

Simulated streaming ${\mathit{v}}_0$ at acoustic energy density $E_{\mathrm{ac}}=150\text{J}/{\text{m}}^3$ and frequency $f=0.957$ MHz for the LED power $P=0$, 5, and 50 mW, respectively. The color plots from $0\text{mm}/\text{s}$ (blue) to $v_{\max }$ (yellow) are the in-plane velocity of the respective planes, on the $yz$ plane it is ${(v_{0,y}^2+v_{0,z}^2)}^{1/2}$, and likewise for the $xy$ and $xz$ planes. All arrows are unit vectors showing the direction of ${\mathit{v}}_0$. (a) ${\mathit{v}}_0$ for $P=0$ mW showing the usual four boundary-driven streaming rolls with $v_{\max }=0.2$ mm/s. (b) ${\mathit{v}}_0$ for $P=5$ mW showing a slightly dominant thermoacoustic streaming flow with $v_{\max }=0.9$ mm/s driven by the acoustic body force ${\mathit{f}}_{\mathrm{ac}}^d$ (51) in the bulk. (c) ${\mathit{v}}_0$ for $P=50$ mW completely dominated by the fast streaming flow with $v_{\max }=5.6\text{mm}/\text{s}$ driven by the acoustic body force ${\mathit{f}}_{\mathrm{ac}}^d$ in the bulk.

Figure 9

Convection due to high streaming velocities at $f=0.957$ MHz. (a) The temperature field in the $yz$ plane at $x=0$ generated by the light absorption from a LED of power $P=5\text{mW}$ ranging from $T_0=20.0{}^{\circ }\text{C}$ (black) to $T_0=20.8{}^{\circ }\text{C}$ (yellow). (b) Same as (a) but for $P=50\text{mW}$ and a color range from $T_0=20.0{}^{\circ }\text{C}$ (black) to $T_0=27.3{}^{\circ }\text{C}$ (yellow). (c) A line plot of the two normalized temperature fields along a line at $x=y=0$ shown in the inset. (d) A line plot of the two normalized temperature fields along a line at $x=z=0$ shown in the inset. The differences in the two temperature fields are due to convection.