Ultrasound resonance in coflowing immiscible liquids in a microchannel

We study ultrasonic resonance in a coflow system comprising a pair of immiscible liquids in a microchannel exposed to bulk acoustic waves. We show using an analytical model that there are two resonating frequencies corresponding to each of the coflowing liquids, which depend on the speed of sound and stream width of the liquid. We perform a frequency domain analysis using numerical simulations to reveal that resonance can be achieved by actuating both liquids at a single resonating frequency that depends on the speeds of sound, densities, and widths of the liquids. In a coflow system with equal speeds of sound and densities of the pair of fluids, the resonating frequency is found to be independent of the relative width of the two streams. In coflow systems with unequal speeds of sound or densities, even with matching characteristic acoustic impedances, the resonating frequency depends on the stream width ratio, and the value increases with an increase in the stream width of the liquid with a higher speed of sound. We show that a pressure nodal plane can be realized at the channel center by operating at a half-wave resonating frequency when the speeds of sound and densities are equal. However, the pressure nodal plane is found to shift away from the center of the microchannel when the speeds of sound and densities of the two liquids are unequal. The results of the model and simulations are verified experimentally via acoustic focusing of microparticles suggesting the formation of a pressure nodal plane and hence a resonance condition. Our study will find relevance in acoustomicrofluidics involving immiscible coflow systems.

Figure 1

(a) Schematic of the layered two-liquid system exposed to BAWs considered in the theoretical model. (b) Schematic of the experimental setup used for studying ultrasound resonance in an immiscible coflow system.

Figure 2

Numerical simulation results for different coflow systems: (a) PEG-dextran with $\left(c_a/c_b\right)=1$ and $\left({\rho }_a/{\rho }_b\right)=1$: (a-i) acoustic pressure contours, stream width ratio, $\left(W_a/W\right)=0.35$, and actuation frequency, $f=2.070\mathrm{MHz}$; (a-ii) variation of the maximum pressure amplitude ($P_{\max }$) with actuation frequency $\left(f\right)$, for $\left(W_a/W\right)=0.25,0.45,$ and $0.65$; (a-iii) particle trajectories showing particle focusing at the pressure N-P, $f=2.070\mathrm{MHz}$, $\left(W_a/W\right)=0.35$. (b) Mineral oil-olive oil with $\left(c_a/c_b\right)=1$ and $\left({\rho }_a/{\rho }_b\right)=1$: (b-i) acoustic pressure contours, $\left(W_a/W\right)=0.35$, and $f=1.910\mathrm{MHz}$; (b-ii) variation of $P_{\max }$ with $f$, for $\left(W_a/W\right)=0.25,0.45,$ and $0.65$; (b-iii) particle trajectories showing particle focusing at the pressure N-P, $f=1.910\mathrm{MHz}$, $\left(W_a/W\right)=0.35$. (c) Mineral oil-silicone oil with $\left(c_a/c_b\right)\ne 1$ and $\left({\rho }_a/{\rho }_b\right)\ne 1$: (c-i) total acoustic pressure field contours, $\left(W_a/W\right)=0.35$, and, $f=1.573\mathrm{MHz}$; (c-ii) variation of $P_{\max }$ with $f$ for $\left(W_a/W\right)=0.25,0.45,$ and $0.65$; (c-iii) particle trajectories showing particle focusing at the pressure N-P, $f=1.573\mathrm{MHz}$, $\left(W_a/W\right)=0.35$. (d) Olive oil-silicone oil with $\left(c_a/c_b\right)\ne 1$ and $\left({\rho }_a/{\rho }_b\right)\ne 1$: (d-i) total acoustic pressure field contours, $\left(W_a/W\right)=0.35$, and, $f=1.584\mathrm{MHz}$; (d-ii) variation of $P_{\max }$ with $f$ for $\left(W_a/W\right)=0.25,0.45,$ and $0.65$; (d-iii) particle trajectories showing particle focusing at the pressure N-P, $f=1.584\mathrm{MHz}$, $\left(W_a/W\right)=0.35$.

Figure 3

The 1D first-order acoustic pressure fields plotted across the width of the microchannel for the different coflow combinations: (a) PEG and dextran, $W_a/W=0.35,f_r=2.07$ MHz; (b) mineral oil and olive oil $\frac{W_a}{W}=0.35,f_r=1.91$ MHz; (c) mineral oil and silicone oil, $W_a/W=0.35,f_r=1.573$ MHz; (d) silicone oil and olive oil, $W_a/W=0.35,f_r=1.584$ MHz. The interface location is indicated by a black dotted line with the pressure profiles in liquids a and b indicated by red and blue lines. The location of the nodal plane (NP) is denoted by a brown dotted line for each case.

Figure 4

The experimental images showing particle focusing or nonfocusing, suggesting the formation or absence of pressure NP, in different coflow systems with ultrasound OFF and ON: (a) PEG-dextran, $Q_{\mathrm{dex}}=15\mu \mathrm{l}/\min$, $Q_{\mathrm{PEG}}=10\mu \mathrm{l}/\min$, $f_r=2.018\mathrm{MHz}$; (b) mineral oil-olive oil, $Q_{mi}=10\mu \mathrm{l}/\min$, $Q_{ol}=15\mu \mathrm{l}/\min$, $f=1.905\mathrm{MHz}$; (c) mineral oil-silicone oil, $Q_{mi}=15\mu \mathrm{l}/\min$, $Q_{si}=25\mu \mathrm{l}/\min$, $W_1/W_2=0.72$, $f=1.3\text{–}2.9\mathrm{MHz}$; (d) silicone oil-olive oil, $Q_{si}=5\mu \mathrm{l}/\min$, $Q_{ol}=10\mu \mathrm{l}/\min$, $W_1/W_2=0.54$, $f=1.3\text{–}2.9\mathrm{MH}z$. Trajectories of particle migration towards the pressure NP with time in different coflow systems. (e) PEG-dextran and olive oil-mineral oil; insets show the experimental images of the particle motion towards NP. (f) Mineral oil-silicone oil and silicone oil-olive oil.

Figure 5

Variation of resonance frequency with the stream width ratio for different immiscible coflow systems: mineral oil-silicone oil (■), olive oil-silicone oil (•), PEG-dextran (▲), and olive oil-mineral oil (♦).

Figure 6

Maximum acoustic pressure amplitude contours as a function of actuating frequency and width ratio of liquids in a coflow system: (a) PEG-dextran, (b) mineral oil-olive oil, (c) mineral oil-silicone oil, and (d) olive oil-silicone oil. The maximum pressure amplitude is more significant than ${10}^4$ Pa at the resonance condition, irrespective of the coflow systems and width ratios.

Figure 7

Variation of acoustic pressure field ($p_1$) for the mineral oil-silicone oil case for the stream width ratio, $\frac{W_a}{W}=0.35$. The pressure field contours across the domain and the pressure field variation across the width of the microchannel are shown for the actuation frequency (a) $f=1.5\mathrm{MHz}$, (b) $f=1.573\mathrm{MHz},$ and (c) $f=1.6\mathrm{MHz}$. This shows that the half-wave resonance is obtained at the frequency $f=f_r=1.573\mathrm{MHz},$ the resonating frequency of the coflowing system for the given width ratio, $\frac{W_a}{W}=0.35$. The half wave pressure node is not formed when the system is actuated at frequency, $f=1.5$ MHz and $f=1.6$ MHz.

Figure 8

Variation of maximum pressure amplitude ($P_{\max }$) with the actuation frequency $\left(f\right)$ for a coflowing fluids with impedance match, i.e., $Z_a=Z_b=2.045\times {10}^6\mathrm{kg}/{\mathrm{m}}^2\mathrm{s}$. The density of fluid a and fluid b are ${\rho }_a=1120\mathrm{kg}/{\mathrm{m}}^3$ and ${\rho }_b=1410\mathrm{kg}/{\mathrm{m}}^3$ (i.e., ${\rho }_a\ne {\rho }_b$). The speed of sound in fluid a and fluid b is $C_a=1825.45\mathrm{kg}/{\mathrm{m}}^3$ and $C_b=1410\mathrm{kg}/{\mathrm{m}}^3$ (i.e., $C_a\ne C_b$). The resonating frequency for the coflowing fluids with $W_a/W=0.25,0.45,$ and $0.65$ is given by 1.993, 2.095, and 2.206 MHz respectively. The resonating frequencies lie between the theoretically obtained frequency for the fluid a and fluid b (i.e., $f_a=2.434\mathrm{MHz}>f_r>f_b=1.88\mathrm{MHz}$).

Figure 9

(a) Variation of maximum pressure amplitude ($P_{\max }$) with the actuation frequency $\left(f\right)$ for coflowing fluids as a function of density variations with $\left(c_a/c_b\right)=1$ and $\left(W_a/W\right)=0.45$. The resonating frequency for $\left({\rho }_a/{\rho }_b\right)=0.77,1.00,$ and $1.43$ is given by $f_r=2.027,2.053,$ and $2.090$ MHz, respectively. (b) Variation of the maximum pressure amplitude ($P_{\max }$) with actuation frequency $\left(f\right)$, for $\left(W_a/W\right)=0.25,0.45,$ and $0.65$ with $\left(c_a/c_b\right)=1$ and $\left({\rho }_a/{\rho }_b\right)=0.77.$ The resonating frequency for width ratios $\left(W_a/W\right)=0.25,0.45,$ and $0.65$ is given by $f_r=1.968,2.027,$ and $2.124$ MHz, respectively.

Figure 10

The 1D first-order pressure field plotted against the width of the microchannel for the coflowing fluids PEG and dextran. (a) Second-order modes $m=n=2$, two nodal planes are observed; and (b) third-order modes $m=n=3$, three nodal planes can be seen irrespective of the location of the fluid-fluid interface.

Figure 11

The 1D first-order pressure field plotted against the width of the microchannel for the coflowing fluids mineral oil and olive oil. (a) Second-order modes $m=n=2$, two nodal planes are observed; and (b) third-order modes $m=n=3$, three nodal planes can be seen irrespective of the location of the fluid-fluid interface.

Figure 12

Experimental images of particle focusing towards the center of the microchannel in a single-fluid case: (a) dextran (10% wt), $f=2.018$ MHz, (b) PEG (5% wt), $f=2.02$ MHz, (c) silicone oil, $f=1.375$ MHz, (d) olive oil, $f=1.93$ MHz, (e) mineral oil, $f=1.97$ MHz, (f) variation of the trajectory of polystyrene particles with time in different fluids.

Figure 13

Formation and absence of half wave resonance mode in various immiscible coflow systems represented in terms of the maximum pressure amplitude depending upon the stream width ratios and actuation frequency. The maximum pressure amplitude is more significant than ${10}^4$ at the resonance condition, irrespective of the coflow systems and width ratios: mineral oil-silicone oil (■), olive oil-silicone oil (•), PEG-dextran (▲), and olive oil-mineral oil (♦).