Singular behavior of microfluidic pulsatile flow due to dynamic curving of air-fluid interfaces

Pulsatile pressure gradients are widely present in microfluidic multiphase systems where the movement of fluids is affected by the presence of fluid-fluid interfaces. We present a simple theoretical model that incorporates dynamic interfacial curvatures produced as a response to pulsatile external forcing. Our equations make evident a singular character of the dynamics at low frequencies, due to surface tension. Analytical solution of the model shows the emergence of a resonant behavior for the dynamic permeability. We have designed and implemented microfluidic experiments to observe both the low-frequency dynamics and the resonance. We have studied a fluid slug whose length was chosen in order to look for resonances as predicted by our theoretical model, in the range of operational frequencies of our piezoelectric actuator. We have obtained the experimental dynamic permeability for water and a 70.0% glycerol solution in water and observed agreement with theoretical findings. Our model, validated by experiments, allows us to understand differences of several orders of magnitude in the amplitude of flow velocity at low frequencies, between systems with and without interfaces.

Figure 1

Experimental setup. The oscillations are generated by the periodic movement of a piezoelectric actuator that moves back and forth a liquid slug confined within the microchannel via moving the air trapped between the piezoactuator and the slug. The pressure drop dynamics is registered at both sides of the fluid slug. A high-speed video camera records the interface movement.

Figure 2

System's geometry showing a fluid slug consisting of a volume of fluid and two air-fluid interfaces. The slug has length $L$, and it is confined between rectangular plates separated by a distance $2l$. The fluid displacement occurs in the $x$ direction. Details can be found in Appendix pp1.

Figure 3

Comparison of the local dynamic permeability, $K_L$, with continuous lines, for water (top panel) and 70.0% glycerol solution in water (bottom panel) in the absence of interfaces (blue) and for a slug containing two air-fluid interfaces (black). Dashed lines denote the corresponding global permeabilities, $K_G$. Theoretical parameters used correspond to a slug of water of 1 cm length, confined between rectangular plates, separated by a distance of $310\mu \text{m}$. For water (top panel) fluid parameters are surface tension 73 $\text{mN}/\mathrm{m}$, density 997 $\text{kg}/{\mathrm{m}}^3$, and viscosity 0.001 $\text{Pa}\text{s}$. For 70.0% glycerol solution in water (bottom panel) fluid parameters are surface tension 65.26 $\text{mN}/\mathrm{m}$, density 1181.25 $\text{kg}/{\mathrm{m}}^3$, and viscosity 0.0225 $\text{Pa}\text{s}$. Red dots correspond to experimental data; as explained in Sec. 3, they should be compared with theoretical continuous black lines. The small dots in the horizontal axis indicate the resonance frequency predicted by the model.

Figure 4

Comparison of the theoretical magnitude of water velocity as a function of time, in the absence of interfaces (blue curve), where $\sigma =0$ and for a fluid slug containing two interfaces (black curve). Curves are obtained from Eqs. (14) and (8). Experimental data (red and cyan dots) are for two independent experiments for a water slug with pressure drop frequency of 1.0 Hz and pressure drop amplitude $\mathrm{\Delta }{p}_{\mathrm{air}}^o=204.0$ Pa.

Figure 5

Left column: Example of pressure drop, position, and interface velocity data, as a function of time, for a water slug driven periodically in the microfluidic channel. Pressure data were acquired at a sampling rate of 5000 data per second (black line) and fitted to a sine function (red line). Details are described in Appendix pp3. In the position versus time graph, raw experimental data points are shown in black; in red are the smoothed data obtained by using a moving average filter window of size four. The instantaneous interface velocity was obtained by differentiating numerically the smoothed curve of position vs time. Right column: Frequency spectra of the signals in the left column, showing a single dominant mode. Dashed lines are a guide to the eye. For position spectra, raw and smoothed signals overlap. For this example, a slug of water is oscillating at 20.0 Hz.

Figure 6

Experimental amplitudes of pressure drop (top row), velocity (middle row), and permeability (bottom row), computed from experimental data, for 70.0% glycerol solution in water (left column) and water (right column). Solid lines are a guide to the eye.

Figure 7

Comparison between experimental and theoretical results for the dynamic permeability of a 70.0% glycerol solution in water slug containing two interfaces in a rectangular microchannel. For the figure, we have used the experimental parameters for 70.0% glycerol solution in water as in Fig. 3. Red circles: Experimental data. Black continuous line: Theoretical permeability, using a surface tension of 65.26 $\text{mN}/\mathrm{m}$, reported in the literature [38]. Theoretical resonance is predicted at $169.5\pm 29.4$ Hz (blue dot in the frequency axis with an error bar). Black dashed line: Error bars for the theoretical permeability, accounting for uncertainty in slug length and channel dimensions. Red dashed line: Theoretical threshold above which the linear approximation is not valid.

Figure 8

Comparison between experimental and theoretical results for the dynamic permeability of a water slug containing two interfaces in a rectangular microchannel. Red circles: Experimental data. Black continuous line: Theoretical permeability, using a surface tension of 73.0 $\text{mN}/\mathrm{m}$, reported in the literature. Theoretical resonance is predicted at $195.2\pm 33.8$ Hz (blue dot in the frequency axis with an error bar). Black dashed line: Error bars for the theoretical permeability, accounting for uncertainty in slug length and channel dimensions. Red dashed line: Theoretical threshold above which the linear approximation is not valid.

Figure 9

Top: Comparison between theoretical local and global dynamic permeabilities for water around resonance. Global permeability has been rescaled by a factor of 3/2. The coincidence of both curves goes beyond the linear regime observed in Fig. 3. Bottom: Ratio of both curves around resonance. Parameters are as in Fig. 3, top.

Figure 10

Comparison between experimental and theoretical results for pure glycerol dynamic permeability reported by Castrejón Pita et al. [19] for a cylindrical geometry. Glycerol properties: Surface tension 63.4 mN/m, density 1250.0 kg/${\mathrm{m}}^3$, viscosity 1.0 $\mathrm{Pa}\mathrm{s}$. Tube properties: Inner radius 25.0 mm, fluid column length 500.0 mm. The viscous frequency for this system is ${\omega }_{\eta }=\frac{\eta }{\rho R^2}=1.28$ rad/s. Inset figure, zoom at small frequencies. Red dots are experimental data taken from [19]. Blue curve is the single fluid theoretical local permeability; black curve is the local permeability in the presence of one fluid interface. The inset shows that resonance for this macroscopic system would happen at frequencies much lower than the ones that were studied in the macroscopic experiment.

Figure 11

Comparison between experimental data for 70.0% glycerol solution in water below 70.0 Hz (7 points) and a linear fit to the theoretical local dynamic permeability at small frequencies. The fitting leads to a surface tension value of $45.8\pm 17.1 \text{mN}/\mathrm{m}$, with correlation $R^2=0.996$.

Figure 12

Top: Illustration of pressure at the interfaces on both sides of the fluid slug for a hydrophilic surface. Curvature of the interface on the left, ${\kappa }_{\mathrm{wetting}}^1$, and curvature of the interface at the right, ${\kappa }_{\mathrm{wetting}}^2$, have opposite sign and approximately the same magnitude. Bottom: Illustration of pressure at the interfaces on both sides of the fluid slug forced periodically and neutral wetting, with instantaneous positive curvatures. Curvature of the interface on the left, ${\kappa }_{\mathrm{forcing}}^1$, and curvature of the interface at the right, ${\kappa }_{\mathrm{forcing}}^2$, have the same sign and approximately the same value.

Figure 13

Local dynamic permeabilities, at the center of the channel, illustrating the singular nature of surface tension at low frequencies, for zero surface tension (red lines) and air-water surface tension (blue lines). (a) Real parts; (b) imaginary parts.