Aerodynamically stabilized Taylor cone jets

We introduce a way to stabilize steady micro/nanoliquid jets issuing from Taylor cones together with coflowing gas streams. We study the dripping-jetting transition of this configuration theoretically through a global stability analysis as a function of the governing parameters involved. A balance between the local radial acceleration to the surface tension gradient, the mass conservation, and the energy balance equations enable us to derive two coupled scaling laws that predict both the minimum jet diameter and its maximum velocity. The theoretical prediction provides a single curve that describes not only the numerical computations but also experimental data from the literature for cone jets assisted with gas coflow. Additionally, we performed a set of experiments to verify what parameters influence the jet length. We adopt a very recent model for capillary jet length to our configuration by combining electrohydrodynamic effects with the gas flow through an equivalent liquid pressure. Due to diameters below 1 $\mu \text{m}$ and high speeds attainable in excess of 100 m/s, this concept has the potential to be utilized for structural biology analyses with x-ray free-electron lasers at megahertz repetition rates as well as other applications.

Figure 1

Sketch of the aerodynamically assisted electrified jet nozzle.

Figure 2

(a) Numerical and (b) experimental stable flows for $4Q_{\ell }/\pi D_i^2{v}_c=0.023$; ${\text{Re}}_g=26.7$; $C=3.0382$; $\mu =0.0001625$; $\rho =0.00104167$; $\chi =8$; $\beta =40$; $\alpha =124479$. The critical eigenvalue of the base flow is $0.0269-0.0356i$ which corresponds to about $v_j\approx 100$ m/s and $D_j\approx 200$ nm and the eigenvalue spectrum are depicted in the embedded figure of (a). In (a), the right vertical axis is the dimensionless electrical current level $I$ for both bulk conduction (dashed line) and surface convection (dotted line). Streamlines of both flows are plotted in each domain. In (b), the liquid cone jet is shown to the right of the white line and the nozzle to the left.

Figure 3

(a) Critical points from the global stability analysis (blue circles), ${\left(v_j/v_{\mu }\right)}^2=c_d{l}_{\mu }/D_j$ for $c_d=11.4$ (solid line) with $R^2=0.973$. Experimental data from the literature for both flow focusing (FF) [28] and electrospray (ES) [19] are also plotted: 500-cSt silicone oil (FF, open red triangle), 100-cSt silicone oil (FF, open red circle), 5-cSt silicone oil (FF, open red square), water (FF, open red diamond), 1-decanol (ES, green triangle) 1-octanol $D_i=210$ $\mu \mathrm{m}$ (ES, green diamond), 1-octanol $D_i=1100$ $\mu \mathrm{m}$ (ES, green star), 3-ETG$+\mathrm{LiCa}$ $0.00005\mathrm{M}$ (ES, green square), 3-ETG$+\mathrm{LiCa}$ 0.0005M (ES, green cross). The inset shows the dimensional relation in micrometers. (b) Power balance equation (solid line) and critical stability points around it (blue circles) with $R^2=0.966$. The inset illustrates the influence of ${\text{Re}}_g$ over $Q_{\ell \text{min}}$.

Figure 4

Gañán-Calvo et al. jet length scaling law (dashed line) and their supporting experimental data (gray) [30] together with experimental realizations for this aero-electro configuration (color). In the inset, the experimental points collected are plotted in terms of the gaseous Reynolds number and the dimensionless voltage. Mark color: ${\text{We}}_{\ell }=6.24$ (black), 10.80 (yellow), 15.28 (blue), 19.72 (green), 44.11 (white framed diamonds). Mark shape: $\left\{\widehat{H_p},{\mathrm{{\rm Y}}}_g\right\}=\left\{0.326,1.245\right\}$ (circle), $\left\{0.7,1.98\right\}$ (triangle), $\left\{0,0.847\right\}$ (square), $\left\{0,1.779\right\}$ (diamond).

Figure 5

Coaxial, miscible cone jet. The sample (pure water) is fed from a capillary tube of 30 $\mu \mathrm{m}$ of diameter, placed at 60 $\mu \mathrm{m}$ upstream from the protruded tip. Conditions: $Q_c=300$ nL/min, $Q_s=50$ nL/min, $\mathrm{\Delta }V=2400$ V, $Q_{mg}=1.86$ mg/min, $\widehat{H_p}=0.7$, ${\mathrm{{\rm Y}}}_g=1.96$, $D_j\approx 260$ nm, and $v_j\approx 110$ m/s.