Stability of a Jet in Confined Pressure-Driven Biphasic Flows at Low Reynolds Numbers

Motivated by its importance for microfluidic applications, we study the stability of jets formed by pressure-driven concentric biphasic flows in cylindrical capillaries. The specificity of this variant of the classical Rayleigh-Plateau instability is the role of the geometry which imposes confinement and Poiseuille flow profiles. We experimentally evidence a transition between situations where the flow takes the form of a jet and regimes where drops are produced. We describe this as the transition from convective to absolute instability, within a simple linear analysis using lubrication theory for flows at low Reynolds number, and reach remarkable agreement with the data.

Figure 1

Flow geometry and notations used in the text.

Figure 2

Map of the flow behavior in the ($Q_i$, $Q_e$) plane. The droplet regime comprises droplets smaller than the capillary (open circle) and pluglike droplets confined by this capillary (filled circle). Jets are observed in various forms: jets with visible peristaltic modulations convected downstream (open square), wide straight jets (filled square) that are stable throughout the $\sim 5\mathrm{cm}$ long channel, and thin jets breaking into droplets at a well-defined location (open diamond). Parameters are $R_c=275\mu \mathrm{m}$, inner viscosity ${\eta }_i=55\mathrm{mPa}.\mathrm{s}$, outer viscosity ${\eta }_e=235\mathrm{mPa}.\mathrm{s}$, surface tension $\Gamma =24\mathrm{mN}/\mathrm{m}$.

Figure 3

Dynamic behavior diagram in the ($x$, $Ka$) plane. The lines are the predictions ${\tilde{v}}_{-}^{*}(Ka,x,\lambda )=0$ for the absolute-convective transition, for viscosity ratios $\lambda$ equal to 10, 1, 0.1, 0.01, and 0.001 (bottom to top). Above the lines, the jet is convectively unstable, whereas below the lines, the jet is absolutely unstable. Regions below the lines correspond to droplets region.

Figure 4

Dynamic behavior in the ($Q_i$, $Q_e$) plane from our simple model. Above the line, the jet is convectively unstable, whereas below the lines it is absolutely unstable and droplets are expected. We vary independently each parameter around the conditions of Fig. 2 chosen as reference: $\Gamma =24\mathrm{mN}/\mathrm{m}$, $R_c=275\mu \mathrm{m}$, ${\eta }_e=235\mathrm{mPa}.\mathrm{s}$ and ${\eta }_i=55\mathrm{mPa}.\mathrm{s}$. (a) From top to bottom, ${\eta }_e=23.5$, 235 and $2350\mathrm{mPa}.\mathrm{s}$. (b) From top to bottom, $\Gamma =50$, 24 and $5\mathrm{mN}/\mathrm{m}$. (c) From top to bottom, $R_c=350$, 275 and $175\mu \mathrm{m}$. (d) From top to bottom, ${\eta }_i=5.5$, 55 and $550\mathrm{mPa}.\mathrm{s}$. The gray shading indicates where the average Reynolds number is larger than unity [24].

Figure 5

Experimental data (symbols) and theoretical predictions (continuous line) displaying the effect on the dynamical behavior of an increase of the capillary radius $R_c$ [$(\mathrm{a}1)\to (\mathrm{a}2)$] and a decrease of the surface tension $\Gamma$ [$(\mathrm{b}1)\to (\mathrm{b}2)$]. Gray symbols (open gray circle and filled gray circle) correspond to droplets and the black ones (open square, filled square, and open diamond) to jets (see Fig. 2). For (a1) and (a2), $\Gamma =24\mathrm{mN}/\mathrm{m}$, ${\eta }_e=0.235\mathrm{Pa}.\mathrm{s}$ and ${\eta }_i=0.055\mathrm{Pa}.\mathrm{s}$, with $R_c=275\mu \mathrm{m}$ for (a1) and $R_c=435\mu \mathrm{m}$ for (a2). For (b1) and (b2), $R_c=275\mu \mathrm{m}$, ${\eta }_e=3\mathrm{mPa}.\mathrm{s}$ and ${\eta }_i=1\mathrm{mPa}.\mathrm{s}$, with $\Gamma =12\mathrm{mN}/\mathrm{m}$ in (b1), and a decreased value using surfactants $\Gamma =0.12\mathrm{mN}/\mathrm{m}$ in (b2). The lines are obtained without adjustable parameters.

Figure 6

Flow behavior in the ($x$, $Ka$) plane for a given value of the viscosity ratio $\lambda =0.23$. $x$ is calculated from the first order solution using the experimental values of the flow rates and viscosities. The data (symbols as in Figs. 2, 5) correspond to two radii and two surface tensions. The line is the theoretical prediction of our linear analysis for the droplets/jet transition, with no adjustable parameters.