Regime Map and Triple Point in Selective Withdrawal

Entrainment in selective withdrawal occurs when both the top and bottom phases are withdrawn through a capillary tube oriented perpendicular to a flat gravitationally separated liquid-liquid interface. The tube introduces two distinct features to the conditions for fluid entrainment. First, the ratio of the two phases being withdrawn is affected by the region of influence of the flow upstream of the tube’s orifice. Second, a minimum withdrawal flow rate must be reached for entrainment regardless of the distance between the interface and the tube. We show that these phenomena can be understood based on the Reynolds number that governs the external flow field around the capillary tube and the capillary number that regulates the effect of the viscosity and capillarity.

Figure 1

Schematic of experimental setup and definition of the variables. $H>0$ is indicated here. $H<0$ corresponds to the end of the tube being in the lower phase. The “front” and “rear” refer to the front and rear of the capillary.

Figure 2

The phase diagram of selective withdrawal using ATPS with $R_0=150\mu \mathrm{m}$. The numbers in (a)–(d) indicate the sampling location in (e) with corresponding numbers. (a) Only withdrawal of the top phase ($T$). (b) Entrainment ($E$) of both phases at positive $H$. (c) Entrainment ($E$) of both phases at negative $H$. (d) Only withdrawal of the bottom phase ($B$). Scale bar is $400\mu \mathrm{m}$ and $Q_0=8\mathrm{ml}/\min$ for (a)–(d). (e) Phase diagram on the $H\text{−}{Q}_0$ plane. $T$: only withdrawal of the top phase; $B$: only withdrawal of the bottom phase; $E$: entrainment of both phases. $Q_0^{*}$ is the minimum flow rate of the entrainment regime. The dashed line delineates the boundary of the hysteresis region between $T$ and $E$. Typical error bars are the size of the symbols. Data points were obtained by fixing $Q_0$ and monotonically decreasing $H$. The solid lines fit the data points.

Figure 3

(a) Entrainment ratio $\varphi$ versus capillary height $H$ using a capillary with $R_0=150\mu \mathrm{m}$ in ATPS system. Filled symbols represent $H<0$ and hollow symbols represent $H>0$. (b) $\varphi$ versus Q for the same data in (a). Black lines are the simulation result reported by Lister (Eq. 6.4 in Ref. [8]). (c) $Q$ versus ${\mathrm{Re}}_1$ at $\varphi =0$ and $\varphi =1$ for the ATPS and 1-decanol (top)–63 wt % glycerol aqueous solution (bottom) on the boundary of the $E$ regime. ATPS: ${\mu }_1=3.41\mathrm{mPa}\mathrm{s}$, ${\mu }_2=2.86\mathrm{mPa}\mathrm{s}$, $\gamma =0.30\mathrm{mN}/\mathrm{m}$, $\mathrm{\Delta }\rho =310\mathrm{kg}/{\mathrm{m}}^3$; 1-decanol–63 wt % glycerol-water: ${\mu }_1={\mu }_2=12.0\mathrm{mPa}\mathrm{s}$, $\gamma =5.0\mathrm{mN}/\mathrm{m}$, $\mathrm{\Delta }\rho =330\mathrm{kg}/{\mathrm{m}}^3$.

Figure 4

${\text{Ca}}_1$ versus ${\text{Oh}}_1$. The black line indicates the relationship ${\text{Ca}}_1=0.38{\text{Oh}}^{1/2}$. The insets show representative time sequences when the capillary is moved toward the interface. Scale bar $500\mu \mathrm{m}$. Top panels represent the filled upward-pointing triangle where the $E$ regime occurs between the $T$ and $B$ regimes ($Q_0=0.25\mathrm{ml}/\min$); bottom panels represent the open upward-pointing triangle where the withdrawal will jump from $T$ to $B$ regimes without going through entrainment of both fluids ($Q_0=0.20\mathrm{ml}/\min$). Circle, square, and diamond: $\gamma =40\mathrm{mN}/\mathrm{m}$, $\lambda =1$, $R_0=750\mu \mathrm{m}$, $350\mu \mathrm{m}$, $150\mu \mathrm{m}$, respectively. Right-pointing triangle and left-pointing triangle: $\gamma =32\mathrm{mN}/\mathrm{m}$, $\lambda =0.1$, $R_0=750\mu \mathrm{m}$, $350\mu \mathrm{m}$, respectively. Upward-pointing triangle and downward-pointing triangle: $\gamma =0.3\mathrm{mN}/\mathrm{m}$, $\lambda =1$, $R_0=350\mu \mathrm{m}$, $150\mu \mathrm{m}$, respectively. Pentagram and hexagram: $\gamma =28\mathrm{mN}/\mathrm{m}$, $\lambda =0.001$, $R_0=750\mu \mathrm{m}$, $350\mu \mathrm{m}$, respectively.