Top view schematic of the experimental setup. Two syringe pumps push the two fluids at a controlled rate. Water is in inlet pump 1 and the viscous sucrose solution is in inlet pump 2. The volume fraction (of fluid 2) into the test T-junction is given as Φin = Q 2/(Q 1 + Q 2). Fluid is collected from the branch and the volume fraction in the outlet is measured. The outlet pump can be switched to the branch and the experiment repeated. Gravity points into the page.
(a) Simulation geometry. The color (in online version) represents the concentration field. The two fluids are brought in vertically with respect to gravity so that the flow will be pre-stratified in the inlet. (b) Sample velocity profiles for stratified flow as a function of radial distance from the center of the tube. Profiles are shown for viscosity ratios of 5 (dashed-dotted black curve), 15 (dashed red curve), and 100 (solid blue curve). For each case the inlet volume fraction is Φin = 0.5. Notice that the location of the interface between the two fluids moves toward the wall as the viscosity ratio increases. The radial coordinate in the plot is taken along a vertical line through the center of the tube in the direction of gravity. For a viscosity ratio of 1 we would have the classic parabolic velocity profile.
Comparison of experiment and simulation for the phase splitting at a T junction as a function of the overall flow rate. Each inlet pump has a flow rate of (a) 0.5 ml/min, (b) 1 ml/min, (c) 2 ml/min, and (d) 5 ml/min. Each data point represents the mean of six measurements (2 independent measurements on 3 independent but identical networks). The error bars represent the maximum and minimum values measured. The composition of the branch and the run are measured separately, though this measurement is redundant. The triangles represent the mean value in the branch (run) as calculated from the measurements in the run (branch)—the data are consistent. The lighter gray data points (red online) near the dashed curve are measured in the run and the darker gray data points (blue online) near the solid curve are measured in the branch. The solid line is the finite element simulation from the branch and the dashed line is the simulation from the run. The sucrose solution coming from pump 2 is a 50% mass fraction solution which has a viscosity of . The Reynolds number in the T-junction (based on the water viscosity) for each case is approximately (a) 19, (b) 38, (c) 76, and (d) 190. The Froude number in each case is (a) 35, (b) 8.7, (c) 2.2, and (d) 0.35. The other dimensionless parameters are Sc = 2000 and β = 0.23.
Comparison of experiment and simulation. Here the two outlet flows are fixed to always be equal; Q br/Q in = 0.5. In (a) we only vary the total flow rate holding the viscosity ratio at . In (b) we hold the total flow constant at 4 ml/min (Re = 76Fr = 2.2) and vary the viscosity ratio. In all cases Sc = 2000 and β = 0.23. The symbols are the same as in Figure 3 .
Comparison of experiment and simulation for the phase splitting at a T junction. In both experiments the total inlet flow rate is 4 ml/min. In (a) the sucrose inlet is 2.8 and the water is 1.2 ml/min, Φ in = 0.7. In (b) the sucrose is 1.2 and the water is 2.8 ml/min, Φ in = 0.3. The sucrose solution in the inlet pump has a viscosity 15 times that of water. The other dimensionless parameters are Re = 76, Fr = 2.2, β = 0.23, and Sc = 2000. The symbols are the same as in Figure 3 .
Comparison of experimental data for the phase splitting at a T junction compared to the simple 1 parameter model over a range of conditions. The points are experimental data (error bars removed for clarity) and the lines are from the fit function—solid lines for the branch and dashed lines for the run. Color is shown online version. All experimental data are the same as shown in previous figures. The conditions denoted in the figure are: (i) black lines for α = 0.6 and black triangles for 2 ml/min total flow with Φin = 0.5; (ii) red lines for α = 1.1 and red pluses for 4 ml/min total flow with Φin = 0.5; (iii) blue lines for α = 1.5 and blue circles for 10 ml/min total flow with Φin = 0.5; (iv) magenta lines for α = 2.2 and magenta diamonds for 4 ml/min total flow with Φin = 0.3. In all cases the sucrose solution in the inlet pump has a viscosity 15 times that of water.
Top view schematic of the network as constructed in experiment. The results are sensitive to whether the connector between the two outlets is the run (as configured here) or the branch of the T-junction.
(a) Schematic and formulation for calculating the effective viscosity of miscible stratified flow in a tube. We show a cross section of the tube where u is the axial velocity coming out of the page. The boundary conditions are no-slip at the wall and constant stress at the interface between the two fluids. (b) Effective viscosity for miscible stratified laminar flow of two fluids in a tube as a function of volume fraction of the more viscous fluid in the mixture. Sample curves are shown for viscosity contrast of 5, 10, and 15. The normalized diffusion time measured in units of D/d 2 was 0.0025; the final result is very close to the immiscible case.
(a) Predictions of the flow in tube C as a function of the ratio of flow in inlet 1 to the total flow rate Q in,1/(Q in,1 + Q in,2) for values of α = 0, 0.5, 1, and 2 (blue, red, black, and magenta respectively). The viscosity contrast is fixed at (b) Predictions of the flow in tube C for different viscosity contrast of , 4, 20, and 150 (blue, red, black, and magenta, respectively). The parameter for the phase separation function is α = 2. In both figures (a) and (b), the inlet volume fraction is Φin,1 = Φin,2 = 0.5 and the lengths of all tubes in the network are the same.
(a) Flow in exit branch A of the network in Figure 7(b) determined from experiment and theory. Notice the region of bistability around the state where the flow rates on the inlets are equal. Both inlets are a 30% sucrose and 70% water mixture in stratified flow. The viscosity contrast between the sucrose solution in inlet pumps and water is 150. The inset shows the network topology. The error bars on the light gray (red) experimental data points show the maximum and minimum value recorded in three independent trials and the data points shows the mean. The dark gray (blue) data points and error bars were taken on two of the trials where we proceeded from left to right along the flow curve. The black data points were from one trial where we moved from right to left. In (b) we have the exact same setup, only the connection for the network is made by the branches of the two inlet T-junctions—see the inset. In both figures the points are experimental data, the solid line is from the theory, using the fit function with α = 2.2 and the dashed line is from the theory using the phase separation as predicted from the Comsol simulation.
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