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TECHNICAL PAPERS

A Simplified Model for Determining Interfacial Position in Convergent Microchannel Flows

[+] Author and Article Information
D. L. Hitt

University of Vermont, Department of Mechanical Engineering, Burlington, VT 05405

N. Macken

Swarthmore College, Department of Engineering, Swarthmore, PA 19081

J. Fluids Eng 126(5), 758-767 (Dec 07, 2004) (10 pages) doi:10.1115/1.1792272 History: Received February 10, 2003; Revised June 04, 2004; Online December 07, 2004
Copyright © 2004 by ASME
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References

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Carr,  R. T., and Kotha,  S. L., 1995, “Separation Surfaces for Laminar Flow in Branching Tubes—Effect of Reynolds Number and Geometry,” ASME J. Biomech. Eng., 117, pp. 442–447.
Hitt,  D. L., and Lowe,  M. L., 1999, “Confocal Imaging of Flows in Artificial Venular Bifurcations,” ASME J. Biomech. Eng., 121, pp. 170–177.
Peach, J. P., Hitt, D. L., and Dunlap, C. T., 2000, “Three-Dimensional Imaging of Microfluidic Mixing Surfaces Using Dual-Channel Confocal Microscopy,” Microelectromechanical Systems (MEMS)—Proceedings of the 2000 ASME International Mechanical Engineering Congress & Exposition, MEMS Vol. 2, pp. 497–504.
Hitt, D. L., 2002, “Confocal Imaging of Fluidic Interfaces in Microchannel Geometries,” in Science, Technology & Education in Microscopy: An Overview, edited by A. Mendez-Vilas, ed., Vol. 1, 622–9.
Enden,  G., and Popel,  A. S., 1992, “A Numerical Study of the Shape of the Surface Separating Flow Into Branches in Microvascular Bifurcations,” ASME J. Biomech. Eng., 114, pp. 298–404.
Ong,  J., Enden,  G., and Popel,  A. S., 1994, “Converging Three-Dimensional Stokes Flow of Two Fluids in a T-Type Bifurcation,” J. Fluid Mech., 270, pp. 51–71.
Landau, L. D., and Lifshitz, E. M., 1987, Course of Theoretical Physics: Volume 6. Fluid Mechanics, Pergamon Press, New York, USA.
Rosenhead, L., 1963, Laminar Boundary Layers, Oxford University Press, New York, USA.
Petersen,  K. E., 1982, “Silicon as a Mechanical Material,” Proc. IEEE, 2, pp. 420–457.
Hirt,  C. W., and Nichols,  B. D., 1981, “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries,” J. Comput. Phys., 39, pp. 201–225.
Hitt, D. L., and McGarry, M., 2004, “Numerical Simulations of Laminar Mixing Surfaces in Pulsatile Microchannel Flows,” in Mathematics and Computers in Simulation, Elsevier Mathematics, Elsevier, Amsterdam.
Harris, T. R., 2003, Master’s thesis, University of Vermont.
Harris, T. R., and Hitt, D. L., 2003, “Low Reynolds Number Transition to Core-Annular Flow in Converging Microchannels,” Bulletin APS—Division of Fluid Dynamics.
Thorsen,  T., Roberts,  R., Arnold,  F., and Quake,  S., 2001, “Dynamic Pattern Formation in a Vesicle-Generating Microfluid Device,” Phys. Rev. Lett., 86, No. 18, pp. 4163–4166.
Nisisako, T., Fukudome, K., Torii, T., and Higchi, T., 2001, “Nanoliter-Sized Droplet Formation in a Microchannel Network,” Proc. of ISMM2001, pp. 102–103.
Dreyfus,  R., Tabeling,  P., and Willaime,  H., 2003, “Ordered and Disordered Patterns in Two-Phase Flows in Microchannels,” Phys. Rev. Lett., 90, No. 14, pp. 144505-1-4.
Harris, T. R., Hitt, D. L., and Macken, N. A., 2003, “Periodic Slug Formation in Converging Immiscible Microchannel Flows,” Bulletin APS—Division of Fluid Dynamics.
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Figures

Grahic Jump Location
Schematic diagram showing the geometry for the elliptical circular channel analysis. The view is that of a hypothetical downstream cross-section. The fluid from the side branch inlet is represented by the shaded area and enters the main branch from the right.
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Schematic diagram showing the geometry for the rectangular channel analysis. The channel has an aspect ratio of a/b; the square channel corresponds to the special case a=b. The view is that of a hypothetical downstream cross-section. The fluid from the side branch inlet is represented by the shaded area and enters the main branch from the right.
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Schematic diagram showing the geometry for the triangular channel analysis. The view is that of a hypothetical downstream cross-section. The fluid from the side branch inlet is represented by the shaded area and enters the main branch from the right. Each leg of the triangle is represented by a linear equation fi(x),i=1, 2, 3, as dictated by the coordinate system shown.
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The piecewise definition of the flow ratio of the triangular duct for the cases Q*>0.5 (left) and Q*<0.5 (right). As before, the shaded are represents the fluid from the side branch and enters the channel from the right.
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The problem definition for the 2-D analysis of converging fluids of unequal viscosities
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A sample 2-D experimental image of steady converging flow in a square microchannel with a cross-section of approximately 127×127 microns. Arrows indicate flow direction. The fluids are identical and Reynolds number in the outlet branch is ∼2. The image is acquired under fluorescent microscopy with a 10× air objective using FITC (side branch) and resorufin (main branch) as fluorescent labels. For the case shown, the inlet flow rates are identical (Q*=0.5) and the interface is clearly visible in the center of the downstream branch.
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A sample volumetric reconstruction of the separation surface for converging identical fluids in a square microchannel (127×127 micron cross-section). The flow ratio here is Q*=0.5. This volume was reconstructed from a series of horizontal image slices acquired using single-channel laser-scanning confocal microscopy; details on this experimental methodology are described in Hitt 6. For improved clarity of the separation surface, only the main branch fluid has been reconstructed. The fully developed separation surface in the downstream branch is located in the middle of the channel as expected and is virtually planar. This provides experimental support for the planar assumption made in this analysis.
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Plot of the predicted planar interface position as a function of the flow ratio for the various cross sectional geometries. Note the sigmoidal shape common to all of the curves. The maximum discrepancies between the different cases occurs near the walls (L→0,1).
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A comparison of the predicted planar interface positions (solid lines) from Fig. 8 with the weighted-mean interfacial positions as determined by 3-D numerical simulations (symbols). The “error bars” associated with the numerical simulations represent the maximum±lateral extents of the numerical separation surface away from the mean position.
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A table of cross-sectional views of the fully developed separation surface in the outlet branch as obtained from numerical simulations. Shown are the results as a function of flow ratio Q* and cross-sectional geometry. The side inlet fluid is the darker shade and enters from the right. In all cases the Reynolds number is ∼2. Virtually all computed surfaces have some degree of curvature and the triangles show the position of the analytically predicted planar interface. The triangular case offers the most interesting results and features inflection in the curvature.
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A comparison of the predicted planar interface positions (solid lines) for a square microchannel (127×127 micron cross-section) with experimental measurements (symbols) obtained from 2-D fluorescence microscopy. The error bars indicate the uncertainty in estimating the interfacial position from the digital images of the flow. Under 2-D imaging one sees a view that is integrated through the channel depth. Surface curvature thus tends to produce a slightly blurred interface.
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Plot of the predicted planar interface position as a function of the flow ratio and viscosity ratio for a 2-D channel. For a fixed flow ratio, the more viscous fluid must occupy a larger fraction of the channel to satisfy mass conservation.
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A comparison of the predicted planar interface positions (solid lines) from Fig. 12 with experimental measurements (symbols) obtained from 2-D fluorescence microscopy in a square microchannel (127×127 micron cross-section). The Reynolds number in all cases is ∼2.
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Fully developed separation surfaces in the downstream branch as a function of Reynolds number as obtained from 3-D numerical simulations. The results are for 90° converging flows of identical liquids in square microchannels with a unity flow ratio (Q*=1). Also shown for comparison is the planar model prediction (solid line). The validity of the planar assumption degrades with increasing Reynolds number and becomes unacceptable for Re≳10.
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Experimental demonstrations of possible nonstratified flow configurations that can arise for converging microchannel flows. (a) For miscible fluids with a nonunity viscosity ratio, a core-annular configuration can occur at sufficiently high Reynolds number. Shown is a volumetric reconstruction obtained from confocal microscopy for Re∼50, Q*=0.4, and μ* =2. (b) For immiscible fluids, periodic slug formation can be observed at low Reynolds numbers. Shown is a high-speed CCD image for converging flows of aqueous glycerol and octanol which have a surface tension σ=0.034 N/m; here Re∼2, Q*=1, and μ* =1.

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