Korea-Australia Rheology Journal

The Korean Society of Rheology (한국유변학회)
권호 표지

Numerical study on the effect of viscoelasticity on pressure drop and film thickness for a droplet flow in a confined microchannel

  • Chung, Chang-Kwon (School of Chemical and Biological Engineering, Seoul National University) ;
  • Kim, Ju-Min (Department of Chemical Engineering, Ajou University) ;
  • Ahn, Kyung-Hyun (School of Chemical and Biological Engineering, Seoul National University) ;
  • Lee, Seung-Jong (School of Chemical and Biological Engineering, Seoul National University)
  • Published : 2009.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

The prediction of pressure drop for a droplet flow in a confined micro channel is presented using FE-FTM (Finite Element - Front Tracking Method). A single droplet is passing through 5:1:5 contraction - straight narrow channel - expansion flow domain. The pressure drop is investigated especially when the droplet flows in the straight narrow channel. We explore the effects of droplet size, capillary number (Ca), viscosity ratio ($\chi$) between droplet and medium, and fluid elasticity represented by the Oldroyd-B constitutive model on the excess pressure drop (${\Delta}p^+$) against single phase flow. The tightly fitted droplets in the narrow channel are mainly considered in the range of $0.001{\leq}Ca{\leq}1$ and $0.01{\leq}{\chi}{\leq}100$. In Newtonian droplet/Newtonian medium, two characteristic features are observed. First, an approximate relation ${\Delta}p^+{\sim}{\chi}$ observed for ${\chi}{\geq}1$. The excess pressure drop necessary for droplet flow is roughly proportional to $\chi$. Second, ${\Delta}p^+$ seems inversely proportional to Ca, which is represented as ${\Delta}p^+{\sim}Ca^m$ with negative m irrespective of $\chi$. In addition, we observe that the film thickness (${\delta}_f$) between droplet interface and channel wall decreases with decreasing Ca, showing ${\delta}_f{\sim}Ca^n$ Can with positive n independent of $\chi$. Consequently, the excess pressure drop (${\Delta}p^+$) is strongly dependent on the film thickness (${\delta}_f$). The droplets larger than the channel width show enhancement of ${\Delta}p^+$, whereas the smaller droplets show no significant change in ${\Delta}p^+$. Also, the droplet deformation in the narrow channel is affected by the flow history of the contraction flow at the entrance region, but rather surprisingly ${\Delta}p^+$ is not affected by this flow history. Instead, ${\Delta}p^+$ is more dependent on ${\delta}_f$ irrespective of the droplet shape. As for the effect of fluid elasticity, an increase in ${\delta}_f$ induced by the normal stress difference in viscoelastic medium results in a drastic reduction of ${\Delta}p^+$.

Keywords

References

  1. Abkarian, M., M. Faivre and H. A. Stone, 2006, High-speed microfluidic differential manometer for cellular-scale hydrodynamics, PNAS 103(3), 538-542 https://doi.org/10.1073/pnas.0507171102
  2. Abraham, S., E. H. Jeong, T. Arakawa, S. Shoji, K. C. Kim, I. Kim and J. S. Go, 2006, Microfluidics assisted synthesis of well-defined spherical polymeric microcapsules and their utilization as potential encapsulants, Lab Chip 6(6), 752-756 https://doi.org/10.1039/b518006f
  3. Adzima, B. J. and S. S. Velankar, 2006, Pressure drops for droplet flows in microfluidic channels, J. Micromech. Microeng. 16(8), 1504-1510 https://doi.org/10.1088/0960-1317/16/8/010
  4. Bretherton, F. P., 1961, The motion of long bubbles in tubes, J. Fluid Mech. 10, 166-188 https://doi.org/10.1017/S0022112061000160
  5. Bringer M. R., C. J. Gerdts, H. Song, J. D. Tice and R. F. Ismagilov, 2004, Microfluidic systems for chemical kinetics that rely on chaotic mixing in droplets, Philos. T. Roy. Soc. A 362(1818), 1087-1104 https://doi.org/10.1098/rsta.2003.1364
  6. Brooks, A. N. and T. J. R. Hughes, 1982, Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Method Appl. Mech. Engrg. 32(1-3), 199-259 https://doi.org/10.1016/0045-7825(82)90071-8
  7. Chinyoka, T., Y. Y Renardy, A. Renardy and D. B. Khismatullin, 2005, Two-dimensional study of drop deformation undεr simple shear for Oldroyd-B liquids, J. Non-Newtonian Fluid Mech. 130(1), 45-56 https://doi.org/10.1016/j.jnnfm.2005.07.005
  8. Chio, H., M. J. Jensen, X. L. Wang, H. Bruus and D. Attinger, 2006a, On the motion of a bubble through microchannel contractions, NSTI-Nanotech. 2, 497-500
  9. Chio, H., M. J. Jensen, X. L. Wang, H. Bruus and D. Attinger, 2006b, Transient pressure drops of gas bubbles passing through liquid-filled microchannel contractions: an experimental study, J. Micromech. Microeng. 16(1), 143-149 https://doi.org/10.1088/0960-1317/16/1/019
  10. Christopher, G. F. and S. L. Anna, 2007, Microfluidic methods for generating continuous droplet streams, J. Phys. D: Appl Phys. 40(19), R319-R336 https://doi.org/10.1088/0022-3727/40/19/R01
  11. Chu, L. Y, A. S. Utada, R. K. Shah, J. W. Kim and D. A. Weitz, 2007, Controllable monodisperse multiple emulsions, Angew. Chem. Int. Edit. 46(47), 8970-8974 https://doi.org/10.1002/anie.200701358
  12. Chung, C., M. A. Hulsεn, J. M. Kim, K. H. Ahn and S. J. Lee, 2008, Numerical study on the effect of viscoelasticity on drop deformation in simple shear and 5:1:5 planar contraction/expansion microchannel, J. Non-Newtonian Fluid Mech. 155, 80-93 https://doi.org/10.1016/j.jnnfm.2008.06.002
  13. Chung, P. M. Y. and M. Kawaji, 2004, The effect of channel diameter on adiabatic two-phase flow characteristics in microchannels, Int. J. Multiphase Flow 30(7-8), 735-761 https://doi.org/10.1016/j.ijmultiphaseflow.2004.05.002
  14. Cox, B. G., 1962, On driving viscous fluid out of a tube, J. Fluid Mech. 14, 81-96 https://doi.org/10.1017/S0022112062001081
  15. Denn, M. M., 1980, Process fluid mechanics, Prentice-Hall, Inc
  16. Edvinsson, R. K. and S. Irandoust, 1996, Finite-element analysis of Taylor flow, AIChE J. 42(7), 1815-1823 https://doi.org/10.1002/aic.690420703
  17. Fairbrother, F. and A. E. Stubbs, 1935, Studies in electroendosmosis. Part VI. The bubble-tube method of measurements, J. Chem. Soc. 1, 527-529
  18. Fuerstman, M. J., A. Lai, M. E. Thurlow, S. S. Shevkoplyas, H. A. Stone and G. M. Whitesides, 2007, The pressure drop along rectangular microchannels containing bubblεs, Lab Chip 7(11), 1479-1489 https://doi.org/10.1039/b706549c
  19. Giavedoni, M. D. and F. A. Saita, 1997, The axisymmetric and plane cases of a gas phase stedily displacing a Newtonian liquid - A simultaneous solution of governing equations, Phys. Fluids 9(8), 2420-2428 https://doi.org/10.1063/1.869360
  20. Giavedoni, M. D. and F. A. Saita, 1999, The rear meniscus of a long bubble steadily displacing a Newtonian liquid in a capillary tube, Phys. Fluids 11(4), 786-794 https://doi.org/10.1063/1.869951
  21. Griffiths, A. D. and Taqfik, 2003, Directed evolution of an extremely fast phosphotriesterase by in vitro compartmentalization, EMBO J. 22(1), 24-35 https://doi.org/10.1093/emboj/cdg014
  22. Hε， M. Y., J. S. Edgar, G. D. M. Jeffries, R. M. Lorenz, J. P. Shelby and D. T. Chiu, 2005, Selective encapsulation of single cells and subcellular organelles into picolter- and femtolitervolume droplets, Anal. Chem. 77(6), 1539-1544 https://doi.org/10.1021/ac0480850
  23. Heil, M., 2001, Finite Reynolds number effiects in the Bretherton problem, Phys. Fluids 13(9), 2517-2521 https://doi.org/10.1063/1.1389861
  24. Hulsen, M. A., R. Fattal and R. Kupferman, 2005, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newtonian Fluid Mech. 127(1), 27-39 https://doi.org/10.1016/j.jnnfm.2005.01.002
  25. Jensen, M. J., G. Goranovic and H. Bruus, 2004, The clogging presure of bubbles in hydrophilic microchannel contractions, J. Micromech. Microeng. 14(7), 876-883 https://doi.org/10.1088/0960-1317/14/7/006
  26. Kline, T. R., M. K. Runyon, M. Pothiawala and R. F. Ismagilov, 2008, ABO, D blood typing and subtyping using plug-based microfluidics, Anal. Chem. 80(16), 6190-6197 https://doi.org/10.1021/ac800485q
  27. Kreutzer, M. T., F. Kapteijn, J. A. Moulijn, C. R. Kleijn and J. J. Heiszwolf, 2005, Inertial and interfacial effiects on pressure drop of Taylor flow in capillaries, AIChE J. 51(9), 2428-2440 https://doi.org/10.1002/aic.10495
  28. Laser, D. J. and J. G Santiago, 2004, A review of micropumps, J. Micromech. Microeng. 14(6), R35-R64 https://doi.org/10.1088/0960-1317/14/6/R01
  29. Liu, A. W., D. E. Bornside, R. C. Armstrong and R. A. Brown, 1998, Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for mlcrostructure and flow fields, J. Non-Newtonian Fluid Mech. 77(3), 153-190 https://doi.org/10.1016/S0377-0257(97)00067-0
  30. Morris, C. J. and F. K. Forster, 2004, Oscillatory flow in microchannels: comparison of exact and approximate impedance models with experiments, Exp. Fluids 36(6), 928-937 https://doi.org/10.1007/s00348-003-0776-9
  31. Olbricht, W. L., 1996, Pore-scale prototypes of mu1tiphase flow in porous media, Ann. Rev. Fluid Mech. 28, 187-213 https://doi.org/10.1146/annurev.fl.28.010196.001155
  32. Pillapakkam, S. B. and P. Singh, 2001, A level-set method for computing solutions to viscoelastic two-phase flow, J. Comput Phys. 174(2), 552-578 https://doi.org/10.1006/jcph.2001.6927
  33. Queguiner, C. and D. Barthes-Biesel, 1997, Axisymmetric motion of capsules through cylindrical channeIs, J. Fluid Mech. 348, 349-376 https://doi.org/10.1017/S0022112097006587
  34. Ratulowski,J. and H. C. Chang, 1989, Transport of gas bubbles in capillaries, Phys. Fluids A 1(10), 1642-1655 https://doi.org/10.1063/1.857530
  35. Reinelt, D. A , 1987, Thε rate at which a long bubble rises in a vertical tube, J. Fluid Mech. 175, 557-565 https://doi.org/10.1017/S0022112087000521
  36. Secomb, T. W. and R. Hsu, 1995, Red blood cell mechanics and functional capillary density, Int. J. Microcirc. 15(5), 250-254 https://doi.org/10.1159/000179026
  37. Sepp, A , F. Ghadessy and Y. Choo, 2007, Cell-free selection of DNA-binding proteins for future gene therapy applications, Gene Therapy and Regulation 3(1), 51-63 https://doi.org/10.1142/S156855860700006X
  38. Shen, E. I. and K. S. Udell, 1985, A finite element study of low Reynolds number two-phase flow in cylindrical tubes, J. Appl. Mech Trans. ASME 52(2), 253-256 https://doi.org/10.1115/1.3169036
  39. Song, H., J. D. Tice and R. F. Ismagilov, 2003, A microfluidic system for controlling reaction networks in time, Angew. Chem. Int. Edit. 42(7), 768-772 https://doi.org/10.1002/anie.200390203
  40. Tawfik, D. S. and A. D. Griffiths, 1998, Man-made cel1-1ike compartments for molecular evolution, Nat Biotechnol. 16(7), 652-656 https://doi.org/10.1038/nbt0798-652
  41. Taylor, G. I., 1961, Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech. 10, 161-165 https://doi.org/10.1017/S0022112061000159
  42. Tryggvason, G, B. Bunner, A Esmaeeli, D. Juric, N. AI-Rawahi, W. Tauber, J. Han, S. Nas and Y. J. Jan, 2001, A front-tracking method for the computations of multiphasε flow, J. Comput. Phys. 169(2), 708-759 https://doi.org/10.1006/jcph.2001.6726
  43. Wong, H., C. J. Radke and S. Morris, 1995, The motion of long bubbles in polygonal capillaries. 1. thin-films, J. Fluid Mech. 292, 71-94 https://doi.org/10.1017/S0022112095001443
  44. Wong, H., C. J. Radke and S. Morris, 1995, The motion of long bubbles in polygonal capillaries. 2. drag, fluid pressure and fluid flow, J. Fluid Mech. 292, 95-110 https://doi.org/10.1017/S0022112095001455
  45. Yue, P. T., J. J. Feng, C. Liu and J. Shen, 2005, Viscoelastic effiects on drop deformation in steady shear, J. Fluid Mech 540, 427-437 https://doi.org/10.1017/S0022112005006166
  46. Zheng, B., J. D. Tice and R. F. Ismagilov, 2004, Formation of arrayed droplets of soft lithography and two-phase fluid flow, and application in protein crystallization, Adv. Mater. 16(15), 1365-1368 https://doi.org/10.1002/adma.200400590