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Experiments in Fluids

, Volume 21, Issue 3, pp 187–200 | Cite as

An experimental investigation of initial oscillations in a radial Hele-Shaw cell

  • S. B. K. Burns
  • S. G. Advani
Originals

Abstract

Hele-Shaw cell is a laboratory device consisting of two parallel plates of glass separated by a thin gap. In this cell, in the flow of two immiscible fluids, when a fluid of higher viscosity is displaced by a fluid of lower viscosity, the less viscous fluid is observed to form “fingers” into the more viscous one due to the unstable interface. The Saffman-Taylor or viscous finger instability has been examined and modeled for over forty years for the rectilinear Hele-Shaw cell and about half as long for the radial Hele-Shaw cell. In this paper, we study, in detail, the early development of viscous instabilities in a radial Hele-Shaw cell. This source flow configuration has been chosen so that the instability can be monitored precisely. The objective of this study is to examine the onset of fingering, i.e. initial number of fingers that form, and the evolution of interface instability. Our experiments suggest that there may be some order in this formation process and one can model this aspect by considering the unsteady velocity components and predicting temporal changes in wavenumber responsible for the initial number of fingers and may be later accounting for the fingertip oscillations and splitting.

We injected a water-based fluid into an oil in a radial Hele-Shaw cell at constant flow rate and recorded the movement of the less viscous droplet as it evolved. The relative curvature changes on the expanding droplet boundary was plotted with the angular positions about the interface and subtracting out the average radius, resulting in a plot of the change in amplitude with respect to time for the interface configuration. Three unstable configured tests at kinematic viscosity contrast (vO) of 0.34, 0.68, and 0.94 were run at approximately the same flow rate (2π cm2/s). The droplet exhibited oscillatory movement for these unstable configuration. The amplitude and the rate of oscillations were measured from digitized data. The smaller the viscosity difference, the smaller was the amplitude growth rate and resulted in a longer time to form visible finger initiation.

Keywords

Immiscible Fluid Interface Instability Viscosity Difference Viscosity Contrast Interface Configuration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arneodo A; Couder Y; Grasseau G; Hakim V; Rabaud M (1990) Nonlinear evolution of spatio-temporal structures in dissipative continuous systems, pp. 481–488. New York: Plenum PressGoogle Scholar
  2. Beer F Pl; Johnston ER (1984) Vector mechanics for engineers: dynamics. New York: McGraw HillGoogle Scholar
  3. Bird RB; Stewart WE; Lightfoot EN (1960) Transport phenomena. New York: WileyGoogle Scholar
  4. Bouissou Ph; Perrin B; Tabeling P (1990) Nonlinear evolution of spatio-temporal structures in dissipative continuous systems, pp. 475–479. New York: Plenum PressGoogle Scholar
  5. Casademunt J; Jasnow David (1992) Interface equation and viscosity contrast in Hele-Shaw flow. J Mod Phy 6: 1647–1656Google Scholar
  6. Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. London: Oxford University PressGoogle Scholar
  7. Chen JG (1989) Growth of radial viscous fingers in a Hele-Shaw cell. J Fluid Mech 201: 223–242Google Scholar
  8. Chen JG (1987) Radial viscous fingering patterns in Hele-Shaw cells. Exp Fluids 5: 363–371CrossRefGoogle Scholar
  9. Chikhliwala ED; Huang AB; Yortsos YC (1988) Numerical study of the linear stability of immiscible displacement in porous media. Transport in Porous Media 3: 257–276CrossRefGoogle Scholar
  10. Couder Y; Gerard N; Rabaud M (1986) Narrow fingers in the Saffman-Taylor instability. Phys Rev 34: 5175–5178Google Scholar
  11. Couder Y; Michalland S; Rabaud M; Thome H (1990) Nonlinear evolution of spatio-temporal structures in dissipative continuous systems, pp. 487–497. New York: Plenum PressGoogle Scholar
  12. Cowen R (1994) A fresh look at a familiar supernova. Science News 146: 81–96Google Scholar
  13. Degregoria AJ; Schwartz LW (1986) A boundary-integral method for two-phase displacement in Hele-Shaw cells. J Fluid Mech 164: 283–400MathSciNetGoogle Scholar
  14. DiFrancesco MW; Rauseo SN; Maher JV (1987) Viscous fingering as a first step toward understanding dendrites. Superlattices and Microstructures 3: 617–623CrossRefGoogle Scholar
  15. Dougherty A; Kaplan PD; Gollub JP (1987) Development of side branching in dendritic crystal growth. Phys Rev Lett 58: 1652–1655CrossRefGoogle Scholar
  16. Gollub JP (1994) Instabilities of film flows. 8 August 1994 Seminar at University of Delaware, Department of Mechanical EngineeringGoogle Scholar
  17. Greenberg MD (1988) Advanced engineering mathematics. New Jersey: Prentice-HallGoogle Scholar
  18. Hill S (1952) Channelling in packed columns. Chem Eng Sci 1: 247–253Google Scholar
  19. Harris CJ; Miles JF (1980) Stability of linear systems: some aspects of kinematic similarity. London: Academic PressGoogle Scholar
  20. Hoffmann FM; Wolf GH (1974) Excitation of parametric instabilities in statically stable and unstable fluid interfaces. J Appl Phys 45: 3859–3863CrossRefGoogle Scholar
  21. Homsy GM (1987) Viscous fingering in porous media. Ann Rev Fluid Mech 19: 271–311CrossRefGoogle Scholar
  22. Howison SD (1986) Fingering in Hele-Shaw cells. J Fluid Mech 167: 439–453zbMATHMathSciNetGoogle Scholar
  23. Kertesz J (1990) Statistical models for the fracture of disordered media, pp. 261–290. North-Holland: Elsevier Science PublishersGoogle Scholar
  24. Kessler DA; Koplik J; Levine H (1988) Pattern selection in fingered growth phenomena. Adv Phys 37: 255–339CrossRefGoogle Scholar
  25. Khabeev NS; Shagapov V Sh (1986) Oscillations of a gas-vapor bubble in an acoustic field. Izv Akad Nauk USSR Mekh Zhidk Gaza 3: 79–83Google Scholar
  26. Kopf-Sill AR; Homsy GM (1987) Narrow fingers in a Hele-Shaw cell. Phys Fluids 30: 2607–2609CrossRefGoogle Scholar
  27. Marion JB (1970) Classical dynamics of particles and systems, pp. 314–315. New York: Academic PressGoogle Scholar
  28. Matsushita M; Yamada H (1990) Dendritic growth of single viscous finger under the influence of linear anisotropy. J Crystal Growth 99: 161–164Google Scholar
  29. Maxworthy T (1987) The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell. J Fluid Mech 177: 207–232Google Scholar
  30. Mclean JW; Saffman PG (1981) The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J Fluid Mech 102: 455–469Google Scholar
  31. Meakin P; Family F; Vicsek T (1987) Viscous fingering simulated by off-lattice aggregation. J Colloid Interface Sci 117: 394–399Google Scholar
  32. Meiburg E; Homsy GM (1988) Nonlinear unstable viscous fingers in Hele-Shaw flows. Phys Fluids 31: 429–439CrossRefGoogle Scholar
  33. Miller CA; Scriven LE (1968) The oscillations of a fluid droplet immersed in another fluid. J Fluid Mech 32: 417–435Google Scholar
  34. Olabisi O (1982) Mechanisms of the structural web process. Pol Eng Rev 2: 29–70Google Scholar
  35. Paterson L (1981) Radial fingering in a Hele-Shaw cell. J Fluid Mech 113: 513–529Google Scholar
  36. Prigogine I; Stengers I (1984) Order out of chaos: man's new dialog with nature. NY: Bantam BooksGoogle Scholar
  37. Prosperetti A (1980) Free oscillations of drops and bubbles: the initial-value problem. J Fluid Mech 100: 333–347zbMATHGoogle Scholar
  38. Properetti A (1987) The linear stability of general two-phase flow models — II. J Multiphase Flow 13:] 161–171Google Scholar
  39. Rabaud M; Couder Y; Gerard N (1988) Dynamics and stability of anomalous Saffman-Taylor fingers. Phys Rev A 37: 935–947CrossRefGoogle Scholar
  40. Rauseo SN; Barnes Jr PD; Maher JV (1987) Development of radial fingering patterns. Phys Rev A 35: 1245–1251Google Scholar
  41. Reinelt DA (1987) The effect of thin film variations and transverse curvature on the shape of fingers in a Hele-Shaw cell. Phys Fluids 30: 2617–2623CrossRefzbMATHGoogle Scholar
  42. Saffman PG (1959) Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell. Quart J Mech Appl Math 12: 146–151zbMATHMathSciNetGoogle Scholar
  43. Saffman PG; Taylor GI (1958) The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc Roy Soc A 245: 312–329MathSciNetGoogle Scholar
  44. Saffman PG (1986) Viscous fingering in Hele-Shaw cells. J Fluid Mech 173: 73–94zbMATHMathSciNetGoogle Scholar
  45. Saffman PG (1991) Selection mechanisms and stability of fingers and bubbles in Hele-Shaw cells. IMA J Appl Math 46: 137–145zbMATHMathSciNetGoogle Scholar
  46. Sarkar SK (1987) The effect of velocity-dependent boundary conditions on pattern formation. Superlattices and Microstructures 3: 589–594CrossRefGoogle Scholar
  47. Sarkar SK (1990) Scaling dynamics of immiscible radial viscous fingering. Phys Rev Lett 65: 2680–2683CrossRefGoogle Scholar
  48. Sarkar S; Jasnow D (1987) Quantitative test of solvability theory for the Saffman-Taylor problem. Phys Rev A 35: 4900–4903Google Scholar
  49. Schwartz LW; Degregoria J (1988) Two-phase flow in Hele-Shaw cells: numerical studies of sweep efficiency in a five-spot pattern. J Austral Math Soc Ser B 29: 375–400MathSciNetGoogle Scholar
  50. Schwartz L (1986) Stability of Hele-Shaw flows: the wetting layer effect. Phys Fluids 29: 3086–3088CrossRefGoogle Scholar
  51. Streeter VL; Wylie EB (1975) Fluid mechanics. Tokyo: McGraw-Hill KogakushaGoogle Scholar
  52. Tryggvason G; Aref H (1983) Numerical experiments on Hele-Shaw flow with a sharp interface. J Fluid Mech 136: 1–30Google Scholar
  53. Vanden-Broeck JM (1983) Fingers in a Hele-Shaw cell with surface tension. Phys Fluids 26: 2033–2034MathSciNetGoogle Scholar
  54. Wilson SDR (1975) A note on the measurement of dynamic contact angles. J Colloid Interface Sci 51: 532–534CrossRefGoogle Scholar
  55. Weinstein SJ; Dussan VEB; Ungar LH (1990) A theoretical study of two-phase flow through a narrow gap with a moving contact line: viscous fingering in a Hele-Shaw cell. J Fluid Mech 221: 53–76Google Scholar
  56. Wei-Shen Dai; Kadanoff LP; Su-Min Zhou (1991) Singularities in complex interface dynamics. Growth and form. New York: Plenum PressGoogle Scholar
  57. Chia-Shun Yih (1977) Fluid mechanics: a concise introduction to the theory. Michigan: West River PressGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • S. B. K. Burns
    • 1
  • S. G. Advani
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of DelawareNewarkUSA

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