Abstract
Displacements of a viscous fluid by a miscible fluid of a lesser viscosity and density in cylindrical tubes were investigated experimentally. Details of velocity and Stokes streamline fields in vertical tubes were measured using a DPIV (digital particle image velocimetry) technique. In a reference frame moving with the fingertip, the streamline patterns around the fingertip obtained from the present measurements confirm the hypothesis of Taylor (1961) for the external patterns, and that of Petitjeans and Maxworthy (1996) for the internal patterns. As discussed in these papers, the dependent variable, m, a measure of the volume of viscous fluid left on the tube wall after the passage of the displacing finger, is a parameter that determines the flow pattern. When m>0.5 there is one stagnation point at the tip of the finger; when m<0.5 there are two stagnation points on the centerline, one at the tip and the other inside the fingertip, and a stagnation ring on the finger surface with a toroidal recirculation in the fingertip between the two stagnation points. The finger profile is obtained from the zero streamline of the streamline pattern.
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References
Chen CY, Meiburg E (1996) Miscible displacements in a capillary tube. Part II. Numerical simulations. J Fluid Mech 326:57–90
Chen CY, Meiburg E (2002) Miscible displacements in a capillary tubes: influence of Korteweg stresses and divergence effects. Phys Fluids 14(7):2052–2058
Clift R, Grace JR, Weber ME (1978) Bubbles, drops, and particles. Academic Press, New York
Cox BG (1962) On driving a viscous fluid out of a tube. J Fluid Mech 14:81–96
Fernandez J, Kurowski P, Petitjeans P, Meiburg E (2002) Density-driven unstable flows of miscible fluids in a Hele-Shaw cell. J Fluid Mech 451:239–260
Fincham AM, Spedding GR (1997) Low cost, high resolution DPIV for measurement of turbulent fluid flow. Exp Fluids 23:449–462
Joseph D, Renardy Y (1993) Fundamentals of two-fluid dynamics. Part II. Lubricated transport, drops and miscible liquids. Springer, Berlin Heidelberg New York
Korteweg DJ (1901) Sur la forme que prennant les equations du mouvement des fluides si l’on tient compte des forces capillaires causees par des variations de densite. Arch Neerl Sci Ex Nat (ii), 6, 1
Kuang J, Maxworthy T, Petitjeans P (2003) Miscible displacement between silicone oils in capillary tubes. Eur J Mech B–Fluid 22:271–277
Lajeunesse E, Martin J, Rakotomalala N, Salin D, Yortsos YC (1999) Miscible displacement in a Hele-Shaw cell at high rates. J Fluid Mech 398:299–319
Payr M, Vanaparthy SH, Meiburg E (2003) Influence of variable viscosity on density-driven instabilities in capillary tubes. To appear in J Fluid Mech
Petitjeans P, Maxworthy T (1996) Miscible displacements in a capillary tube. Part 1. Experiments. J Fluid Mech 326:37–56
Rakotomalala N, Salin D, Watzky P (1997) Miscible displacement between two parallel plates: BGK lattices gas simulations. J Fluid Mech 338:277–297
Reinelt DA, Saffman PG (1985) The penetration of a finger into a viscous fluid in a channel and tube. SIAM J Sci Stat Comp 6:542–561
Taylor GI (1961) Deposition of a viscous fluid on the wall of a tube. J Fluid Mech 10:161–165
Acknowledgements
This work was supported by the NASA Microgravity Program under grant Number NAG3–2125 to the University of Southern California.
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An erratum to this article can be found at http://dx.doi.org/10.1007/s00348-004-0842-y
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Kuang, J., Petitjeans, P. & Maxworthy, T. Velocity fields and streamline patterns of miscible displacements in cylindrical tubes. Exp Fluids 37, 301–308 (2004). https://doi.org/10.1007/s00348-004-0824-0
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DOI: https://doi.org/10.1007/s00348-004-0824-0