Impulsive displacement of a liquid in a pipe at high Reynolds numbers
Author: Gerald G. Kleinstein
Journal: Quart. Appl. Math. 69 (2011), 157-176
MSC (2000): Primary 76B99, 76D99, 76E17
Published electronically: December 23, 2010
MathSciNet review: 2807983
Abstract: We consider the problem of an impulsive displacement of a liquid, originally at rest in a circular pipe, which is displaced by another liquid. The purpose of this analysis is to show that at a sufficiently high inertia the initial essentially inviscid motion can be extended to cover the entire displacement process, thus creating an inviscid window to which an inviscid analysis can be applied. We simplify the problem first, by considering a 1-liquid problem where the displacing liquid and displaced liquid are the same. We identify two characteristic times in this problem: the time it takes an inviscid liquid to be displaced, and the time it takes a viscous liquid to attain a steady state. Taking the ratio of the two defines the Reynolds number for the problem and we show that the motion becomes essentially inviscid once the Reynolds number is sufficiently high. We obtain the general solution of the 1-liquid problem which determines the nondimensional viscous displacement time as a function of the Reynolds number. We derive from the general solution: a critical Reynolds number above which the motion remains unsteady throughout the entire displacement process, and a formula which determines quantitatively whether applying an inviscid analysis to the 1-liquid viscous problem at a given Reynolds number is admissible within an acceptable error tolerance. We also show that at the limit the Reynolds number approaches infinity the viscous displacement time approaches the inviscid displacement time and that the velocity profile and the shape of the material surface separating the displacing from the displaced liquid approach their counterpart in the inviscid solution. Second, based on these results we propose that an inviscid solution is applicable to the 2-liquid viscous problem once the condition of a high Reynolds number is independently met by the two participating liquids. We obtain the solution to the inviscid 2-liquid displacement problem and calculate various examples. Finally, we present a stability analysis of the flat interface between the two inviscid liquids, which shows which of the examples is stable, neutrally stable, or unstable. The paucity of data for an impulsive displacement in the high Reynolds number range makes quantitative comparisons difficult. However, the excellent agreement obtained between the critical Reynolds number derived in this analysis and the result obtained in a numerical analysis of the viscous 2-liquid problem elsewhere constitutes at least a partial validation of the theory. Additional confirmation is obviously recommended.
- 1. G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
- 2. Dimakopoulos, Y. and Tsamopoulos, J., 2003. Transient displacement of a Newtonian fluid by air in a straight or suddenly constricted tubes, Physics of Fluids Vol. 15, No 7.
- 3. Gerald G. Kleinstein, On the nonpersistence of irrotational motion in a viscous heat-conducting fluid, Arch. Rational Mech. Anal. 101 (1988), no. 2, 95–105. MR 921933, https://doi.org/10.1007/BF00251455
- 4. D. A. Reinelt and P. G. Saffman, The penetration of a finger into a viscous fluid in a channel and tube, SIAM J. Sci. Statist. Comput. 6 (1985), no. 3, 542–561. MR 791184, https://doi.org/10.1137/0906038
- 5. P. G. Saffman and Geoffrey Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London. Ser. A 245 (1958), 312–329. (2 plates). MR 0097227, https://doi.org/10.1098/rspa.1958.0085
- 6. Geoffrey Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I, Proc. Roy. Soc. London. Ser. A. 201 (1950), 192–196. MR 0036104, https://doi.org/10.1098/rspa.1950.0052
- 7. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
- 8. Szymanski, F., 1932, Quelques solutions exactes des equations de l'hydrodynamique de fluide visqueux dans le cas d'un tube cylinderique. J. Math. Pures Appliqués, Series 9, 11, 67.
- Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge University Press. MR 1744638 (2000j:76001)
- Dimakopoulos, Y. and Tsamopoulos, J., 2003. Transient displacement of a Newtonian fluid by air in a straight or suddenly constricted tubes, Physics of Fluids Vol. 15, No 7.
- Kleinstein, G. G., 1988, On the non-persistence of irrotational motion in a viscous, heat-conducting, fluid, Arch. Ratl. Mech. Anal. 101 (No. 2), pp. 95-105. MR 921933 (89c:76024)
- Reinlet, D. A. and Saffman, P. G., 1985, The penetration of a finger into a viscous fluid in a channel and tube, SIAM J. Sci. Stat. Comput. Vol. 6, No. 3, July 1985. MR 791184 (87a:76015)
- Saffman, P. G. and Taylor, G.I., 1958, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. A, 245, pp. 312-329. MR 0097227 (20:3697)
- Taylor, G.I., 1950, The stability of liquid surfaces when accelerated in the direction perpendicular to their planes, Proc. Roy. Soc. (London) A, 201, 192-196. MR 0036104 (12:58f)
- Watson. G.N. 1952, A Treatise on the Theory of Bessel Functions, Cambridge at the University Press. MR 1349110 (96i:33010)
- Szymanski, F., 1932, Quelques solutions exactes des equations de l'hydrodynamique de fluide visqueux dans le cas d'un tube cylinderique. J. Math. Pures Appliqués, Series 9, 11, 67.
Gerald G. Kleinstein
Affiliation: Dept. of Mechanical and Industrial Eng., Northeastern University, Boston, MA 02115
Keywords: Impulsive displacement, liquids, high Reynolds numbers, displacement times, interface stability
Received by editor(s): August 13, 2009
Published electronically: December 23, 2010
Additional Notes: The author wishes to thank Professor Lu Ting of the Courant Institute of Mathematical Sciences and Dr. George Waldman for many valuable discussions.
Article copyright: © Copyright 2010 Brown University