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Miscible displacements in capillary tubes. Part 2. Numerical simulations

Published online by Cambridge University Press:  26 April 2006

Ching-Yao Chen  and
Eckart Meiburg
Ching-Yao Chen
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles, CA 90089-1191, USA
Eckart Meiburg
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles, CA 90089-1191, USA
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Abstract

Numerical simulations are presented which, in conjunction with the accompanying experimental investigation by Petitjeans & Maxworthy (1996), are intended to elucidate the miscible flow that is generated if a fluid of given viscosity and density displaces a second fluid of different such properties in a capillary tube or plane channel. The global features of the flow, such as the fraction of the displaced fluid left behind on the tube walls, are largely controlled by dimensionless quantities in the form of a Péclet number Pe, an Atwood number At, and a gravity parameter. However, further dimensionless parameters that arise from the dependence on the concentration of various physical properties, such as viscosity and the diffusion coefficient, result in significant effects as well.

The simulations identify two distinct Pe regimes, separated by a transitional region. For large values of Pe, typically above O(10), a quasi-steady finger forms, which persists for a time of O(Pe) before it starts to decay, and Poiseuille flow and Taylor dispersion are approached asymptotically. Depending on the strength of the gravitational forces, we observe a variety of topologically different streamline patterns, among them some that leak fluid from the finger tip and others with toroidal recirculation regions inside the finger. Simulations that account for the experimentally observed dependence of the diffusion coefficient on the concentration show the evolution of fingers that combine steep external concentration layers with smooth concentration fields on the inside. In the small-Pe regime, the flow decays from the start and asymptotically reaches Taylor dispersion after a time of O(Pe).

An attempt was made to evaluate the importance of the Korteweg stresses and the consequences of assuming a divergence-free velocity field. Scaling arguments indicate that these effects should be strongest when steep concentration fronts exist, i.e. at large values of Pe and At. However, when compared to the viscous stresses, Korteweg stresses may be relatively more important at lower values of these parameters, and we cannot exclude the possibility that minor discrepancies observed between simulations and experiments in these parameter regimes are partially due to these extra stresses.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 326 , 10 November 1996 , pp. 57 - 90
Copyright
© 1996 Cambridge University Press

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References

Brandt, A. 1977 Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166.Google Scholar
Cox, B. G. 1962 On driving a viscous fluid out of a tube. J. Fluid Mech. 14, 81.Google Scholar
Davis, H. T. 1988 A theory of tension at a miscible displacement front. In Numerical Simulation in Oil Recovery. IMA Volumes in Mathematics and Its Applications 11 (ed. M. Wheeler). Springer.
Fletcher, C. A. J. 1988 Computational Techniques for Fluid Dynamics, Vol. 1. Springer.
Horne, R. H. & Rodriguez, F. 1983 Dispersion in tracer flow in fractured geothermal systems. Geophys. Res. Lett. 10, 289.Google Scholar
Joseph, D. D. 1990 Fluid dynamics of two miscible liquids with diffusion and gradient stresses. Eur. J. Mech. B/Fluids 9, 565.Google Scholar
Joseph, D. D. & Hu, H. 1991 Interfacial tension between miscible liquids. Army High Performance Computing Research Centre, University of Minnesota, preprint 91-58.
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Part II. Lubricated Transport, Drops, and Miscible Liquids. Springer.
Korteweg, D. 1901 Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité. Arch. Neerl. Sci. Ex. Nat., (II), 6, 1.Google Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in a capillary tube. Part 1. Experiments. J. Fluid Mech. 326, 37 (referred to herein as Part 1).Google Scholar
Rakotomalala, N., Salin, D. & Watzky, P. 1996 Miscible displacement between two parallel plates: BGK Lattice gas simulations. Submitted to J. Fluid Mech.Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Statist. Comput. 6, 542.Google Scholar
Rogerson, A. M. & Meiburg, E. 1993a Shear stabilization of miscible displacement processes in porous media. Phys. Fluids A 5, 1344.Google Scholar
Rogerson, A. M. & Meiburg, E. 1993b Numerical simulation of miscible displacementprocesses in porous media flows under gravity. Phys. Fluids A 5, 2644.Google Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: Rectilinear flow. Phys. Fluids 29, 3549.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly througha tube. Proc. R. Soc. Lond. A 219, 186.Google Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161.Google Scholar
Yang, Z. & Yortsos, Y. C. 1996 Asymptotic solutions of miscible displacements ingeometries of large aspect ratio. Submitted to Phys. Fluids.Google Scholar