Variable density and viscosity, miscible displacements in capillary tubes

The displacement of a more viscous fluid by a miscible, less viscous one of lower density in a horizontal capillary tube is studied by means of Stokes flow simulations. Both axisymmetric and three-dimensional simulations are conducted at Peclet numbers up to 104, in order to resolve discrepancies between earlier simulations by [C.Y Chen, E. Meiburg, Miscible displacements in capillary tubes. Part 2.. Numerical simulations, J. Fluid Mech. 326 (1996) 571 and corresponding experiments of [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] and [J. Kuang, T. Maxworthy, P. Petitjeans, Miscible displacements between silicone oils in capillary tubes, Eur. J. Mech. B Fluids 22 (2003) 271-277]. An initial set of simulations addresses the influence of different viscosity-concentration relations on the quasisteady finger tip velocity. The results indicate that steeper relations generally result in a higher tip velocity. However, the effect is too small to explain the above discrepancies. Further simulations show that a concentration-dependent diffusion coefficient results in a slight reduction of the tip velocity at moderate Pe, but again the effect is too small to fully account for the observed differences. Three-dimensional simulations that include gravitational forces yield a much more significant effect. Consistent with the experiments of [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 371, at moderate Pe the tip slows down as the gravity parameter increases, an effect that becomes more pronounced as Pe decreases. However, the three-dimensional simulations do not produce the longitudinal splitting phenomenon observed by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37]. In order to check for the existence of gravitational instabilities that might cause such a splitting, additional two-dimensional simulations are conducted in cross-sections of the tube. A comparison of these two-dimensional results with corresponding three-dimensional simulations demonstrates that for a wide range of parameters the evolution of the trailing finger sections is governed by a two-dimensional balance between gravitational and viscous forces. However, a gravitational instability along the lines suggested by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] was not observed. On the other hand, for some parameter combinations the evolution of a 'dimple' is observed on the lower side of the finger, and close to its tip. This dimple may signal the evolution of a splitting phenomenon after long times, which are beyond the reach of the current simulations. Taken together, the two- and three-dimensional simulations suggest that the splitting phenomenon observed by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] likely is caused by the gravity-induced modification of the flow around the tip of the finger, rather than by a gravitational instability per se. (c) 2007 Elsevier Masson SAS. All rights reserved.