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 Péclet 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) 57] 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) 37], 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.