MISCIBLE DISPLACEMENTS IN CAPILLARY TUBES .2. NUMERICAL SIMULATIONS

Numerical simulations are presented which, in conjunction with the acc ompanying experimental investigation by Petitjeans & Maxworthy (1996), are intended to elucidate the miscible flow that is generated if a fl uid of given viscosity and density displaces a second fluid of differe nt such properties in a capillary tube or plane channel. The global fe atures of the how such as the fraction of the displaced fluid left beh ind on the tube walls, are largely controlled by dimensionless quantit ies in the form of a Peclet number Pe, an Atwood number At, and a grav ity parameter. However, further dimensionless parameters that arise fr om the dependence on the concentration of various physical properties, such as viscosity and the diffusion coefficient, result in significan t effects as well. The simulations identify two distinct Pe regimes, s eparated by a transitional region. For large values of Pe, typically a bove O(10(3)), a quasi-steady finger forms, which persists for a time of O(Pe) before it starts to decay, and Poiseuille flow and Taylor dis persion are approached asymptotically. Depending on the strength of th e gravitational forces, we observe a variety of topologically differen t 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 o f 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 d ecays from the start and asymptotically reaches Taylor dispersion afte r a time of O(Pe). An attempt was made to evaluate the importance of t he Korteweg stresses and the consequences of assuming a divergence-fre e velocity field. Scaling arguments indicate that these effects should be strongest when steep concentration fronts exist, i.e. at large val ues of Pe and At. However, when compared to the viscous stresses, Kort eweg stresses may be relatively more important at lower values of thes e parameters, and we cannot exclude the possibility that minor discrep ancies observed between simulations and experiments in these parameter regimes are partially due to these extra stresses.