Residual mass and flow regimes for the immiscible liquid–liquid displacement in a plane channel Jackson F. Freitas a , Edson J. Soares a,⇑ , Roney L. Thompson b a LFTC, Department of Mechanical Engineering, Universidade Federal do Espirito Santo, Avenida Fernando Ferrari, 514, Goiabeiras 29075-910, ES, Brazil b LFTC, LMTA-PGMEC, Department of Mechanical Engineering, Universidade Federal Fluminense, Rua Passo da Patria 156, 24210-240 Niteroi, RJ, Brazil article info Article history: Received 10 May 2010 Received in revised form 21 February 2011 Accepted 3 March 2011 Available online 9 March 2011 Keywords: Plane channel Liquid–liquid displacement Finite element method Residual mass abstract The motion of two immiscible liquids in a plane channel is analyzed for the case in which the flow con- ditions and the interactions between the liquids and the solid surface maintain the displaced fluid attached to the wall. The Galerkin Finite Element Method is used to compute the velocity field and the configuration of the interface between the two fluids. We compare the residual mass fraction left on the wall with its two counterparts in capillary tubes, namely residual mass fraction and dimensionless layer thickness of the displaced fluid. The main result of this comparison was that although there is a qualitative similarity concerning the layer thickness between the two cases, the residual fraction of mass presented an important difference, showing that when the aspect ratio of the capillary passage is large there is an increase in the displacement efficiency. The thickness of the displaced liquid film attached to the channel walls is a function of the capillary number (Ca) and the viscosity ratio (N l ). A map of streamlines in the Cartesian space (Ca, N l ) with the different flow regimes of the problem is presented. We also showed that we can adapt the available analytical results obtained for gas-displacement in cap- illary tubes to the plane channel case, for low values of Ca. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction We investigate in the present work the liquid–liquid displace- ment in a plane channel. Examples of applications where this kind of problem is important are the oil recovery from porous media and the cementation of oil wells. The liquid film of the displaced material that remains attached to the walls plays a significant role in the efficiency of these operations. As a consequence of the small length scale, the capillary forces have a fundamental importance on the physics of the phenomenon. The complete 3-D liquid–liquid displacement analysis of oil recovery in porous media depends on the shape of the cross sec- tional area of the pore. When the aspect ratio of this area is around unity the tube is a reasonable approximation. However, when the aspect ratio of the cross sectional area of these internal passages is large, a plane 2-D geometry is preferred to represent the geomet- rical features of the pore. There are two different outputs of interest in the fluid–fluid dis- placement problem depicted in Fig. 1. One, is the fraction of mass of Phase 2 attached to the wall, m, calculated as m ¼ Area occupied by Phase 2 at Region IV Area occupied by Phase 2 at Region I : ð1Þ The other output is the dimensionless thickness of the same layer, h / , which can be defined as h  ¼ Layer thickness of Phase 2 attached to the wall Half of the distance between the plates : ð2Þ For a 2-D Cartesian case, a plane channel, it happens that these two entities lead to the same quantity. This coincidence does not occur in a tube, where m > h / . In fact one can find, in the literature of dis- placement in tubes, the output result emphasizing the first, e.g. (Taylor, 1961), or the second, e.g. (Westborg and Hassager, 1989), quantity. It should be mentioned that, depending on the problem considered, one result or the other is of particular interest. For example, in the oil recovery process, the fraction of residual mass is more adequate to measure a displacing efficiency, while in the cementing of a oil well or in a coating process, the layer thickness is the quantity to be controlled. Concerning the liquid–liquid displacement in a capillary space, it seems that the first investigation was made by Goldsmith and Mason (1963). They conducted an experimental analysis in a cap- illary tube with immiscible Newtonian liquids. Thus, besides the Capillary number, Ca  l V r (the ratio of viscous forces to the inter- face tension forces), they investigate the ratio of viscosities of the displaced fluid to the displacing one, N l  l 2 l 1 . In their experiments, 0301-9322/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmultiphaseflow.2011.03.003 ⇑ Corresponding author. Tel.: +55 27 40092162; fax: +55 40092851. E-mail address: edson@ct.ufes.br (E.J. Soares). International Journal of Multiphase Flow 37 (2011) 640–646 Contents lists available at ScienceDirect International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow