J. Non-Newtonian Fluid Mech. 142 (2007) 162–182 Transient displacement of Newtonian and viscoplastic liquids by air in complex tubes Yannis Dimakopoulos, John Tsamopoulos ∗ Laboratory of Computational Fluid Dynamics, Department of Chemical Engineering, University of Patras, Patras 26500, Greece Received 13 March 2006; received in revised form 14 July 2006; accepted 4 August 2006 Abstract We study the transient displacement of Newtonian and viscoplastic liquids by highly pressurized air in cylindrical tubes of finite length with an expansion followed by a contraction in their cross section. Papanastasiou’s formula is employed to regularize the discontinuous Bingham model. For both fluid models considered, the distribution of the remaining film on the inner tube wall is non-uniform and only partly follows the tube geometry: it is thinner in the expanding section of the tube, thicker in the contracting one, and as thin as observed in relevant experiments for straight tube segments, if these are long enough. The effect of changes in the diameter of the narrow introductory tube on the width of the remaining film depends on liquid inertia and yield stress. It is confined near the expansion corner for small Reynolds numbers, but causes extended distortions on the free surface for larger ones due to the development of ‘lip’ or ‘corner’ vortices on the expanding side of the tube. The increased viscosity of viscoplastic materials, especially where velocity gradients are smaller, reduces the extent of flow and interface fluctuations. Unyielded regions appear along the film remaining on the tube wall, the core area of the main and exit tubes and around the concave corners of the tube. These cause the thinning of the remaining material inside the entrance tube, around the centerline of the main section and the flattening of the bubble front. The size of the solid-like areas increases at higher Reynolds numbers. The tip velocity for both viscous and viscoplastic materials increases along the introductory tube until it attains a maximum, then decreases down to a plateau while it moves inside the main tube, and finally increases again as it approaches the contraction corner. Its values decrease as the Bingham number increases. © 2006 Elsevier B.V. All rights reserved. Keywords: Viscoplastic fluids; Free boundary problems; Elliptic grid generation; Liquid displacement by air 1. Introduction Due to its numerous technological and scientific applications, the displacement of a liquid by gas has been attracting the inter- est of many researchers for a long time. The most important early works were performed by Taylor [1] and Cox [2,3] who mea- sured the thickness of the remaining liquid film as a function of the capillary number inside very long tubes initially com- pletely filled with a liquid in order to investigate the efficiency of a tube clearance by air. Since then, new research efforts have been undertaken due to important applications in enhanced oil recovery [4], multiphase monolith reactors [5], enhanced micro- filtration [6], operation of the airways in the lungs [7], and, finally, the gas-assisted injection molding (GAIM) [8]. The lat- ∗ Corresponding author. Fax: +30 2610 996 178. E-mail addresses: dimako@chemeng.upatras.gr (Y. Dimakopoulos), tsamo@chemeng.upatras.gr (J. Tsamopoulos). ter is a modification of the conventional injection molding [9], where the direct application of mechanical power is replaced by highly pressurized air or inert gas (usually N 2 ), in order to reduce the energy consumption and improve the quality of the produced plastic articles. GAIM is different from the other applications in at least three aspects: it is an inherently transient operation with a much higher applied gas pressure, resulting in larger velocities and effec- tive capillary number, Ca > 10, and the liquid being displaced is non-Newtonian. So we will try not to expand on the exist- ing literature, when these conditions are not met. Motivated by the need to simulate it accurately and efficiently, Dimakopou- los and Tsamopoulos [10,11] developed a quasi-elliptic mesh- generating scheme for the transient displacement of viscous fluids in straight and suddenly constricted tubes of finite length. Their results gave a quite complete picture of the evolution of the bubble surface and the flow field ahead and behind its tip. In straight tubes, they verified Cox’s experimental observations in the high Ca and low Reynolds number, Re, regime, whereas they 0377-0257/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2006.08.002