A numerical study on the stability of the steady displacement of a liquid plug along a small conduit

DIEGO M. CAMPANA; SEBASTIÁN UBAL; MARÍA D. GIAVEDONI; FERNANDO A. SAITA

Venecia, Italia

Congreso; 8th. World Congress on Computational Mechanics / 5th. European Congress on Computational Methods in Applied Sciences and Engineering (WCCM8/ECCOMAS 2008); 2008

International Association for Computational Mechanics (IACM) / European Community on Computational Methods in Applied Sciences (ECCOMAS)

Liquid plugs are commonly encountered in a large number of technological applicationssuch as oil recovery and micro-channel reactors; also, they may form in the respiratorytree either naturally from an instability of the liquid film lining the walls of the smallestconduits during the expiration process in certain pathological conditions or from theinstillation of a liquid for therapeutic purposes.A prototype of this problem is the motion of a certain volume of liquid inside a capillarytube coated by a film of the same fluid. When the propagation is steady, the thicknessof this film (the precursor film) must be equal to the thickness of the film left by thebubble travelling behind the plug (the deposited or trailing film). This steady motionmight be the result of drawing out the front bubble with constant velocity or, morecommonly, of applying a constant pressure drop between the front and the rear gasphases.When the liquid plug is large, the gas phases can be regarded as semi-infinite bubblestravelling alone and the flow problem can be split into two smaller ones (one for theleading and the other for the trailing bubble); these problems are linked by the filmthickness which is solely determined by the leading bubble.The steady motion of a semi-infinite bubble in a capillary tube or between two closelyspaced parallel plates initially filled with a liquid, has been extensively studiedanalytically, numerically, and experimentally since the pioneering works by Taylor[1]and Bretherton.[2]When the distance between the bubble tips is small, the propagation of the plug isaffected by the interaction of the gas phases, and the velocity and pressure fields mustbe simultaneously computed in the whole domain (the central core region and the rearand trailing menisci). The works on this subject are considerably less numerous andmany of them were carried out at the group leaded by Professor Grotberg, mainlymotivated by the transport of liquid plugs in the pulmonary airways. These studiestheoretically or numerically investigated the effects of propagation speed[3],surfactants[4,5] and gravity[5] on the motion and splitting of a liquid plug.An important point of one of these works[3] concerns the stability of the steady statescomputed. In fact, the authors conjecture that the lack of convergence of their numericalalgorithm within certain range of the parameters might be due to the non existence ofstable steady states; i.e., if the steady state solution is perturbed, the distance betweenthe menisci would either continuously increase or decrease until the collapse of theplug.The aim of this work is to analyze the stability of the steady state displacement of aliquid plug. We study two particular situations: (a) the leading gas phase is forced tomove at a constant speed or, (b) a constant pressure difference between the bubblesdrives the motion of the plug.In order to conduct the study, the numerical solution of the Navier-Stokes equations isrequired. We employed an algorithm based on the Galerkin/finite element methodcombined with the parameterization of the free surface by means of spines for thespatial discretization of the governing equations and their boundary conditions. A finitedifference scheme and an automatic step-size control are used to march on time.[6]Graphs of the steady state dimensionless film thickness (H), as a function of thedimensionless plug length (LP) within a large range of the parameters, are built. Weshow by simple physical arguments that the stability of the system may be inferred fromthe shape of these curves; these results were verified by performing transientsimulations of the system. For case (a), we found a critical Reynolds number beyondwhich the steady state solutions are unstable. In case (b), the steady state solutions areunstable except in bounded regions in the H?LP plane, located for values of LP < 1approximately.REFERENCES[1] Taylor, G. I. ?Deposition of viscous fluid on the wall of a tube?. J. Fluid Mech.,Vol. 10, pp. 161-165, (1961).[2] Bretherton, F. P. ?The motion of long bubbles in tubes?. J. Fluid Mech., Vol. 10,pp. 166-188, (1961).[3] Fujioka, H., Grotberg, J. B. ?Steady propagation of a liquid plug in a twodimensionalchannel?. ASME J. Biomech. Eng, Vol. 126, pp. 567-577, (2004)[4] Fujioka, H., Grotberg, J. B. ?The steady propagation of a surfactant-laden liquidplug in a two-dimensional channel?. Phys. Fluids, Vol. 17, p. 082102, (2005).[5] Zheng, Y., Fujioka, H., Grotberg, J. B. ?Effects of gravity, inertia, and surfactanton steady plug propagation in a two-dimensional channel?. Phys. Fluids, Vol. 19,p. 082107, (2007).[6] Ubal, S.; Giavedoni, M. D.; Saita, F. A. ?A numerical analysis of the influence ofthe liquid depth on two dimensional Faraday waves?. Phys. Fluids, Vol. 15, pp.3099-3113, (2003).