Gravity-driven slug motion in capillary tubes

Abstract

The velocity of a liquid slug falling in a capillary tube is lower than predicted for Poiseuille flow due to presence of menisci, whose shapes are determined by the complex interplay of capillary, viscous, and gravitational forces. Due to the presence of menisci, a capillary pressure proportional to surface curvature acts on the slug and streamlines are bent close to the interface, resulting in enhanced viscous dissipation at the wedges. To determine the origin of drag-force increase relative to Poiseuille flow, we compute the force resultant acting on the slug by integrating Navier-Stokes equations over the liquid volume. Invoking relationships from differential geometry we demonstrate that the additional drag is due to viscous forces only and that no capillary drag of hydrodynamic origin exists ͑i.e., due to hydrodynamic deformation of the interface͒. Requiring that the force resultant is zero, we derive scaling laws for the steady velocity in the limit of small capillary numbers by estimating the leading order viscous dissipation in the different regions of the slug ͑i.e., the unperturbed Poiseuille-like bulk, the static menisci close to the tube axis and the dynamic regions close to the contact lines͒. Considering both partial and complete wetting, we find that the relationship between dimensionless velocity and weight is, in general, nonlinear. Whereas the relationship obtained for complete-wetting conditions is found in agreement with the experimental data of Bico and Quéré ͓J. Bico and D. Quéré, J. Colloid Interface Sci. 243, 262 ͑2001͔͒, the scaling law under partial-wetting conditions is validated by numerical simulations performed with the Volume of Fluid method. The simulated steady velocities agree with the behavior predicted by the theoretical scaling laws in presence and in absence of static contact angle hysteresis. The numerical simulations suggest that wedge-flow dissipation alone cannot account for the entire additional drag and that the non-Poiseuille dissipation in the static menisci ͑not considered in previous studies͒ has to be considered for large contact angles.

Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy

“…(1), (7), and (10)). The method has proven able to numerically simulate jet breakup and splashing [33,32,41] and to deal with the contact angle in a rather natural way, capturing all relevant mechanisms, which include hysteresis of the contact angle and film flow [23,26,41]. It consists of three main ingredients: an appropriate model of surface tension; a technique to advect the fluid-function that controls the numerical diffusion; and a Navier-Stokes solver.…”

“…(13). Note that this does not imply that the curvature of the interface remains constant if the interface is moving: the normal vector is fixed only on the faces adjacent to the solid whereas the velocity field can deform the interface in the other cells (see also [26]). The dynamic effects associated with the constant contact angle are therefore taken into account.…”

“…In the CSF-VOF method the interaction between the fluids and the solid is easily included as a boundary condition for the surface normal vector at the solid wall [24][25][26]41]: when Eq. (23) is used to compute the curvature in cells adjacent to the solid obstacle, Eq.…”

“…Interface tracking methods [21,22] are based on Lagrangian algorithms and are normally not well suited for flow with interfaces undergoing complex deformations (as in case of coalescence or break up) [23]; in contrast, interface capturing methods are based on Eulerian algorithms and are more appropriate for complex interface motion. Examples of such methods are Level-Set (LS) [31], Volume Of Fluid (VOF) [19,[23][24][25][26] and phase field methods [27].…”

Infiltration into Two-Layered Soil: The Green–Ampt and Averyanov Models Revisited

“…1 at z > f 0 ) soil is dry (or more accurately, the immobile moisture content is at an irreducible value), as discussed in Kacimov and Obnosov (2013), Lunati and Or (2009), Ali et al (2013), Assouline (2013) and Valiantzas (2010). In this sense, the GA model is similar to the Washburn-Lukas (WL) model on the level of capillary channels (see, e.g., Lunati and Or 2009;Hilpert and Glantz 2013) and both models-by solving a Cauchy problem for a nonlinear ODE-mathematically predict a finite (although, generally speaking, discontinuous) speed of front/meniscus propagation through a dry soil or empty capillary tube. As numerous experiments and field studies showed (see, e.g., Siemens et al 2013), the GA fronts propagating downward through a practically dry soil leave a "tail" of air-filled pore space at z > d in our Fig.…”