Multiscale level-set method for accurate modeling of immiscible two-phase flow with deposited thin films on solid surfaces

Abstract

We developed a multiscale sharp-interface level-set method for immiscible two-phase flow with a pre-existing thin film on solid surfaces. The lubrication approximation theory is used to model the thin-film equation efficiently. The incompressible Navier–Stokes, level-set, and thin-film evolution equations are coupled sequentially to capture the dynamics occurring at multiple length scales. The Hamilton–Jacobi level-set reinitialization is employed to construct the signed-distance function, which takes into account the deposited thin-film on the solid surface. The proposed multiscale method is validated and shown to match the augmented Young–Laplace equation for a static meniscus in a capillary tube. Viscous bending of the advancing interface over the precursor film is captured by the proposed level-set method and agrees with the Cox–Voinov theory. The advancing bubble surrounded by a wetting film inside a capillary tube is considered, and the predicted film thickness compares well with both theory and experiments. We also demonstrate that the multiscale level-set approach can model immiscible two-phase flow with a capillary number as low as ${10}^{-6}$.

Introduction

Wetting phenomena are important in determining the behavior of a wide range of physical processes. Examples include two-phase flow in porous media (enhanced oil recovery [41] and CO₂ sequestration [9]), microfluidic devices [71], coating flows [37], and inkjet printing [40]. There are several challenges in modeling multiphase flow with wetting dynamics from both the physical and numerical standpoints. The physics of problems involving wetting dynamics is fundamentally multiscale and can include effects that originate at the molecular scale [59]. With the conventional hydrodynamics assumptions of no-slip, Newtonian fluids, and a rigid solid, there will be an unbounded stress singularity when the fluid interface comes in contact with the solid surface (also known as the Moving Contact Line (MCL) problem) [30].

To circumvent the stress singularity associated with MCLs, several mechanisms that have plausible physical basis have been proposed including interface diffusion [58], and evaporation/condensation processes [70]. The most common approaches include slip-based models [20], and film models based on intermolecular surface forces [13]. The parameters associated with the film and slip MCL models have a small length-scale, and that places severe constraints on our ability to resolve the flow physics at scales of practical interest, such as natural porous media. Here, we study MCLs whereby the stress singularity is regularized by assuming that a pre-existing thin-film covers the solid surface. The film may be a precursor film due to Van der Waals forces (between 1 and 100 nanometer), or a thick film (micrometer). Thin films that cover the solid surfaces have been observed experimentally [26], [36], and they can successfully remove the MCL singularity. However, the details of how the contact line moves and the underlying physical mechanisms that govern the dynamics are topics of ongoing research [3].

Theoretical hydrodynamic models based on the thin-film assumption for the MCL have been developed using matched asymptotic expansions to relate the observed macroscopic contact-angle to the inner (nanoscale) region. In the thin-film model, the meniscus curvature, or the tube radius serves as the macroscopic scale, while the inner scale represents the thickness of the ‘thin’ film. For a precursor film due to Van der Waals forces, Voinov [69] derived the asymptotic solution for an advancing fluid interface as shown in Fig. 1(a). In the Voinov model, the macroscopic contact-angle deviates from the microscopic contact-angle of zero (complete wetting) due to viscous bending at an intermediate length scale. This intermediate scale lies between the macroscopic and the inner regions. De Gennes et al. [23] generalized the asymptotic solution to a finite equilibrium microscopic contact-angle (partial wetting) in the presence of a precursor film. The asymptotic solution of the macroscopic contact-angle in the precursor-film-based model is similar to the well known slip-based Cox–Voinov model [68], [11]. The primary difference in the asymptotic solution between the two models is the length scale of the inner region [21].

Although slip and thin-film models based on the matched asymptotic method have shown good agreement with experimental observations [47], [29], they do not fully capture wetting transitions from a finite contact angle to a thin-film on the solid surface. The thin-film regime is important in corners, rough surfaces, and receding interfaces (i.e., drainage), whereby the nonwetting fluid displaces the wetting fluid (Fig. 1(b)). Thin films are thought to play a critical role in dictating the spatial distributions of the fluid phases in natural porous media [10]. The snap-off phenomena associated with immiscible two-phase flow in porous media are closely tied to thin-film dynamics, whereby the nonwetting phase gets fully surrounded (trapped) by the wetting fluid in the pores [41]. In addition to film flow, the stability of thin films plays a key role in the dynamics of menisci. Intermolecular forces, such as Van der Waals and electrostatic interactions, can cause thin films to collapse, or break-up into drops. For example, wetting film rupture allows for direct contact between the solid surface and the nonwetting fluid (oil or CO₂). As a result, dissolved mineral in the nonwetting phase, such as asphaltene in the oil, adsorbs to the rock surface causing wettability alteration [39], [57]. The thickness of wetting thin films can be on the order of a nanometer, which again underscores the multiscale nature of MCLs. Thin-film dynamic models, including those that account for intermolecular forces, have been used extensively to predict film stability [12]; however, these models cannot be used to simulate general multiphase problems, such as flow in multiple pores, or channels, with complex geometry. The suitable continuum model is the set of two-phase Navier–Stokes equations, which is a free-boundary problem that requires an interface tracking method such as front tracking [67], diffuse interface [32], [17], volume-of-fluid (VOF) [28], and level-set [63]. Numerical methods for the Navier–Stokes with MCLs are reviewed by Sui et al. [61].

The Van der Waals intermolecular force has been considered to model collision of two droplets [34], and as a precursor film model for MCL problems using VOF method [45]. The approach in [45] has been improved to enable the modeling of complex dynamics, such as thin-film rupture [46]. Although VOF is attractive because of its mass conservation property, curvature estimation using VOF can be tedious. This is due to the fact that the VOF is a discontinuous function. Inaccurate interface curvature computation is problematic for two-phase flows, especially at low capillary numbers. In the small capillary-number regime, spurious currents due to errors in the numerical representation of the surface tension force can degrade the computed solutions severely. The capillary number, defined as $Ca=\mu u/\sigma$ where μ, σ, and u are viscosity, surface tension, and characteristic local velocity, respectively, can reach ${10}^{-6}$, or even lower, for microfluidic applications [71] and multiphase flow in natural porous media [41]. Also, the accurate representation of fluid interfaces is important in problems involving interfacial oscillations and jumps, which have been shown to play a significant role in how interfaces move in confined domains [55], [50]. Improvement in the geometric representation of interfaces in VOF is an active area of research [52].

In this work, a level-set approach is used to track the interface that separates the two immiscible fluids. We assume that a persistent film covers the solid surface, which has a strong physical basis for the problems considered. The thin-film is captured efficiently using a subgrid model described by the lubrication approximation [51], [12]. The recent work in [46], [45] uses adaptive mesh refinement (AMR) with the inclusion of Van der Waals forces to model the precursor film. However, the small grid cells clustered near the thin-film impose a severe time-step restriction, especially for low capillary-number flows, whereby the capillary force determines the size of the allowable time-step. For numerical stability, the time-step restriction due to the surface tension force has to be satisfied, even if the capillary force is implicitly treated [14].

The new multiscale approach developed in this work couples the Navier–Stokes and level-set equations with the lubrication approximation. The Navier–Stokes and level-set equations are solved on a grid, while the lubrication approximation is applied whenever the height of the thin-film dips below the grid resolution. Coupling the thin-film model and the Navier–Stokes equations for a drop colliding a solid surface has been accomplished previously using a front tracking method [66]. The thin-film model used in [66] is a simplified model that does not take account for surface tension, or intermolecular forces. In the proposed multiscale method based on the level-set approach, both the surface tension and intermolecular forces are considered. The proposed method is based on a sharp treatment of the surface tension force using the ghost fluid method [42]. The level-set reinitialization to a signed distance function is performed using high-order subgrid fix approach [18]. Although the numerical code is developed for 2-D and axisymmetric problems, the approach can be easily extended to 3-D.

The paper is organized as follows. The level-set, Navier–Stokes, and lubrication approximation are summarized in section 2. In section 3, the discretization scheme and the numerical method for the coupling of the Navier–Stokes, the subgrid thin-film, and the level-set reinitialization are described. Section 4 includes validation examples of the proposed method based on comparisons with both theory and experiments for capillary tube problems. The simulation examples include the augmented Young–Laplace equation, moving interfaces, and a moving bubble in a capillary tube. The conclusion and future work are stated in section 5.