Low Capillary Number Regime Research Articles

Steady moving contact line of water over a no-slip substrate

The movement of the triple contact line plays a crucial role in many applications such as ink-jet printing, liquid coating and drainage (imbibition) in porous media. To design accurate computational tools for these applications, predictive models of the moving contact line are needed. However, the basic mechanisms responsible for movement of the triple contact line are not well understood but still debated. We investigate the movement of the contact line between water, vapour and a silica-like solid surface under steady conditions in low capillary number regime. We use molecular dynamics (MD) with an atomistic water model to simulate a nanoscopic drop between two moving plates. We include hydrogen bonding between the water molecules and the solid substrate, which leads to a sub-molecular slip length. We benchmark two continuum methods, the Cahn–Hilliard phase-field (PF) model and a volume-of-fluid (VOF) model, against MD results. We show that both continuum models reproduce the statistical measures obtained from MD reasonably well, with a trade-off in accuracy. We demonstrate the importance of the phase-field mobility parameter and the local slip length in accurately modelling the moving contact line.

Direct numerical simulation of trapped-phase recirculation at low capillary number

Understanding multiphase flow in porous media, especially how velocity is distributed at the pore-scale, has been the aim of several studies. However, these studies address the recirculation behavior inside the trapped phase experimentally without any comprehensive numerical study of the impact of different governing mechanisms related to the fluid configurations and properties, including drag force and capillary number analysis at low capillary number regime. In this study, we analyzed the recirculation phenomenon inside the trapped phase for various displacement mechanisms, fluid configurations, and dynamic properties. To simulate the pore-scale displacement at low capillary number, we used a filtered surface-force formulation of volume of fluid method, which was implemented in a separately available solver for OpenFoam package. The results showed that within the ranges of capillary number of invading phase analyzed in this study (in the order of 1 × 10−7 to 1 × 10−2), the recirculation phenomenon exists in trapped phases. During the imbibition mechanism, two stagnant regions are created adjacent to the fluid-fluid interface inside the invading fluid. Drag-force analysis on fluid-fluid interfaces shows that during imbibition the maximum force is exerted near the center of the interface, whereas during drainage more force is applied on two elongated interface tails on a solid surface. The centroids are elongated parallel to the interface during drainage and perpendicularly during imbibition, which is in concordance with drag-force distribution along with the interface. The existence of a solid surface near the fluid-fluid interface affects the recirculation process in a way that one or more centroids can be created depending on displacement mechanisms. When the ratio of trapped-phase radius to cavity depth is lower, two simultaneous recirculation zones are formed inside the invading and trapped phases While the changes in viscosity ratio and interfacial tension shifted the centroid location inside the trapped zone, the center of rotation seems to be independent of injection velocity. The average velocity of trapped phase is individually a logarithmic function of the surface tension and fluids viscosity ratio. The stationariness of centroid results in a linear relationship between the average velocity inside the trapped zone and injection velocity. For all ranges of viscosity ratios, a linear relationship between the capillary number of invading and trapped phase is obtained. The findings of this study lead to a better understanding of trapping and mobilization mechanisms in microchannels where various forces are acting on fluid-fluid interfaces.

Stable and Efficient Time Integration of a Dynamic Pore Network Model for Two-Phase Flow in Porous Media

We study three different time integration methods for a dynamic pore network model for immiscible two-phase flow in porous media. Considered are two explicit methods, the forward Euler and midpoint methods, and a new semi-implicit method developed herein. The explicit methods are known to suffer from numerical instabilities at low capillary numbers. A new time-step criterion is suggested in order to stabilize them. Numerical experiments, including a Haines jump case, are performed and these demonstrate that stabilization is achieved. Further, the results from the Haines jump case are consistent with experimental observations. A performance analysis reveals that the semi-implicit method is able to perform stable simulations with much less computational effort than the explicit methods at low capillary numbers. The relative benefit of using the semi-implicit method increases with decreasing capillary number $\mathrm{Ca}$, and at $\mathrm{Ca} \sim 10^{-8}$ the computational time needed is reduced by three orders of magnitude. This increased efficiency enables simulations in the low-capillary number regime that are unfeasible with explicit methods and the range of capillary numbers for which the pore network model is a tractable modeling alternative is thus greatly extended by the semi-implicit method.

A Flexible Coupled Level Set and Volume of Fluid (flexCLV) method to simulate microscale two-phase flow in non-uniform and unstructured meshes

An innovative Flexible Coupled Level Set (LS) and Volume of Fluid (VOF) algorithm (flexCLV) to simulate two-phase flows at the microscale on unstructured and non-uniform meshes is proposed. The method combines the advantages of the VOF method in terms of mass conservation and the LS method in terms of accuracy of the surface tension implementation and can handle both 2D and 3D domains discretized by either structured hexaedra or unstructured tetrahedral grids with high aspect ratio elements, thus guaranteeing flexibility and robustness. The method is implemented within the VOF-based OpenFOAM’s solver interFoam, which is retained as the base algorithm for the interface advection, while the surface tension force is calculated by using the level set function reconstructed from the VOF’s fraction. The method is first validated in static flow conditions by simulating a circular bubble at equilibrium and then in dynamic flow conditions by studying a freely bubble rising in both 2D and 3D domains discretized by both structured and unstructured meshes. The proposed flexCLV algorithm is then used to simulate the dynamics of confined bubbles in circular microchannels in the low capillary number regime. 2D and 3D mesh grids with high aspect ratio elements are utilized to discretized the liquid film at the tube’s walls. The numerical results are compared with the available literature and simulations performed with the original interFoam solver in terms of bubble shape and velocity, thickness of the liquid film and amplitude of the bubble tail oscillations. Results compare very well with the experimental measurements and demonstrate the superior accuracy of the coupled flexCLV method with respect to the original VOF method when surface tension and accurate interface representation play a fundamental role. Importantly, the present study also provides a precious insight on the time-dependent patterns appearing on the bubble surface in the visco-inertial regime, which could be here investigated in detail.

Influence of disjoining pressure on the dynamics of steadily moving long bubbles inside narrow cylindrical capillaries.

We study the influence of disjoining pressure for moving long bubbles inside cylindrical capillaries. Towards that end, consistent thin-film equations, for the annular region separating the bubble from the channel surface, are presented with specific emphasis on three different attributes: (a) the van der Waals interaction, formalized by the classical Lifshitz form of disjoining pressure; (b) the nonuniformity in film thickness, accommodated by the necessary corrections in the disjoining pressure; and (c) the electrostatic component of disjoining pressure, reminiscent of the electrostatic interactions in the presence of surface charges. The present thin-film analysis appositely uncovers the existence and the breakdown of the two-thirds power law for minimum film thickness behavior. This is attributed to the alteration in the characteristic length scales governing the underlying physics, as quantitatively established by our consistent scaling analysis. In the breakdown regimes, the characteristic length scales are found to be composed of the suitable combinations of the capillary number and the physics driven intrinsic length scales. The characteristics of the breakdown regime reported by us appear to match excellently with reported experimental data in the low capillary number regime. Towards unveiling the possible implications of slope and curvature dependence of disjoining pressure, our analysis reveals that the consequent correction term endorses an order two-thirds power of the capillary number contribution without alerting the governing length scales. The subsequent asymptotic analysis reveals that this correction may be neglected to the leading order approximation. Finally, we consider the electrostatic component of the disjoining pressure which may be realized in the presence of surface charges. Our analysis reveals that the significance of the electrostatic interaction is realized over a very small capillary number regime, leading towards the departure from the two-thirds power law type behavior. Reasonably good agreement is obtained with reported experimental data over this regime.

The effect of contact line dynamics and drop formation on measured values of receding contact angle at very low capillary numbers

Owing to contact angle hysteresis, the contact angle that a sessile drop adopts on a surface can be quite different depending on the process of drop formation. This is particularly noticeable for receding contact angle because it is most susceptible to the details of the measuring method. We performed low-rate dynamic contact angle experiments with steadily moving contact lines and millimeter-size drops in the very low-capillary number regime. We employed a quadratic volumetric flow rate (V∝t3) to achieve the steady motion of contact lines such as happens in the Wilhelmy balance. The values of receding contact angle provided with the quadratic flow rate were stable and time-independent over a larger area of the polymer surfaces studied. Next, we monitored receding sessile drops with equal initial volume but different static contact angle on the same surface. This procedure allowed us to scan the contact angle hysteresis range up to the minimum observable value of contact angle. We probed with distilled water the different response of surfaces of polymer (PMMA, PC and PTFE) and metal oxide (titanium). We found two behaviors according to the substrate employed: a constant value of receding contact angle regardless of the static contact angle and a decrease of receding contact angle as the static contact angle decreased.

Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.

Modeling and simulation of wetting and spreading

We present a mathematical model for the slow three-dimensional motion of a liquid coating on a substrate with wetting and de-wetting edges. The equilibrium contact angle is considered to be a material property of the liquid-substrate system. Substrate chemical heterogeneity, or physical roughness, may also be an important determinant of edge motion. Conversely, dynamic contact angle information is not required by the model; it is predicted as part of the solution. Calculated results are compared with experimental observation with good agreement. Many industrial applications involve liquid coating and wetting considerations are usually quite important. References Deryaguin, B. V., Theory of the capillary condensation and other capillary phenomena taking into account the disjoining effect of long-chain molecular liquid films, Zhur. Fiz. Khim. 14 , 137 (1940) [In Russian]. Eres, M. H., Schwartz, L. W., and Roy, R. V., Fingering phenomena for driven coating films, Physics of Fluids 12 , 1278--1295, 2000. Frumkin, A. N., On the phenomena of wetting and sticking of bubbles, Zhur. Fiz. Khim. 12 , 337 (1938) [In Russian]. Haze, R. and Lammers, J. H., ``Liquid height measurement through light absorption,'' Philips Research Laboratory, Report No. UR 819/99, Eindhoven, Netherlands, 1999. Huh, C. and Scriven, L. E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci. 35 , 85, 1971 . Levich, V. G., Physiochemical Hydrodynamics , 1962, Prentice--Hall, Englewood Cliffs. Moriarty, J. A. and Schwartz, L. W., Effective slip in numerical calculations of moving-contact-line problems, J. Engineering Math. 26 81--86, 1992. Peaceman, D. W. and Rachford, H. H., The numerical solution of parabolic and elliptic differential equations, SIAM J. 3 , 28, 1955. Podgorski, T., Flesselles, J.-M., and Limat, L., Corners, cusps, and pearls in running drops, Phys. Rev. Lett. 87 , 1--4, 2001. Schwartz, L. W., Roy, R. V., Eley, R. R. and Petrash, S. Dewetting Patterns in a Drying Liquid Film, J. Colloid Interface Science 234 , 363--374, 2001. Atherton, R. W. and Homsy, G. M., On the derivation of evolution equations for interfacial waves, Chem. Eng. Comm. 2 , 57, 1976. Schwartz, L. W. and Roy, R. V., Theoretical and numerical results for spin coating of viscous liquids, Physics of Fluids 16 , 569--584, 2004. Schwartz, L. W., Roux, D. and Cooper-White, J. J., On the shapes of droplets that are sliding on a vertical wall, Physica D 209 , 236--244, 2005. Seeberg, J. E. and Berg, J. C.. Dynamic wetting in the low capillary number regime. Chem. Eng. Sci. 47 4455--4464, 1992. Sherman, F. S., Viscous Flow , 1990, McGraw--Hill, New York. Benney, D. J., Long waves on liquid films, J. Math. and Phys. 45 , 150, 1966. Blake, T. D., ``Dynamic contact angles and wetting kinetics.'' In: J. C. Berg (ed.) Wettability , Marcel Dekker, New York, 251--309, 1993. Cazabat, A. M, Heslot, F., Carles, P. and Troian, S. M., Hydrodynamic fingering instability of driven wetting films, Adv. Colloid Interf. Sci. 39 , 61--75, 1992.

MIXING DRIVEN BY RAYLEIGH–TAYLOR INSTABILITY IN THE MESOSCALE MODELED WITH DISSIPATIVE PARTICLE DYNAMICS

In the mesoscale, mixing dynamics involving immiscible fluids is truly an outstanding problem in many fields, ranging from biology to geology, because of the multiscale nature which causes severe difficulties for conventional methods using partial differential equations. The existing macroscopic models incorporating the two microstructural mechanisms of breakup and coalescence do not have the necessary physical ingredients for feedback dynamics. We demonstrate here that the approach of dissipative particle dynamics (DPD) does include the feedback mechanism and thus can yield much deeper insight into the nature of immiscible mixing. We have employed the DPD method for simulating numerically the highly nonlinear aspects of the Rayleigh–Taylor (R–T) instability developed over the mesoscale for viscous, immiscible, elastically compressible fluids. In the initial stages, we encounter the spontaneous, vertical oscillations in the incipient period of mixing. The long-term dynamics are controlled by the initial breakup and the subsequent coalescence of the microstructures and the termination of the chaotic stage in the development of the R–T instability. In the regime with high capillary number, breakup plays a dominant role in the mixing whereas in the low capillary number regime, the flow decelerates and coalescence takes over and causes a more rapid turnover. The speed of mixing and the turnover depend on the immiscibility factor which results from microscopic interactions between the binary fluid components. Both the speed of mixing and the overturn dynamics depend not only on the mascrocopic fluid properties but also on the breakup and coalescent patterns, and most importantly on the nonlinear interactions between the microstructural dynamics and the large-scale flow.

Viscous drag of a solid sphere straddling a spherical or flat surface

The aim of this paper is to compute the friction felt by a solid particle, of radius a, located across a flat or spherical interface of radius R, and moving parallel to the interface. This spherical interface can be a molecular film around an emulsion or aerosol droplet, the membrane of a vesicle or the soap film of a foam bubble. For simplicity, the acronym VDB is used to refer to either vesicle, drop, or bubble. The theory is designed as a tool to interpret surface viscosimetry experiments involving spherical probes attached to films or model membranes, taking care of the finite-size effects when the film encompasses a finite fluid volume. The surface of the VDB is a two-dimensional fluid, characterized by dilational (ηsdil) and shear (ηssh) surface viscosities. The particle intercepts a circular disc in the interface, whose size depends on the particle penetration inside the VDB. The three-dimensional fluids inside and outside the interface may be different. The analysis holds in the low Reynolds number and low capillary number regime. A toroidal (x1,x2,φ) coordinate system is introduced, which considerably simplifies the geometry of the problem. Then the hydrodynamic equations and boundary conditions are written in x1,x2,φ. The solution is searched for the first-order Fourier component of the velocity field in the radial angle φ. Reformulating the equations in “two-vorticity-one-velocity” representation, one basically ends up with a set of equations in x1,x2 only. This set is numerically solved by means of the Alternating-Direction-Implicit method. Numerical results show that the particle friction is influenced both by the viscosity and by the finiteness of the VDB volume. Finite-size effects have two origins: a recirculation effect when a/R is not very small, and an overall rotation of the VDB-particle complex when ηs is very large. In principle, the theory allows for a quantitative determination of ηs whatever a/R, including the limit a/R=0 (flat interface).

Dynamic wetting in the low capillary number regime

An understanding of dynamic wetting is necessary when considering processes in which one fluid is forced to displace another fluid from a solid substrate. This work examines the forced wetting behavior exhibited by a series of solid-liquid systems in the low capillary number regime. In particular, the effects of specific chemical (i.e. acid-base) interactions between the liquid and the solid and of solid surface roughness on dynamic wetting are investigated. Measurement of the dynamic contact angle as a function of the interline velocity is performed using the technique of microtensiometry. The data are compared with empirical correlations in order to test the hypothesis that a universal correlation may be used to describe wetting behavior. The results suggest that two correlation having the same functional form but different constants are necessary to adequately describe wetting behavior above and below a capillary number of the order of 10 −3. No significant differences are observed between the wetting behavior of systems with only dispersive interactions between the liquid and solid and systems with acid-base interactions, implying that the static contact angle adequately accounts for the effect of acid-base interactions. Surface roughness produces noticeable stick-slip effects in the low capillary number regime when the static contact angle is larger than approximately 15°. While these effects produce a large degree of scatter in the dynamic wetting measurements, they do not alter the constants from those found in the case of smooth interline motion.