Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface

When boundary conditions arising from the usual hydrodynamic assumptions are applied, analyses of dynamic wetting processes lead to a well-known nonintegrable stress singularity at the dynamic contact line, necessitating new ways to model this problem. In this paper, numerical simulations for a set of representative problems are used to explore the possibility of providing material boundary conditions for predictive models of inertialess moving contact line processes. The calculations reveal that up to Capillary number Ca=0.15, the velocity along an arc of radius 10Li (Li is an inner, microscopic length scale) from the dynamic contact line is independent of the macroscopic length scale a for a>103Li, and compares well to the leading order analytical “modulated-wedge” flow field [R. G. Cox, J. Fluid Mech. 168, 169 (1986)] for Capillary number Ca<0.1. Systematic deviations between the numerical and analytical velocity field occur for 0.1<Ca<0.15, caused by the inadequacy of the leading order analytical solution over this range of Ca. Meniscus shapes produced from calculations in a truncated domain, where the modulated-wedge velocity field [R. G. Cox, J. Fluid Mech. 168, 169 (1986)] is used as a boundary condition along an arc of radius R=10−2a from the dynamic contact line, agree well with those using two inner slip models for Ca<0.1, with a breakdown at higher Ca. Computations in a cylindrical geometry reveal the role of azimuthal curvature effects on velocity profiles in the vicinity of dynamic contact lines. These calculations show that over an appropriate range of Ca, the velocity field and the meniscus slope in a geometry-independent region can potentially serve as material boundary conditions for models of processes containing dynamic contact lines.

Simulation of moving contact lines in two-phase polymeric fluids

The interaction of pure capillary waves with boundaries that constrain the contact line are of interest for problems involving liquids contained by minimal solid contact for applications in low gravity and at small scales in normal gravity. Time-harmonic capillary waves on a liquid cylinder axially incident on and scattered by an infinitesimal concentric barrier are considered theoretically in the inviscid limit. The barrier is taken to be infinitesimally small in the sense that its immersed depth is of the order of the amplitude of contact-line motion. Edge conditions on the barrier that are investigated include a pinned contact line and a moving contact line by an effective-slip model, assuming that contact-line velocity is proportional to the deviation of the contact angle from equilibrium multiplied by a slip coefficient. The incident waves are taken to be those with wavelengths short enough to be stable on the liquid cylinder. Scattering and dissipation by the contact line are determined as a function of wavenumber and slip coefficient. Zero transmission is approached in the long-wave limit. The short-wave limit agrees with established results for the scattering of planar gravity–capillary waves on a deep liquid by a surface-piercing vertical barrier in the limit of zero barrier depth and zero gravity. We find that contact-line dissipation at the barrier is a maximum for incident waves whose phase speed is of the order of the slip coefficient, which is interpreted as an effect of impedance matching. Transmission past an infinitesimal barrier is found to be low over all parameter space, illustrating the importance of contact-line constraints.

The dynamics of the triple gas-liquid-solid contact line is analyzed for the case where the gas is the saturated vapor corresponding to the liquid. For partial wetting conditions, a nonstationary contact line problem where the contact line motion is caused by evaporation or condensation is treated. It is shown that the Navier slip condition alone is not sufficient to relax the hydrodynamic contact line singularity: the Marangoni term is equally important when the heat transfer is involved. The transient heat conduction inside the heater is accounted for. A multiscale problem of drop evaporation with freely moving contact line is solved in the lubrication approximation as an illustration of the proposed approach.

Deposition of colloidal material in evaporating droplets is important in many applications including DNA sequencing and medical diagnostic testing. When colloidal droplets evaporate, the majority of material is often deposited at the periphery of the resultant deposition in a coffee-ring pattern. Formation of this pattern is the result of contact line pinning and the interplay between evaporative and surface tension effects in the droplet. When the contact line is pinned and the evaporative flux in the droplet is highest at the periphery, a radially outward flow is generated to conserve mass that deposits particles in the fluid at the contact line. Evaporation at the contact line can also create a temperature gradient across the droplet. This gradient gives rise to a surface tension driven flow that can resuspend particles in the droplet. When the evaporative flow dominates, particles are deposited at the contact line in a coffee-ring pattern. The presence of the coffee-ring pattern is undesirable in many printing and medical diagnostic processes. Suppression of the coffee-ring effect has been achieved by addition of surfactant, enhancement of surface tension flow, surface modification, alteration of particle shape, and application of an electric field. Manipulation of the coffee-ring effect has been achieved through the application of both AC and DC electric fields. One result of the presence of this field is the electrowetting force at the contact line which acts to reduce the contact angle and increase contact area. Since this force acts at the contact line, it may disrupt typical contact line dynamics, including evaporative dynamics, which are responsible for the formation of the coffee-ring effect. This work will experimentally examine contact line dynamics of evaporating droplets in the presence of DC electric fields. Droplets of water will be desiccated on a device with a photolithographically patterned electrode covered with a thin layer of SU8-3005. Experimental cases with applied DC fields will be compared with unactuated control cases to examine changes in transient interface shape and contact diameter.

Dynamic wetting characteristics during droplet formation in a microfluidic T-junction

The movement of the contact line in liquid-liquid-solid systems is a major phenomenon in natural and industrial processes. In particular, n-alkanes are widely occurring in the oil, soil pollution, and chemical industries, yet there is little knowledge on the effects of molecular chain length on the contact line movement. Here, we studied the effects of molecular chain length on the contact line movement in water/n-alkane/solid systems with different surface wettabilities. We used n-heptane (C7), n-decane (C10), and n-hexadecane (C16) as alkanes and α-quartz as the solid surface. We calculated the time-variation contact line moving velocity and also analyzed the jump frequency and the mean distance of the molecular displacement occurring within the contact line zone by molecular-kinetic theory. Molecular dynamics simulation results show that the contact line velocity decreases with increasing the chain length, originally caused by the decreasing the jump frequency and mean distance. These variations with the molecular chain length are related to the more torsions and deformations of the molecules with a longer chain length. In addition, the moving mechanism of the contact line on the same solid surface does not change at different molecular chain lengths, implying that the moving mechanism mainly depends on the three-phase wettability.

Numerical simulation of bubble formation on orifice plates with a moving contact line

When a solid surface makes contact with a fluid-fluid interface, a three-phase contact line is formed. This contact line may move along the surface, driven by both the external fluid flow and the thermodynamically-determined contact angle that the fluid-fluid interface makes with the solid surface. Although the steady motion of contact lines has been studied extensively, the formation and unsteady motion of contact lines has not. The process of contact-line formation is of obvious interest in any physical situation where contact lines exist. In order to study the formation and motion of contact lines we have analysed a particular model problem: the interaction of an initially flat interface between two inviscid fluids with a thin, semi-infinite, flat plate. We extend the results obtained by Billingham and King (J. Fluid Mech 296 1995), where this problem was studied for the case of normal incidence, to incidence at an arbitrary initial angle.

Aiming at the process of the droplet motion on a slanted smooth substrate surface driven by horizontal airflow velocity, a numerical simulation model of droplet on a slanted smooth substrate surface was established based on an arbitrary Lagrange-Euler method. Combined with the numerical simulation and experimental analysis, the motion state of the droplet and the distribution characteristics of the airflow velocity in the direction perpendicular to the slanted smooth surface near the bottom contact line at the lower end of the droplet were studied at different horizontal airflow velocity. It is considered that there were four regions in the airflow velocities: the low-velocity non-linear region close to the wall, the linear zone, non-linear transition zone and high-speed linear zone, of which h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = 0.25 mm, h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> = 0.9 rnm, h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> = 1.4 mm. The results show that the droplet moved in a spreading-shrinking manner on the slanted smooth substrate surface under the action of gravity and wind force. The experimental results verify the correctness of the spreading-shrinking movement of the droplet on a slanted smooth surface. Based on the numerical simulation and experimental analysis, it is considered that the instability of liquid-gas free interface affects the movement of the three-phase contact line, the deformation of the droplet on the windward side was large and on the leeward side was small.

In two-phase flows, the interface intervening between the two fluid phases intersects the solid wall at the contact line. A classical problem in continuum fluid mechanics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a nonintegrable stress singularity. Recently, various diffuse-interface models have been proposed to explain the contact line motion using mechanisms missing from the sharp-interface treatments in fluid mechanics. In one-component two-phase (liquid-gas) systems, the contact line can move through the mass transport across the interface while in two-component (binary) fluids, the contact line can move through diffusive transport across the interface. While these mechanisms alone suffice to remove the stress singularity, the role of fluid slip at solid surface needs to be taken into account as well. In this paper, we apply the diffuse-interface modeling to the study of contact line motion in one-component liquid-gas systems, with the fluid slip fully taken into account. The dynamic van der Waals theory has been presented for one-component fluids, capable of describing the two-phase hydrodynamics involving the liquid-gas transition [A. Onuki, Phys. Rev. E 75, 036304 (2007)]. This theory assumes the local equilibrium condition at the solid surface for density and also the no-slip boundary condition for velocity. We use its hydrodynamic equations to describe the continuum hydrodynamics in the bulk region and derive the more general boundary conditions by introducing additional dissipative processes at the fluid-solid interface. The positive definiteness of entropy production rate is the guiding principle of our derivation. Numerical simulations based on a finite-difference algorithm have been carried out to investigate the dynamic effects of the newly derived boundary conditions, showing that the contact line can move through both phase transition and slip, with their relative contributions determined by a competition between the two coexisting mechanisms in terms of entropy production. At temperatures very close to the critical temperature, the phase transition is the dominant mechanism, for the liquid-gas interface is wide and the density ratio is close to 1. At low temperatures, the slip effect shows up as the slip length is gradually increased. The observed competition can be interpreted by the Onsager principle of minimum entropy production.

Contact line relaxation of sessile drops on PDMS surfaces: A methodological perspective

A comprehensive numerical and experimental investigation on micrometer-sized water droplet impact dynamics and evaporation on an unheated, flat, dry surface is conducted from the standpoint of spray-cooling technology. The axisymmetric time-dependent governing equations of continuity, momentum, energy, and species are solved. Surface tension, wall adhesion effect, gravitational body force, contact line dynamics, and evaporation are accounted for in the governing equations. The explicit volume of fluid (VOF) model with dynamic meshing and variable-time stepping in serial and parallel processors is used to capture the time-dependent liquid-gas interface motion throughout the computational domain. The numerical model includes temperature- and species-dependent thermodynamic and transport properties. The contact line dynamics and the evaporation rate are predicted using Blake's and Schrage's molecular kinetic models, respectively. An extensive grid independence study was conducted. Droplet impingement and evaporation data are acquired with a standard dispensing/imaging system and high-speed photography. The numerical results are compared with measurements reported in the literature for millimeter-size droplets and with current microdroplet experiments in terms of instantaneous droplet shape and temporal spread (R/D(0) or R/R(E)), flatness ratio (H/D(0)), and height (H/H(E)) profiles, as well as temporal volume (inverted A) profile. The Weber numbers (We) for impinging droplets vary from 1.4 to 35.2 at nearly constant Ohnesorge number (Oh) of approximately 0.025-0.029. Both numerical and experimental results show that there is air bubble entrapment due to impingement. Numerical results indicate that Blake's formulation provides better results than the static (SCA) and dynamic contact angle (DCA) approach in terms of temporal evolution of R/D(0) and H/D(0) (especially at the initial stages of spreading) and equilibrium flatness ratio (H(E)/D(0)). Blake's contact line dynamics is dependent on the wetting parameter (K(W)). Both numerical and experimental results suggest that at 4.5 < We < 11.0 the short-time dynamics of microdroplet impingement corresponds to a transition regime between two different spreading regimes (i.e., for We < or = 4.5, impingement is followed by spreading, then contact line pinning and then inertial oscillations, and for We > or = 11.0, impingement is followed by spreading, then recoiling, then contact line pinning and then inertial oscillations). Droplet evaporation can be satisfactorily modeled using the Schrage model, since it predicts both well-defined transient and quasi-steady evaporation stages. The model compares well with measurements in terms of flatness ratio (H/H(E)) before depinning occurs. Toroidal vortices are formed on the droplet surface in the gaseous phase due to buoyancy-induced Rayleigh-Taylor instability that enhances convection.

Theoretical model of a finite force at the moving contact line

Dipole model of vorticity at the moving contact line