Session A27: Surface Tension: Analytical and Numerical Advances

Pressure-dependent surface viscosity and its surprising consequences in interfacial flows

The surface shear viscosity of a surfactant monolayer almost always depends strongly on surface pressure, and this oft-ignored rheological feature significantly alters fluid flow and dynamics of particles on the interface. In order to illustrate the qualitatively new phenomena that arise out of pressure-dependent rheology, we focus here on a series of analytically tractable yet paradigmatic examples of lubrication geometries. Thin-gap flows naturally amplify pressure changes, and thus exemplify the effects of pressure-dependent viscosity. We show that much of the mathematical machinery from Newtonian lubrication analyses can be modified in a relatively straightforward manner in such systems. Our analysis reveals novel features such as a self-limiting flux when a surfactant is pumped through a narrow channel, a maximum approach velocity in squeeze flows due to divergent inter-particle forces, and forces perpendicular to the direction of motion that breaks symmetries associated with Newtonian analogs. We discuss the broader implications of these phenomena, especially with regard to interfacial suspension mechanics for which these lubrication geometries provide a convenient limit.   

Thermo/Soluto-capillary instabilities in evaporating bi-component liquid layers using DNS

We investigate the stability, flow dynamics and evaporation kinetics of bi-component miscible liquid layers subject to a horizontal temperature gradient by means of two-phase direct numerical simulation. Flow is dominated by surface tension and driven by both thermal and solutal Marangoni effects, in which thermophoresis and mixture thermodynamics play a role. We employ a 3D model based on the Volume-of-Fluid method to account for the deformable liquid-gas interface. We note that the addition of a second species to the liquid phase affects the stability of the laterally heated layer over the single component case. We focus on systems with low Prandtl numbers (Pr \textless 1) and find that the layer is unstable for a certain critical Marangoni number, exhibiting the so-called hydrothermal waves. The structure of internal flow is a strong function of the instability and this we are also able to determine. Phase-separation/segregation within the liquid is more pronounced in evaporating systems, as is its effects on the hydrothermal waves. Our results show that solutal Marangoni convection is not only stronger in evaporating systems but also has a destabilising effect on the layer.   

Spreading and mixing of drops on a miscible liquid of different surface tension

We carry out Volume-of-Fluid based numerical simulations of a Marangoni-driven spreading of isopropyl alcohol (IPA) drops placed on water--air interface. The two fully miscible liquids create a spatially varying surface tension, leading to the spreading of the IPA drop on the water surface. We study the spreading of drops as IPA concentration is varied. In particular, we compute the spreading velocity and show that the scaling of the front position, $L$, with time, $t$, is given by $L\sim t^{0.7}$. We observe that while the surface tension difference between the two liquids controls the spreading velocity, it only slightly alters the power-law behavior for the range of considered IPA concentrations. We also provide detailed insight of the mixing of the IPA and water, and show the time evolution of liquid--air surface tension distribution. We show that the mixing results in a volume flux in a thin region on the surface, generating a vortical flow underneath the spreading front; we investigate the details of these flow patterns and show the time evolution of the circulation within the water. The numerical results are supported by new experimental observations reported separately.   

Draining Capillary Liquids from Containers with Interior Corners

A new solution is found for the late stage draining of a wetting capillary fluid in an interior corner. A formulation for slender flow along the interior base-corner of a right circular cylinder is presented, where a separation of variables solution offers a method to predict drain rates for this and related double-drain geometries. It is shown the maximum volumetric liquid removal rate is $Q \sim t^{-3}$, volume removed is $V \sim t^{-2}$, and nominal liquid depth is $h \sim t^{-1}$. Representative experimental results are presented to assess the quantitative value of the approach.   

Dynamic Wetting Failure and Hydrodynamic Assist in Curtain Coating

Dynamic wetting failure in curtain coating of Newtonian liquids is studied in this work. A hydrodynamic model accounting for air flow near the dynamic contact line (DCL) is developed to describe two-dimensional (2D) steady wetting and to predict the onset of wetting failure. A hybrid approach is used where air is described by a one-dimensional model and liquid by a 2D model, and the resulting hybrid formulation is solved with the Galerkin finite element method. The results reveal that the delay of wetting failure in curtain coating---often termed hydrodynamic assist---mainly arises from the hydrodynamic pressure generated by the inertia of the impinging curtain. This pressure leads to a strong capillary-stress gradient that pumps air away from the DCL and thus increases the critical substrate speed for wetting failure. Although the parameter values used in the model are different from those in experiments due to computational limitations, the model is able to capture the experimentally observed non-monotonic behavior of the critical substrate speed as the feed flow rate increases [T. D. Blake et al., Phys. Fluids, 11(8), 1995 (1999)].   

Marangoni flows induced by A + B → C reaction fronts with arbitrary diffusion coefficients

We consider horizontal aqueous solutions in contact with air where three reacting species A, B, and C can affect the surface tension of the solution, thereby driving Marangoni flows. When the two reactants A and B, that are initially separated, are brought into contact, a reaction front producing species C is formed and evolves in time due to diffusion, convection and reaction processes. The resulting dynamics is studied by numerically integrating the incompressible Navier-Stokes equations coupled to reaction-diffusion-convection equations for the three chemical species. For equal initial concentrations of reactants and equal diffusion coefficients, we have explained how chemically-driven Marangoni flows can lead to complex dynamics of the front propagation. Here we extend such results for arbitrary values of the diffusion coefficients and initial concentrations of reactants. We give the general classification of the surface tension profiles as a function of the Marangoni numbers quantifying the effect of each species on the surface tension, the ratio of initial concentrations of reactants and the ratios of diffusion coefficients. Such a classification allows us then to study the resulting structure of the convective rolls as well as the nonlinear dynamics of the reaction front.   

Dynamics of two-phase interfaces and surface tensions: A density-functional theory perspective

Classical density functional theory (DFT) is a statistical mechanical framework for the description of fluids at the nanoscale, where the inhomogeneity of the fluid structure needs to be carefully accounted for. By expressing the grand free-energy of the fluid as a functional of the one-body density, DFT offers a theoretically consistent and computationally accessible way to obtain two-phase interfaces and respective interfacial tensions in a ternary solid-liquid-gas system. The dynamic version of DFT (DDFT) can be rigorously derived from the Smoluchowsky picture of the dynamics of colloidal particles in a solvent. It is generally agreed that DDFT can capture the diffusion-driven evolution of many soft-matter systems. In this context, we use DDFT to investigate the dynamic behaviour of two-phase interfaces in both equilibrium and dynamic wetting and discuss the possibility of defining a time-dependent surface tension, which still remains in debate.   

Variational Methods For Sloshing Problems With Surface Tension.

We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in a container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and we seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displacement. We derive a new variational formulation of the coupled problem and establish the existence of solutions using the direct method from the Calculus of Variations. In the limit of zero surface tension, we recover the variational formulation of the classical Steklov eigenvalue problem, as derived by B. A. Troesch. For the particular case of an axially symmetric container, we propose a finite element numerical method for computing the sloshing modes of the coupled system. The scheme is implemented in FEniCS and we obtain a qualitative description of the effect of surface tension on the sloshing modes.