A cut finite element method for coupled bulk-surface problems on time-dependent domains

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh

Phase-field modeling droplet dynamics with soluble surfactants

Using lattice Boltzmann approach, a phase-field model is proposed for simulating droplet motion with soluble surfactants. The model can recover the Langmuir and Frumkin adsorption isotherms in equilibrium. From the equilibrium equation of state, we can determine the interfacial tension lowering scale according to the interface surfactant concentration. The model is able to capture short-time and long-time adsorption dynamics of surfactants. We apply the model to examine the effect of soluble surfactants on droplet deformation, breakup and coalescence. The increase of surfactant concentration and attractive lateral interaction can enhance droplet deformation, promote droplet breakup, and inhibit droplet coalescence. We also demonstrate that the Marangoni stresses can reduce the interface mobility and slow down the film drainage process, thus acting as an additional repulsive force to prevent the droplet coalescence

Numerical approximation of a phase-field surfactant model with fluid flow

Modelling interfacial dynamics with soluble surfactants in a multiphase system is a challenging task. Here, we consider the numerical approximation of a phase-field surfactant model with fluid flow. The nonlinearly coupled model consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes equation. With the introduction of two auxiliary variables, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semi-explicitly. By certain subtle explicit-implicit treatments to stress and convective terms, we construct first and second-order time marching schemes, which are extremely efficient and easy-to-implement, for the transformed governing system. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of phase-field variables, velocity and pressure are fully decoupled. We further establish a rigorous proof of unconditional energy stability for the first-order scheme. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow, where the increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence

A level-set model for two-phase flow with variable surface tension: thermocapillary and surfactants

An unstructured conservative level-set method for two-phase flow with variable surface tension is introduced. Surface tension is a function of temperature or surfactant concentration on the interface. Consequently, the called Marangoni stresses induced by temperature gradients or surfactant concentration gradients on the interface lead to a coupling of momentum transport equation with thermal energy transport equation or interface surfactant transport equation. The finite-volume method discretizes transport equations on 3D collocated unstructured meshes. The unstructured conservative level-set method is employed for interface capturing, whereas the multiple marker approach avoids the numerical coalescence of fluid particles. The fractional-step projection method solves the pressure-velocity coupling. Unstructured flux-limiters are proposed to discretize the convective term of transport equations. A central difference scheme discretizes diffusive terms. Gradients are evaluated by the weighted least-squares method. Verifications and validations are reportedThe main author, N. Balcazar-Arciniega, as a Serra-Húnter Fellow (UPC-LE8027), acknowledges the Catalan Government for the financial support through this programme. Simulations were executed using computing time granted by the RES (IM-2021-1-0013, IM-2020-2-0002) and PRACE 14th Call (2016153612) on the supercomputer MareNostrum IV based in Barcelona, Spain. The authors acknowledge the financial support of the MINECO, Spain (PID2020-115837RB-100).Peer ReviewedPostprint (published version

A Two-Field Formulation for Surfactant Transport within the Algebraic Volume of Fluid Method

Surfactant transport plays an important role in many technical processes and industrial applications such as chemical reactors, microfluidics, printing and coating technology. High fidelity numerical simulations of two-phase flow phenomena reveal rich insights into the flow dynamics, heat, mass and species transport. In the present study, a two-field formulation for surfactant transport within the algebraic volume of fluid method is presented. The slight diffuse nature of representing the interface in the algebraic volume of fluid method is utilized to track the concentration of surfactant at the interface as a volumetric concentration. Transport of insoluble and soluble surfactants is investigated by tracking two different concentrations of the surfactant, one within the bulk of the liquid and the other one at the interface. These two transport equations are in turn coupled by source terms considering the ad-/desorption processes at a liquid-gas interface. Appropriate boundary conditions at a solid-fluid interface are formulated to ensure surfactant conservation, while also enabling to study the ad-/desorption processes at a solid-fluid interface. The developed numerical method is verified by comparing the numerical simulations with well-known analytical and numerical reference solutions. The presented numerical methodology offers a seamless integration of surfactant transport into the algebraic volume of fluid method, where the latter has many advantages such as volume conservation and an inherent ability of handling large interface deformations and topological changes

A computational fluid dynamics study of two-phase flows in the presence of surfactants

Drop formation in co-flowing fluids and drops rising in a tube are important in applications such as microencapsulation and enhanced oil recovery. A hybrid volume-of-fluid method with a front-tracking scheme is developed to study two-phase flows in the presence of surfactants at finite Reynolds numbers. Both fluids can be Newtonian or shear-thinning, and surfactants are soluble in the adsorption-desorption limit. A drop in the co-flowing geometry typically breaks up at the primary neck. The drop breaks faster with smaller volumes as the outer flow rate increases or the drop viscosity decreases. When surfactants are present, they accumulate in the neck region resulting in Marangoni stresses that slow down the neck thinning rate. This results in longer breakup times with larger drop volumes. At high surfactant coverages, the primary neck formation slows down enough and breakup occurs at the secondary neck. Increasing outer co-flowing flow weakens the retarding effect of the high surfactant coverage leading to breakup again at the primary neck. The adsorption-desorption kinetics also affects the neck breakup position, and the primary drop volume and breakup time depend non-linearly on the Biot number. The presence of a confining wall may lower the value of the critical equilibrium fractional coverage required for the drop to enter the no-necking regime. As the drop becomes more shear-thinning, the drop breaks up faster with a shorter remnant drop length. Multiple satellite drops are observed at breakup with strongly shear-thinning drop fluid at high coverage of soluble surfacants. The buoyancy-driven motion of drops in a tube is investigated by determining the steady shapes and velocities of the drops as a function of the drop size. Higher buoyancy force leads to larger deformation of drops and increased terminal velocities. Higher inertia increases the terminal velocity of drops and results in the development of negative curvatures at the rear of the drop. The non-uniform distribution of surfactants at the interface gives rise to Marangoni stresses that retard the drop motion though the drop shapes remain unaffected

Interfacial dynamics with soluble surfactants: A phase-field two-phase flow model with variable densities

 In this work, we present a hydrodynamics coupled phase-field surfactant model with variable densities. Two scalar auxiliary variables are introduced to transform the original free energy functional into an equivalent form, and then a new thermodynamically consistent model can be obtained. In this model, evolutions of two phase-field variables are described by two Cahn-Hilliard-type equations, and the fluid flow is dominated by incompressible Navier-Stokes equation. The finite difference method on staggered grid is used to solve the above model. Then a classical droplet rising case and a droplet merging case are used to validate our model. Finally, we study the effect of surfactants on droplet deformation and merging. A more prolate profile of droplet is observed under a higher surfactant bulk concentration, which verifies the effect of surfactant in reducing the interfacial tension. Increases in surface Peclet number and initial surfactant bulk concentration can enhance the non-uniformity of surfactant distribution around the interface, which will arise the Marangoni force. The Marangoni force acts as an additional repulsive force to delay the droplet merging.Cited as: Zhu, G., Li, A. Interfacial dynamics with soluble surfactants: A phase-field two-phase flow model with variable densities. Advances in Geo-Energy Research, 2020, 4(1): 86-98, doi: 10.26804/ager.2020.01.0