Analytical and Computational Methods for Two-Phase Flow with Soluble Surfactant

Xu, Kuan; Booty, Michael R.; Siegel, Michael

doi:10.1137/120881944

Cited by 10 publications

(22 citation statements)

References 47 publications

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“…See e.g. the conditions in [36,2,42,43], [25, (5)], and the kinetic condition in [17, (2.14)]; see also [17, (23)]. In fact, in §6.1.1 we show that close to steady state (2.19) is related to the classical condition, which was studied e.g.…”

Section: Governing Equationsmentioning

confidence: 59%

“…In [39,1] the volume of fluid (VOF) method was used, which approximates the characteristic function of one of the phases. The level set method, which describes the interface as the level set of a function, was considered in [43]. Numerical computations based on diffuse interface models have been presented in [35,42,24,26].…”

Section: Introductionmentioning

confidence: 99%

“…A front tracking method for soluble surfactants has been introduced in [36,41]. In addition we mention the arbitrary Lagrangian-Eulerian approach in [25], the segment projection method in [33] and the hybrid methods studied in [12,43]. For more references and an introduction to numerical methods for two-phase flow we refer to the book [30].…”

Section: Introductionmentioning

confidence: 99%

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Stable finite element approximations of two-phase flow with soluble surfactant

Barrett

Garcke

Nürnberg

2015

Journal of Computational Physics

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Please cite this article in press as: J.W. Barrett et al., Stable finite element approximations of two-phase flow with soluble surfactant, J. Comput. Phys. (2015), http://dx.doi.org/10. 1016/j.jcp.2015.05.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractA parametric finite element approximation of incompressible two-phase flow with soluble surfactants is presented. The Navier-Stokes equations are coupled to bulk and surfaces PDEs for the surfactant concentrations. At the interface adsorption, desorption and stress balances involving curvature effects and Marangoni forces have to be considered. A parametric finite element approximation for the advection of the interface, which maintains good mesh properties, is coupled to the evolving surface finite element method, which is used to discretize the surface PDE for the interface surfactant concentration. The resulting system is solved together with standard finite element approximations of the Navier-Stokes equations and of the bulk parabolic PDE for the surfactant concentration. Semidiscrete and fully discrete approximations are analyzed with respect to stability, conservation and existence/uniqueness issues. The approach is validated for simple test cases and for complex scenarios, including colliding drops in a shear flow, which are computed in two and three space dimensions.

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Section: Governing Equationsmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stable finite element approximations of two-phase flow with soluble surfactant

Barrett

Garcke

Nürnberg

2015

Journal of Computational Physics

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“…39 The mesh-based method for the transition layer is second order accurate in space and uses an explicit, first order Euler method for the time update, although in principle it can be made higherorder accurate (a spectrally accurate implementation in space in 2D is given in Ref. 23). Equispaced collocation points N j = jN M /M, j = 0, .…”

Section: Mesh-based Methods For the Transition Layermentioning

confidence: 99%

“…The surface and bulk Peclet numbers are often of the same order of magnitude, but unlike the treatment of bulk diffusion, the surface diffusion term Pe −1 s ∇ 2 s has sufficiently small influence that it is often neglected. 23 An equation representing conservation of the total amount of surfactant is obtained by integrating (21) over the drop surface. The integral of the left-hand side is equal to the time rate of change of the total amount of surfactant, d dt S d S, 10 …”

Section: The Infinite Bulk Peclet Number Limitmentioning

confidence: 99%

Numerical simulation of drop and bubble dynamics with soluble surfactant

2014

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Numerical computations are presented to study the effect of soluble surfactant on the deformation and breakup of an axisymmetric drop or bubble stretched by an imposed linear strain flow in a viscous fluid. At the high values of bulk Peclet number Pe in typical fluid-surfactant systems, there is a thin transition layer near the interface in which the surfactant concentration varies rapidly. The large surfactant gradients are resolved using a fast and accurate "hybrid" numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the free boundary problem. The method is used to investigate the dependence of drop deformation on parameters that characterize surfactant solubility. We also compute resolved examples of tipstreaming, and investigate its dependence on parameters such as flow rate and bulk surfactant concentration. C 2014 AIP Publishing LLC. [http://dx. Wang, Siegel, and Booty Phys. Fluids 26, 052102 (2014) alone. Under the simplification of low Reynolds number flow, the evolution can be described completely by interface quantities and solved by surface-based methods such as the boundary integral method. This is among the most accurate and efficient numerical methods for solving free and moving boundary problems. Boundary integral simulations of the effect of insoluble surfactant on the deformation and breakup of axisymmetric and 3D drops in an imposed flow have been given These characterize the dependence of drop deformation on the capillary number Ca = μ 2 GR 0 /γ 0 , where μ 2 is the suspending fluid viscosity, G is the imposed strain rate, R 0 is the undeformed radius of the drop, and γ 0 is the surface tension of a clean interface. Tipstreaming is observed in the boundary integral computations of Bazhlekov, Anderson, and Meijer, 12 and Eggleton, Tsai, and Stebe 13 when the ratio λ = μ 1 /μ 2 of drop to suspending fluid viscosity is small and the capillary number is above a critical value. To the best of our knowledge, these are the only numerical simulations that evoke the tipstreaming observed in experiments. Other boundary integral studies of drop and bubble evolution with insoluble surfactant are, e.g., Refs. 14-19. Volume of fluid numerical simulations 20, 21 of low viscosity ratio drops with insoluble surfactant in extensional flow also show droplets or small drop fragments emitted from pointed bubble ends. However, these are on the scale of the mesh spacing and are therefore not well resolved, so that tipstreaming may have been introduced as a numerical artifact.A soluble surfactant advects and diffuses in the bulk fluid, and there is an exchange or transfer between its dissolved form in the bulk and its adsorbed form on the interface. We adapt a "hybrid" numerical method for the study of surfactant solubility effects in interfacial flow that was introduced in Refs. 22 and 23. The method applies for small bulk diffusion of surfactant, or equivalently for large bulk Peclet number Pe. The value of Pe in ...

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A highly accurate boundary integral equation method for surfactant-laden drops in 3D

Sorgentone

Tornberg

2018

Journal of Computational Physics

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The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.

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Analytical and Computational Methods for Two-Phase Flow with Soluble Surfactant

Cited by 10 publications

References 47 publications

Stable finite element approximations of two-phase flow with soluble surfactant

Stable finite element approximations of two-phase flow with soluble surfactant

Numerical simulation of drop and bubble dynamics with soluble surfactant

A highly accurate boundary integral equation method for surfactant-laden drops in 3D

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