A level-set method for interfacial flows with surfactant

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Abstract

A level-set method for the simulation of fluid interfaces with insoluble surfactant is presented in two-dimensions. The method can be straightforwardly extended to three-dimensions and to soluble surfactants. The method couples a semi-implicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [J. Xu, H. Zhao. An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput. 19 (2003) 573–594] with the immersed interface method originally developed by LeVeque and Li and [R. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994) 1019–1044] for solving the fluid flow equations and the Laplace–Young boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component mass and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, drop–drop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to non-uniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the level-set method has been used to simulate fluid interfaces with surfactant.

Introduction

In this paper, we propose a level-set/immersed interface method for the evolution of deformable fluid interfaces with insoluble surfactant in two-dimensions. The method can be straightforwardly extended to three-dimensions and to soluble surfactants. Surfactants are surface-active molecules that selectively adhere to interfaces. Surfactants typically consist of a hydrophilic head and a hydrophobic tail – detergents are common examples. Surfactants play a critical role in numerous important industrial and biomedical applications ranging from enhanced oil recovery (e.g. [42]) to pulmonary function (e.g. [16]).

Surfactants are advected and diffused along interfaces by the motion of the fluid and by molecular mechanisms, respectively [9]. The surface tension depends on the surfactant distribution through the equation of state – regions of higher surfactant concentration have lower surface tension. Non-uniform surfactant concentration along an interface creates non-uniform Capillary (normal) and Marangoni (tangential) forces in the fluid. This in turn affect the fluid velocity that then couples back to affects the surfactant distribution. For example, the convection of surfactant toward the stagnation points at the tip of a drop tends to lower the surface tension there and increase the drop deformation. On the other hand, Marangoni forces resist the convection of surfactant toward the drop tip and thus restrain the deformation of the drop. Compression/stretching of the interface results in a corresponding increase/decrease in the surfactant concentration.

Computing the motion of interfacial flows with surfactant is challenging. The Navier–Stokes equations must be solved in a complex, multiply connected moving domain with prescribed jumps in the normal (Capillary) and the tangential (Marangoni) stress across the interface separating the domains. The moving interface must be accurately simulated and topology transitions may occur as interfaces reconnect or break-up. Further, as surfactant is advected and diffused along the interface there may be adsorption/desorption of surfactant from/to the bulk to/from the interface [9]. For simplicity, we focus here on the case of insoluble surfactant so that the surfactant remains bound to the interface.

In this paper, a level-set method for the simulation of fluid interfaces with insoluble surfactant is presented in two-dimensions. The method couples a semi-implicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [62] with the immersed interface method originally developed by LeVeque and Li [31] for solving the fluid flow equations and the Laplace–Young boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component (domain) volume and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, drop–drop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to non-uniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the level-set method has been used to simulate fluid interfaces with surfactant.

There are now a number of different numerical methods that have been developed to simulate the motion of surface-tension mediated interfacial flows (e.g., see [26]). Popular approaches include boundary integral methods (e.g., see the reviews [20], [48]) where the flow equations are mapped to the interface, front-tracking/continuum surface force (CSF) methods (e.g., see the reviews [15], [47], [60]) where the flow equations are solved in the volume domain, a separate mesh is used to describe the interface and nearly singular surface forces (continuum surface force) are introduced to approximate the singular surface tension force, volume-of-fluid/CSF methods (e.g., see the review [50]) where a volume-fraction function is used to identify the interface, level-set/CSF methods (e.g., see the reviews [43], [44], [51]) where the interface is characterized by the zero contour of a level-set function and phase-field methods where a concentration field is introduced to identify fluid components (e.g., see the review [3], [25], [64]). A number of hybrid methods now exist including level-set/volume-of-fluid methods [56], [58], particle level-set methods [12], [18], marker/volume-of-fluid methods [4] and level-contour front tracking methods [52].

In addition to CSF methods, other flow solvers have been developed that directly account for the Laplace–Young surface tension jump conditions without smoothing. Advantages of such an approach include (1) no introduction of intermediate non-physical states near the interface since the interface condition is sharp, (2) higher-order accuracy can be achieved as opposed to CSF based methods which are generally only first-order accurate. Methods without smoothing include the method developed by Helenbrook et al. [17], the ghost-fluid (GF) method (e.g. [13], [38]) and the immersed interface method (IIM, e.g. [19], [30], [32], [35]). These algorithms have the common feature that standard finite difference schemes are used at grid points away from interfaces while the finite difference schemes are modified at grid points near interfaces. In the GF algorithm, subcell resolution is used to mark the interface position and the values of discontinuous quantities are artificially extended to grid points neighboring the interface via extrapolation. A fully second-order accurate GF method for moving interfaces with geometric boundary conditions has recently been developed [40]. A fourth-order GF method for the Laplace and heat equations has also been developed recently [14]. In the IIM, which is the approach we use here together with a level-set method, a local coordinate system is introduced to explicitly incorporate jump conditions and discontinuous coefficients into second-order accurate finite difference schemes. Advantages of this approach include its high-order accuracy, the ease of implementation and the fact that fast solvers (e.g., the FFT) can be used to invert the discrete systems.

Despite the vast literature on studies of drops and interfaces in multiphase flows, there are relatively few works in which the effects of surfactants are incorporated. Much of the previous work on surfactants has utilized the boundary integral method for axisymmetric (e.g., see [11], [41], [55]) and 3D (e.g., see [33], [48], [63]) Stokes flows. Recently, CSF-based methods have been developed for interfacial flows with surfactants using immersed boundary/front tracking methods [5], [22] and volume-of-fluid methods [10], [21], [49]. We remark that in [21], an algorithm was developed to conserve both component mass and surfactant mass and is capable of simulating an arbitrary equation of state for the surfactant.

Recently, Xu and Zhao in [62] and Adalsteinsson and Sethian [2] presented methodologies to simulate transport and diffusion along deformable interfaces in conjunction with a level-set method. In the former work, Xu and Zhao applied their algorithm to study specifically the evolution of surfactant although they did not couple their method to a flow solver. Several test cases were presented in [62] in which a velocity field is prescribed. Here, we build upon this work by coupling the transport algorithm of Xu and Zhao to an IIM flow solver. We introduce modifications to conserve component and surfactant mass and we examine the effects of non-uniform Capillary forces and Marangoni forces on the evolution of interfaces in Stokes flow.

The remainder of this paper is organized as follows. The governing equations are presented in Section 2. The numerical method is described in Section 3, which includes the IIM for solving incompressible Stokes flow and the evolution schemes for the surfactant concentration and the level-set function. Numerical simulations are presented in Section 4 to illustrate the performance of the method. Conclusions and future directions are discussed in Section 5.

Section snippets

The Navier–Stokes equations

Consider an incompressible two-phase flow consisting of fluids 1 and 2 in a fixed domain Ω = Ω1  Ω2 where an interface Σ separates Ω1 from Ω2. In each region, the Navier–Stokes equations govern the fluid motionρiuit+(ui·)ui=(·Ti)T+ρiginΩiand·ui=0inΩi,where i = 1, 2 denotes the fluid region, Ti=-piI+μi(ui+uiT) is the stress tensor, pi is the pressure, ρi is the density, μi is the viscosity and g is the gravitational acceleration.

In the far-field, we assume thatu=uonΩ.Across the interface Σ,

The immersed interface method

The Stokes equations are solved using the IIM. Let {xi,j = (xi, yj): 0  i  N, 0  j  N1} denote a uniform Cartesian mesh. Let h be the step size in both x- and y-directions. The resulting scheme is an approximate projection method in that the velocity field is not exactly divergence-free on the discrete level.

A grid point xi,j is called irregular if the level-set function ϕ changes the sign from xi,j to its four neighbors xi+1,j, xi−1,j, xi,j+1, and xi,j−1, otherwise it is called regular. At an

Numerical results

In this section, we present 2D simulations illustrating the effect of surfactant on the evolution of a single drop in shear flow, drop–drop interactions among two drops as well as interactions among multiple drops.

Conclusions

In this paper, we have presented a level-set method for the simulation of fluid interfaces with insoluble surfactant in two-dimensions. The method can be straightforwardly extended to three-dimensions and to soluble surfactants. The method couples the IIM for solving the fluid flow equations with a modification of scheme recently developed by Xu and Zhao [62] for the surfactant transport equation. Novel techniques were developed to accurately conserve component (domain) volume and surfactant

Acknowledgements

The authors thank Mike Siegel and Vittorio Cristini for helpful discussions. The authors acknowledge the support of the Network and Academic Computing Services at the University of California, Irvine. The authors also thank the computing facilities of the Department of Mathematics and the Department of Biomedical Engineering at the University of California, Irvine. J. Xu acknowledges the support of a PIMS fellowship. Z. Li was partially supported by NSF grants DMS-0201094 and DMS-0412654, and

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