A conservative interface-interaction model with insoluble surfactant
Introduction
Flows of two incompressible, immiscible, viscous fluids with surface tension frequently are encountered in industry and nature [1], [2], [3], [4], [5]. At their interface the fluids exchange mass, momentum, and energy. The presence of surface active agents (surfactants) effects capillary phenomena and needs to be taken into account [3], [6], [7]. In many classical two-phase flows, such as viscous fingering [8], drop break-up and coalescence [9], tip-streaming [10], and buoyancy-driven bubble-motion [11], surfactant related effects are significant.
Surfactants accumulate at fluid interfaces [12]. They are transported along, adsorbed to or desorbed from the interface. Their distribution along the interface modulates surface tension: higher concentration of surfactant implies lower surface tension. Inhomogeneous surface tension entails interfacial stresses in interface-normal and tangential directions, denoted as capillary and Marangoni stresses, respectively.
Numerical methods for solving interfacial flows with surfactants may be categorized into interface tracking and interface capturing methods. Interface tracking methods either use an interface-adapted grid or marker particles to represent the interface. Boundary integral methods employ a surface mesh to track the interface. In the context of surfactants, a boundary integral method for studying the effect of insoluble surfactants on drop deformation was developed in [13], [14]. Another interface-tracking method is the front-tracking method [15], where a fixed grid is used to compute the flow, while a set of connected marker particles tracks the interface and surfactant on the interface. A front-tracking method for insoluble surfactant was developed in [16]. A related front tracking method is the immersed boundary method [17], which was used to simulate interfacial flows with insoluble surfactants by the surfactant-conserving marker- and cell (MAC) algorithm [18]. A ghost-cell immersed boundary method was introduced in [19], and employed to study the effects of diffusion-controlled surfactant on a viscous drop injected into a viscous medium [20].
A hybrid level-set/front-tracking approach was used to study the dynamics of capillary waves with insoluble surfactant [21]. Another front-tracking method which combines a finite element methodology with adaptive body-fitted meshes served to simulate the deformation and breakup of axisymmetric liquid bridges [22] and thin filaments [23] with insoluble surfactants. Interface tracking methods are very accurate, yet, especially for topological changes and in three dimensions, the implementation-effort can be overwhelming. Possible drawbacks of marker Lagrangian approaches include difficulties with evaluating topological changes, the need to remove parts of the evolving front (delooping) to characterize the viscosity solution correctly, the need to adaptively add and remove points, and complexities in three dimensions [24].
With interface capturing methods the interface is implicitly defined by an auxiliary function, such as a level-set, color or phase-field function. This simplifies gridding, discretization and handling of topological changes. For example, a volume-of-fluid (VOF) method [4], for insoluble surfactants was developed in [25]. Hameed et al. [26] have used the Arbitrary Lagrangian–Eulerian (ALE) method combined with a coupled level-set and volume of fluid method to simulate flows containing fluid interfaces with insoluble surfactant.
With the diffuse-interface, or phase-field method the interface of a multi-fluid domain is represented by a phase-field function, which is an approximation of the characteristic function of the bulk fluid domain [27]. In [28] Teigen et al. develop and apply the diffuse interface approach to simulate flows in the presence of soluble and insoluble surfactants.
A level-set method [5], [29] for solving the surfactant transport equation has been presented by Adalsteinsson et al. [30] and Xu et al. [31], and coupled with the immersed-interface method (IIM) [32] in [33]. With the IIM, the interface jump conditions are handled explicitly by modifying the discretization stencils near the interface. As a simple and robust alternative to IIM Xu et al. have modeled interface forces within the level-set framework by a continuous surface force (CSF). A common property of these incompressible level-set methods, which they share with phase-field methods, is the smoothing of material properties, such as density and viscosity, at the interface across several grid points. This implies their main drawbacks, the lack of discrete conservation and ineffectiveness at large density and viscosity ratios.
A fully conservative level-set based sharp-interface method (SIM) for compressible flows [34], robust even for large topological interface changes [35], has been applied to two-phase flows where each phase may obey different equations of state, and large density and viscosity ratios [36]. It has been further developed to model viscous, incompressible two-phase flows [37] by incorporating a weakly compressible fluid model [37], [38]. High accuracy has been demonstrated for the buoyancy driven motion of viscous, immiscible flows [39].
Key idea of the SIM framework employed in our model, where the Navier–Stokes equations are solved on a Cartesian grid, is the modification of finite volumes that are cut by the interface in order to allow explicit application of interface-jump conditions, including interactions due to capillary and Marangoni stresses. In Ref. [37] Luo et al. propose interaction terms that consider capillary forces for a constant surface-tension coefficient and viscous interactions. This formulation of the viscous momentum exchange, however, does not allow for a jump condition in interface-tangential direction, occurring in the general case of non-vanishing Marangoni stresses [40], [41].
In this paper we develop a robust and consistent interface interaction model, incorporating inviscid, viscous, capillary, and Marangoni stresses into the SIM framework. The resulting interface flux is derived from an interface Riemann problem. The model is simpler than the CSF approach and the IIM-based approach of [28], [33]. Explicit time integration is applied for the evolution of surfactant concentration, level-set, and fluid phases. Efficiency is enhanced by a multi-resolution (MR) algorithm [35]. Level-set transport and surfactant-concentration transport are evaluated only within a narrow band near the interface. Within the SIM framework, local interface-segment lengths/areas are computed, avoiding the need for a smoothed delta-function, for approximation of the interface length. We show that surfactant mass and interface fluxes are generally predicted more accuratly than in previous works [28], [31]. Costly propagation of the interface area as an additional variable, as proposed in [25], also is avoided.
In Section 2, the governing equations are given. Section 3 describes the numerical discretization. In Section 4 we present simulation results demonstrating the capability of the method. Validation simulations for passive transport of surfactant are presented in section 4.1. A thermocapillary flow is considered for demonstrating the correct prediction of Marangoni and viscous stresses at the interface, Sec. 4.2. For demonstrating robustness and performance of the method when also inviscid and capillary stresses are present at the interface, a two-dimensional and a three-dimensional test configuration is selected, Sec. 4.3. A drop in a shear-flow serves to demonstrate the performance of the model for interfaces evolving under external shear forces, Sec. 5.
Section snippets
Basic equations
Consider the system of two immiscible fluid phases, as sketched in Fig. 1. The fluid phases occupy two non-overlapping subdomains and . Ω is bounded by , corresponding boundaries exist for the subdomains. For each phase, the system of conservation equations for weakly compressible fluids in differential, non-dimensional form holds [42], where ρ denotes the non-dimensional density, and the vector of non-dimensional
Discretization of conservation equations for fluid transport
Applying Gauss' theorem, the integral form of Eq. (1) becomes where is the normal on Γ, see Fig. 1. The flux vectors include advective and viscous components , .
The advective (superscript a) and viscous (superscript ν) fluxes are In a neighborhood of the interface the conservation equations for mass
Transport mechanisms of surfactant concentration
In this section, the accuracy of our scheme for -transport by advection, diffusion and interface evolution is investigated. A test case proposed by Xu and Zhao [31] is considered. An analytical solution exists in the case of pure advection or diffusion.
Single drop in linear shear flow
In this section we study the evolution of a drop in shear flow, similar to [66], as a demonstration application of our method. The interest is on the transient deformation behavior. We initialize a circular drop of and at the center of a computational domain, , immersed in a bulk fluid of . If not noted otherwise, , which corresponds to , also , and , i.e. or , and . The domain
Concluding remarks
In this paper, we have developed and validated a conservative interface-interaction method for viscous flows with surface tension and insoluble surfactant based on an interface-interaction method of Hu et al. [34]. We employ a level-set sharp-interface formulation to include inviscid, viscous, capillary and Marangoni stresses at the interface. Surfactant mass conservation is reproduced even for low resolution and severe interface deformation. Embedding the surfactant-conservation correction
Constructing sub-cell corrected of the interface curvature
Let the radius of the osculating circle at a point on the interface be r. Its curvature, the reciprocal of its radius, , equals the principal curvature of the local interface segment. In two-dimensional space the mean curvature is . By Taylor-series expansion of , one obtains a first-order accurate approximation to the interface curvature Considering that may also be negative, Eq. (63) is modified as
Acknowledgements
We acknowledge the Deutsche Forschungsgemeinschaft (DFG) for funding this work under Grant No. AD 186/7-2. Felix S. Schranner is a member of the TUM Graduate School. The Munich Centre of Advanced Computing (MAC) has provided the computational resources. V. Rozov and D. Azarnykh have contributed by invigorating discussions and proposing tests.
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2018, Journal of Computational PhysicsCitation Excerpt :Understanding these flows is important since they are relevant in microfluidics [2], heat pipe flows [3], motion of drops or bubbles in materials processing applications that include heating or cooling [4], evolution of metal films of nanoscale thickness melted by laser pulses [5,6], and in a variety of other thin film flows, see [7,8] for reviews. Numerical methods for studying variable surface tension flows include front tracking [9], level set [10], diffuse interface [11], marker particle [12,13], immersed boundary [14], boundary integral [15], interface-interaction [16], and Volume-of-Fluid (VOF) [17–19] methods. The VOF method is efficient and robust for tracking the topologically complex evolving interfaces.