Arbitrary Lagrangian–Eulerian finite-element method for computation of two-phase flows with soluble surfactants Sashikumaar Ganesan a,⇑ , Lutz Tobiska b a Numerical Mathematics and Scientific Computing, Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore 560012, India b Institute of Analysis and Numerical Mathematics, Otto-von-Guericke University, PF 4120, D-39016 Magdeburg, Germany article info Article history: Received 20 September 2010 Received in revised form 3 January 2012 Accepted 13 January 2012 Available online 1 February 2012 Keywords: Finite-elements ALE approach Interfacial fluid flows Soluble surfactant Navier–Stokes equations abstract A finite-element scheme based on a coupled arbitrary Lagrangian–Eulerian and Lagrangian approach is developed for the computation of interface flows with soluble surfactants. The numerical scheme is designed to solve the time-dependent Navier–Stokes equations and an evolution equation for the surfactant concentration in the bulk phase, and simulta- neously, an evolution equation for the surfactant concentration on the interface. Second- order isoparametric finite elements on moving meshes and second-order isoparametric surface finite elements are used to solve these equations. The interface-resolved moving meshes allow the accurate incorporation of surface forces, Marangoni forces and jumps in the material parameters. The lower-dimensional finite-element meshes for solving the surface evolution equation are part of the interface-resolved moving meshes. The numer- ical scheme is validated for problems with known analytical solutions. A number of com- putations to study the influence of the surfactants in 3D-axisymmetric rising bubbles have been performed. The proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The presence of surface active agents (surfactants) significantly alters the dynamics of multiphase flows. Surfactants low- er the surface tension on the interface and since, in general, their concentration along the interface is not uniform, Marangoni forces are induced. These properties of surfactants offer the possibility of controlling the dynamics of multiphase flow systems. Surfactant-controlled multiphase flow systems are widely used in scientific, engineering and biomedical applications. For example, surfactants can be used to manipulate very small droplets and bubbles [6,11] which is useful in flow-focusing de- vices [2,29]. The presence of surfactants in pulmonary alveoli is essential for the proper functioning of the defense mecha- nism of lungs [9,17,25]. A lack of pulmonary surfactants in premature neonates causes the respiratory distress syndrome (RDS) [3]. A mathematical model describing interface flows with soluble surfactants consists of the time-dependent Navier–Stokes equations coupled with the bulk and the surface evolution equations for the concentration of surfactants in the bulk fluid phase and on the interface, respectively. Since the interface has to be captured/tracked during the computations, the solution of the surface evolution equation on the deforming interface is one of the main challenges in the computation of flows with 0021-9991/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2012.01.018 ⇑ Corresponding author. E-mail addresses: sashi@serc.iisc.in, sashi@serc.iisc.ernet.in (S. Ganesan), tobiska@ovgu.de (L. Tobiska). URLs: http://www.serc.iisc.ernet.in/~sashi/ (S. Ganesan), http://www-ian.math.uni-magdeburg.de/home/tobiska (L. Tobiska). Journal of Computational Physics 231 (2012) 3685–3702 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp