A Robust Lattice Boltzmann Method for Interface-Bound Transport of a Passive Scalar: Application to Surfactant-Laden Multiphase Flows

Abstract: The transport of a passive scalar restricted on interfaces, which is advected by the fluid motions have numerous applications in multiphase transport phenomena. A prototypical example is the advection-diffusion of the concentration field of an insoluble surfactant along interfaces. A sharp-interface model of the surfactant transport on the interface (Stone, Phys. Fluids A, 1990) has been extended to a diffuse-interface formulation based on a delta-function regularization by Teigen et al. (in Comm. Math. Sci., 2009). However, the latter approach involves singular terms which can compromise its numerical implementation. Recently, Jain and Mani (in Annual Research Briefs, CTR, Stanford University, 2022) circumvented this issue by applying a variable transformation, which effectively leads to a generalized interface-bound scalar transport equation with an additional interfacial confining flux term. The resulting formulation has similarities with the conservative Allen-Cahn equation (CACE) used for tracking of interfaces. In this paper, we will discuss a novel robust central moment lattice Boltzmann (LB) method to simulate the interface-bound advection-diffusion transport equation of a scalar field proposed in Teigen et al. by applying Jain and Mani's transformation. It is coupled with another LB scheme for the CACE to compute the evolving interfaces, and the resulting algorithm is validated against some benchmark problems available in the literature. As further extension, we have coupled it with our central moment LB flow solver for the two-fluid motions, which is modulated by the Marangoni stresses resulting from the variation of the surface tension with the local surfactant concentration modeled via the Langmuir isotherm. This is then validated by simulating insoluble surfactant-laden drop deformation and break-up in a shear flow at various capillary numbers.