A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface

Abstract

Abstract. A method is presented to solve two-phase problems involving a material quantity on an interface. The interface can be advected, stretched, and change topology, and material can be adsorbed to or desorbed from it. The method is based on the use of a diffuse interface framework, which allows a simple implementation using standard finite-difference or finite-element techniques. Here, finite-difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Interfacial flow with soluble surfactants is used as an example of the application of the method, and several test cases are presented demonstrating its accuracy and convergence.

Turbulent Flows With Drops and Bubbles: What Numerical Simulations Can Tell Us—Freeman Scholar Lecture

“…The first equation describes the surfactant transport on the interface, while the second one the surfactant transport in the bulk of the phases (D is the bulk diffusion coefficient of the surfactant); _ S w and _ S C , are the coupling terms between the two separate domains. Similar to the formulations for insoluble surfactants, the surfactant concentration is defined at the interface [30,[201][202][203] or in a thin layer across it [204,205]. When the interfacial surfactant concentration is defined only on a two-dimensional interface (e.g., using a front-tracking method), surfactant exchanges between the interface and the bulk occur in a thin adsorption layer adjacent to the interface [30,[201][202][203]: the surfactant mass Fig.…”

Immersed Boundary Simulations of Active Fluid Droplets

“…This then results in a profile where all points outside the droplet are approximately equal to zero. However, as discussed in [48] this can lead to numerical issues when dividing by H, and so we redefine the level set function at all points as H = √ H 2 + 2 setting = 10 −6 as a sufficiently small discrepancy to avoid noticeable 'leaking' of the internal concentration.…”

“…The polarisation is updated using the coupled fourth order Runge-Kutta method at the same time as the boundary is updated using Eqs (48) and (49), detailed in the "Algorithm Overview" section.…”

Diffusion of tangential tensor fields: numerical issues and influence of geometric properties

“…Similar to section 3.3 the discrete covariant derivative ∇ S is described along a componentwise description and extended to the embedding space by using the extended geometric quantities n , P , and H , cf. (18). Remark 6.…”

“…The DI method, see [23,18,20], considers an approximation of eq. ( 17), which is a classical problem in the embedding space R 3 and thus leads to a set up where established standard volume FEM can be applied.…”