Modelling and simulation of interfacial flows featuring surface viscous effects*

Interfacial flows are accompanied by stresses which dilate, compress, or shear the deforming interface. Constitutive assumptions are commonly made to describe the transmission of the interfacial stresses to the bulk phases. The simplest interface corresponds to an inviscid surface with an extensible line tension in which stress is transmitted via a normal pressure-jump across the interface. Complexity arises in the presence of surface-active species that spatiotemporally alter the line tension as a function of surfactant concentration leading to an elastic effect at the surface. Increasing the surfactant concentration leads to complex structures that respond to deformation and shear, leading to the emergence of interfacial rheology. The most utilised surface-viscous interface model is the Newtonian surface commonly known as the Boussinesq-Scriven interface [1]. Two major features of the surface rheology are the shear and dilatational surface viscosities, which can be understood in terms of their bulk analogues. In this work, we first review the Boussinesq-Scriven fluid interface utilised in different coordinate systems in the literature to draw attention to the correct choice of mathematical model formulation. Next, we present a numerical method based on the use of our in-house multiphase code, BLUE[2], which exploits the advantages of a Lagrangian interface embedded within a Eulerian grid to construct the surface viscous stresses. Finally, we present benchmark problems, such as drop deformation in simple shear flows, and capillary-gravity wave formation, to elucidate the significance of the interfacial rheology in the nonlinear regime.   

References:

[1] J. C. Slattery, L. Sagis, E.-S. Oh, Interfacial transport phenomena, Springer Science & Business Media, 2007.

[2] S. Shin, J. Chergui, D. Juric, L. Kahouadji, O. K. Matar, and R. V. Craster, “A hybrid interface tracking–level set technique for multiphase flow with soluble surfactant,” Journal of Computational Physics, vol. 359, pp. 409–435, 2018