A model of flow and surfactant transport in an oscillatory alveolus partially filled with liquid Equations of the interfacial convection and convection-diffusion describing the transport of surfactants, and more general interfacial balance laws, in the context of a three-dimensional incompressible two-phase flow are considered. Here, the interface is represented implicitly by a zero level set of an appropriate function. All interfacial quantities and operators are extended from the interface to the threedimensional domain. In both convection and convection-diffusion settings, infinite families of conservation laws that essentially involve surfactant concentration are derived, using the direct construction method. The obtained results are also applicable to the construction of the general balance laws for other excess surface physical quantities. The system of governing equations is subsequently rewritten in a fully conserved form in the three-dimensional domain. The latter is essential for simulations using modern numerical methods. C 2012 American Institute of Physics.
In the research literature there exist very rare analytical solutions of the surfactant transport equation on an interface. In the present article, we derive sets of exact solutions to interfacial convection-diffusion equations which describe the interfacial transport of insoluble surfactants in a two-phase flow. The investigated model is based on a Stokes flow setting where a spherical shaped inner phase is dispersed in an outer phase. Under the assumption of the small capillary number, the deformation of the spherical phase interface is not taken into account. Neglecting the dependence of the surface tension on the interfacial surfactant concentration, hence neglecting the Marangoni effect, general exact solutions to the surfactant conservation law on the spherical surface with both convective and diffusive terms are provided by means of Heun’s confluent function. For the steady case, it is shown that these solutions collapse to a simple exponential form. Furthermore, for the purely diffusive problem, exact solutions are constructed using Legendre polynomials. Such analytical solutions are very valuable as benchmark problems in numerical investigations.
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