Mathematical study on the thickened interface model by viscosity solution of the level-set equation

Abstract

The level-set approach extended for its viscosity solution is investigated to derive a relation to the conservation law of fluid phenomena and the phase-field approach based on the free energy theory. This mathematical approach is useful to consider approximate models for fluid interface problems. Here, this approach is applied to the thickened interface model of the combustion flame problem to derive a new mathematical formulation by the viscosity solution of the level-set equation, which can maintain a flame propagation speed of global solution. This approach reveals a mathematical incompatibility of empirical thickened flame models and proposes some improvements.

A novel approach for wall-boundary immersed flow simulation (proposal of modified Navier-Stokes equation)

“…It has been extensively investigated in mathematics (Chen et al, 1991), and applied to practical problems such as image processing (Li et al, 2011), 3D printing (Cacace et al, 2017, and simulations of interface problems, including a phase interface (Grave et al, 2020) and combustion flame (Liu and Oshima, 2011). Oshima (2016Oshima ( , 2017Oshima ( , 2022) investigated a viscosity solution for the level-set equation as laminated multiple iso-contours of an interface "region" with a finite thickness, instead of a single surface defined by the traditional level-set approach. The interface is also mathematically identical to a physical interface model based on the theory of free energy by phase field approach and to the transport equation of a conservation variable in thermo-fluid problems, which become a mathematical basis of thickened interface models (Oshima, 2022).…”

“…Oshima (2016Oshima ( , 2017Oshima ( , 2022) investigated a viscosity solution for the level-set equation as laminated multiple iso-contours of an interface "region" with a finite thickness, instead of a single surface defined by the traditional level-set approach. The interface is also mathematically identical to a physical interface model based on the theory of free energy by phase field approach and to the transport equation of a conservation variable in thermo-fluid problems, which become a mathematical basis of thickened interface models (Oshima, 2022). Moreover, they are expected to facilitate a novel approach for wall-boundary interface of the velocity "vector" and momentum transport of fluid flow.…”