Capturing the kinematics and dynamics of fluid fronts

TL;DR

This paper introduces the extended dividing hypersurface (EDH), a generalized mathematical framework for representing various fluid fronts such as interfaces, shock fronts, and vortex sheets, capturing their kinematic and dynamic properties.

Contribution

It extends Gibbs' dividing surface concept to a unified hypersurface applicable to different fluid fronts, derived from a continuum approximation of diffused regions.

Findings

EDH accurately models static and dynamic fluid fronts.

The framework relates fluxes across the hypersurface to front properties.

Demonstrated applicability to shock fronts, vortex sheets, and contact lines.

Abstract

Gibbs was the first person to represent a phase interface by a dividing surface. He defined the dividing surface as a mathematical surface that has its own material properties and internal dynamics. In this paper, an alternative derivation to this mathematical surface is provided that generalizes the concept of dividing surface to fluid fronts beyond that of just a phase or material interface. Other fluid fronts being a vortex sheet, shock front, moving contact line, and gravity wavefront, to name a few. Here, this extended definition of dividing surface is referred to as the extended dividing hypersurface (EDH), as it is not just applicable to a surface front but also to a line and a point front. This hypersurface is a continuum approximation of a diffused region with fluid properties and flow parameters varying sharply but continuously across it. This paper shows that the properties and equations describing an EDH can be derived from the equations describing the diffused region by integrating it in the directions normal to the hypersurface. This is equivalent to collapsing the diffused region in the normal direction. Hence, ensuring that the EDH is both kinematically and dynamically equivalent to that of the diffused region. Various canonical problems are examined to demonstrate the ability of the EDH to accurately represent different types of fluid and flow fronts, including static and dynamic interfaces, shock fronts, and vortex sheets. These examples emphasize the EDH's capability to represent various functionalities within a front, the relationship between the flux of quantities and hypersurface quantities, and the importance of considering the mass of front and associated dynamics.