The Extended Dividing Hypersurface: A General Framework for Fluid Fronts

Gibbs originally introduced the concept of a dividing surface as a mathematical construct to model phase interfaces, attributing it with distinct material properties and internal dynamics. Building on this foundation, we present a generalized framework—termed the Extended Dividing Hypersurface (EDH)—that expands this concept beyond classical phase or material interfaces to encompass a broader class of fluid fronts, including shock fronts and vortex sheets. The hypersurface represents a continuum approximation of a diffused region where flow variables and fluid properties transition sharply yet continuously. By integrating the governing equations across the normal direction of such regions, we derive the corresponding governing equations and material properties of the EDH, ensuring that the EDH is both kinematically and dynamically equivalent to that of the diffused region. This "collapse" in the normal direction allows the EDH to capture essential interfacial behavior while reducing complexity. A range of canonical problems, including shock waves, vortex sheets, and moving or static fluid interfaces, are explored to demonstrate the versatility of the EDH framework. These cases underscore the importance of incorporating front-associated mass, interfacial flux balances, and localized dynamics when modeling fluid fronts using hypersurface representations.