Balance Laws and Transport Theorems for Flows with Singular Interfaces

TL;DR

This paper provides a rigorous mathematical framework for describing flows with singular interfaces, including transport theorems and the transition from integral to differential balance laws.

Contribution

It introduces a unified geometric approach and detailed proofs for transport theorems applicable to flows with discontinuous physical states at singular surfaces.

Findings

Provides a rigorous mathematical description of control volumes with singular interfaces

Derives transport theorems for balance laws in such flows

Details the transition from integral to differential equations in this context

Abstract

This paper gives a concise but rigorous mathematical description of a material control volume that is separated into two parts by a singular surface at which physical states are discontinuous. The geometrical background material is summarized in a unified manner. Transport theorems for use in generic balance laws are given with proofs since they provide some insight into the results. Also the step from integral balances to differential equations is given in some detail.