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Balance Laws and Transport Theorems for Flows with Singular Interfaces

Ferdinand Thein, Gerald Warnecke

TL;DR

This work develops a rigorous geometric framework for balance laws in flows separated by singular moving interfaces $\Sigma$, where bulk and interfacial states may be discontinuous. By combining stationary and moving surface calculus with transport theorems, it derives volume and surface balance equations and explicit jump conditions that couple bulk and interfacial dynamics, including curvature effects through the mean curvature $κ_M$. The key contributions include a unified treatment of volume and surface balances, a generalized Reynolds transport theorem for embedded moving surfaces, and a pillbox-based derivation of surface jump conditions that account for normal motion $w_ν$ and tangential motion $w^∥$. The results enable rigorous modeling of phase boundaries, shock fronts, and interfacial transport in continuum mechanics, with practical implications for phase transitions, chemical reactions, and surfactant-laden interfaces. The framework also provides systematic reductions to lower-dimensional cases, linking 3D interfacial theory to 2D curves and 1D points.

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.

Balance Laws and Transport Theorems for Flows with Singular Interfaces

TL;DR

This work develops a rigorous geometric framework for balance laws in flows separated by singular moving interfaces , where bulk and interfacial states may be discontinuous. By combining stationary and moving surface calculus with transport theorems, it derives volume and surface balance equations and explicit jump conditions that couple bulk and interfacial dynamics, including curvature effects through the mean curvature . The key contributions include a unified treatment of volume and surface balances, a generalized Reynolds transport theorem for embedded moving surfaces, and a pillbox-based derivation of surface jump conditions that account for normal motion and tangential motion . The results enable rigorous modeling of phase boundaries, shock fronts, and interfacial transport in continuum mechanics, with practical implications for phase transitions, chemical reactions, and surfactant-laden interfaces. The framework also provides systematic reductions to lower-dimensional cases, linking 3D interfacial theory to 2D curves and 1D points.

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.

Paper Structure

This paper contains 16 sections, 4 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Calculus of Stationary and Moving Surfaces
  3. Stationary Surfaces
  4. Surface Differential Operators
  5. Moving Surfaces
  6. Moving Control Volumes with Singular Moving Surfaces
  7. Generic Balance Laws
  8. Transport Theorems, Volume and Surface Balance Equations
  9. Volume Transport
  10. Divergence Theorem and Transport Theorem for Embedded Moving Surfaces
  11. Separate Differential Balance Equations
  12. A Generalized Transport Theorem
  13. Derivation of Surface Equation with Jump Conditions
  14. Moving Curves in Two Dimensions, Points in One Dimension
  15. Time Dependent Determinants
  16. ...and 1 more sections

Key Result

Theorem 4.1

Let $\psi : I\times\mathbb{R}^n \to \mathbb{R}$ be a continuously differentiable density function and $\mathcal{V}({\textup{t}} ) \in \mathbb{R}^n$ be a closed volume moving with the local particle velocity $\mathbf{v}({\textup{t}} ,\mathbf{x})$. We assume that the boundary $\mathcal{S}({\textup{t}} and

Figures (2)

  • Figure 1: An arbitrary material volume $\mathcal{V}$ separated by an immaterial internal surface $\Sigma$ into the sub-volumes $\mathcal{V}_1$ and $\mathcal{V}_2$.
  • Figure 2: Sketch of the pillbox argument.

Theorems & Definitions (4)

  • Theorem 4.1: The Reynolds Transport Theorem in $\mathbb{R}^n$
  • Theorem 4.2: Divergence Theorem or Gauss-Green Theorem for Surfaces in $\mathbb{R}^3$
  • Theorem 4.3: Transport Theorem for Moving Surfaces in $\mathbb{R}^3$
  • Theorem 4.4: Generalized Reynolds Transport Theorem in $\mathbb{R}^3$