Co-moving volumes and the Reynolds transport theorem for two-phase flows

This paper addresses the ill-posed nature of kinematic differential equations in two-phase flows with discontinuous velocity fields by employing differential inclusions to rigorously define co-moving sets and prove an extended Reynolds transport theorem that accounts for phase change and interfacial slip.

The Gist

Here is an explanation of the paper "Co-moving Volumes and the Reynolds Transport Theorem for Two-Phase Flows" using simple language and creative analogies.

The Big Picture: Tracking a Moving Crowd

Imagine you are a director filming a movie. You want to track a specific group of actors (a "co-moving volume") as they move through a busy city square.

In a normal movie (a single-phase flow), everyone moves smoothly. If you tell your camera to follow a specific group of people, you can easily predict where they will be in one second, one minute, or one hour. The rules of physics (specifically the Reynolds Transport Theorem) tell us exactly how to calculate how the "stuff" inside that group (like their total energy or mass) changes as they move.

The Problem:
Now, imagine the city square is split in half by a magical, invisible wall.

• On the Left Side, people are running fast to the East.
• On the Right Side, people are running fast to the West.
• The Wall (The Interface): This is where the two groups meet.

In the real world, this happens when oil meets water, or when ice melts into water. At the boundary, things get weird:

1. Slip: The people on the left might slide past the people on the right without sticking to them.
2. Phase Change: Some people might suddenly turn into ghosts (evaporate) or appear out of thin air (condense) right at the wall.

The Crisis:
If you try to follow a single actor who is standing exactly on that wall, the math breaks down.

• Should they go Left?
• Should they go Right?
• Should they stay still?
• Or should they split into two versions of themselves?

In standard physics, this creates a "mathematical traffic jam." The equations say there is no single, unique answer for where that person goes. If you can't predict where the actors go, you can't write the script for the movie. The standard tools (the Reynolds Transport Theorem) fail because they assume everyone moves smoothly and predictably.

The Solution: The "Fuzzy" Map

The authors, Dieter Bothe and Matthias Köhne, say: "Don't panic. We can fix this."

Instead of asking, "Where does this one actor go?", they ask, "What is the set of all possible places this actor could go?"

They use a concept called Differential Inclusions. Think of it like this:

• Old Way: A GPS that gives you one specific route. If the road is blocked, the GPS crashes.
• New Way: A GPS that gives you a "cloud" of possible routes. If you are at the wall, the GPS says, "You could go Left, you could go Right, or you could go both ways simultaneously."

By accepting that the path is a "cloud" of possibilities rather than a single line, they can rigorously define what a "moving group of actors" looks like, even when the rules are chaotic.

The New Rulebook (The New Theorem)

Once they have this "fuzzy" map, they derive a new version of the Reynolds Transport Theorem. This is the mathematical rulebook for calculating changes in a moving group.

Their new rulebook handles three tricky scenarios that old rulebooks couldn't:

1. The Slippery Wall: When two fluids slide past each other (like oil on water). The new math accounts for the fact that the boundary itself is moving and sliding, creating new edges in the moving group.
2. The Magic Transformation: When matter changes state (ice to water). The new math accounts for the fact that the group might suddenly gain or lose volume because mass is crossing the boundary.
3. The "Ghost" Scenario: When a particle hits the wall and the math says it could go anywhere. The new math treats the particle as a "cloud" of possibilities, ensuring the total mass and energy are still conserved, even if the path is uncertain.

The "Traffic Jam" Analogy

Imagine a highway with two lanes merging into one, but the merge point is broken.

• Cars from the Left want to go straight.
• Cars from the Right want to turn left.
• The Broken Merge: A car arriving at the merge point doesn't know which lane to pick.

Old Math: "The car must pick one lane. If it can't decide, the traffic system crashes."
New Math (Bothe & Köhne): "The car is a 'probability cloud.' It is 50% in the left lane and 50% in the right lane. We can still calculate the total number of cars passing through the intersection, even if we don't know exactly which lane each specific car chose."

Why Does This Matter?

This paper is crucial for engineers and scientists who model:

• Weather: Clouds forming and raining (water changing phase).
• Engines: Fuel spraying into an engine (liquid turning to gas).
• Biology: Blood flowing through vessels with different properties.

In all these cases, the "interface" (the boundary between phases) is messy, slippery, and changing. The old math tools were too rigid to handle this mess. This paper provides a robust, rigorous way to track these moving, changing, and "slippery" groups of matter, ensuring that the laws of physics (like conservation of mass and energy) still hold true, even when the math gets messy.

Summary in One Sentence

The authors created a new mathematical "fuzzy logic" system that allows us to track moving groups of fluids across messy, slippery, and changing boundaries, ensuring we can still calculate the laws of physics even when the path of a single particle is uncertain.

Technical Summary

Here is a detailed technical summary of the paper "Co-moving Volumes and the Reynolds Transport Theorem for Two-Phase Flows" by Dieter Bothe and Matthias Köhne.

1. Problem Statement

The paper addresses a fundamental gap in the mathematical modeling of two-phase flows with phase change and interfacial slip within the sharp-interface framework.

• The Core Issue: The classical Reynolds Transport Theorem (RTT) relies on the assumption that the velocity field $v$ is sufficiently regular (typically $C^1$ or at least Lipschitz continuous) to ensure the unique solvability of the kinematic ordinary differential equation (ODE) $\dot{x} = v(t, x)$. This unique solvability defines "co-moving volumes" (material volumes) composed of the same fluid elements.
• The Discontinuity: In two-phase flows with phase change (mass transfer across the interface) and slip (tangential velocity jumps), the velocity field is discontinuous at the interface $\Sigma(t)$.
  • Normal Component: Jumps occur due to mass transfer ($\dot{m} \neq 0$).
  • Tangential Component: Jumps occur due to slip between phases.
• Consequence: This discontinuity renders the kinematic ODE ill-posed in the classical sense. Specifically, solutions may not be unique (non-uniqueness) or may not exist in the standard sense when a fluid particle reaches the interface. Consequently, the standard definition of co-moving volumes and the derivation of the RTT fail.

2. Methodology

The authors employ a rigorous mathematical framework combining differential inclusions and geometric measure theory to resolve the ill-posedness of the kinematic equations.

• Differential Inclusions: Instead of treating the velocity field as a single-valued function $v(t,x)$, the authors replace it with a set-valued regularization (convexification) at the interface.
  • For $x \in \Sigma(t)$, the velocity is defined as the convex hull of the one-sided limits: $\hat{v}(t, x) = \text{conv}\{v^+(t, x), v^-(t, x)\}$.
  • This transforms the ODE $\dot{x} = v$ into a differential inclusion $\dot{x} \in \hat{v}(t, x)$.
• Flow Maps for Inclusions: Using the theory of differential inclusions (referencing Filippov and Aubin/Cellina), they define a two-phase flow map $\hat{\Phi}^t_{t_0}$. Unlike the classical map, this is a set-valued map where $\hat{\Phi}^t_{t_0}(x_0)$ represents the set of all possible trajectories (attainable sets) starting from $x_0$.
• Space-Time Divergence Theorem: To prove the transport theorem, the authors avoid the classical "pull-back" method (which requires a smooth flow map). Instead, they utilize a space-time divergence theorem. They construct a space-time domain $\Omega_0$ (the graph of the moving volume) and apply the divergence theorem in $\mathbb{R}^4$ (time + 3D space). This approach is robust enough to handle the lower regularity of the boundary and the set-valued nature of the flow.
• Geometric Analysis of Interfaces: The paper rigorously defines the interface $\Sigma(t)$ as a $C^{1,2}$-family of moving hypersurfaces. It analyzes the "transversality" condition where the initial control volume boundary intersects the interface, ensuring that the convergence of surface integrals holds even as the volume evolves.

3. Key Contributions

A. Rigorous Definition of Co-moving Sets

The paper provides the first rigorous definition of "co-moving volumes" for two-phase flows with discontinuous velocities. By utilizing differential inclusions, the authors define a co-moving set $G(t)$ not as a single evolving domain, but as the union of all possible trajectories starting from an initial set $G_0$. This handles the non-uniqueness of particle paths at the interface mathematically.

B. Generalized Reynolds Transport Theorem (RTT)

The central result is Theorem 5.2, a new version of the RTT for two-phase flows. For a co-moving volume $G(t)$ and a scalar field $\phi$, the rate of change of the integral is given by:
$$\frac{d}{dt} \int_{G(t)} \phi \, dx \bigg|_{t=t_0} = \int_{G_0 \setminus \Sigma_0} (\partial_t \phi + \text{div}(\phi v)) \, dx + \int_{\Sigma_0 \cap G_0} [[\phi (v - v_\Sigma) \cdot n_\Sigma]] \, do$$
Where:

• $v_\Sigma$ is the intrinsic velocity of the interface.
• $[[\cdot]]$ denotes the jump across the interface.
• The formula accounts for both bulk transport and the flux across the moving interface, even when the interface velocity is not uniquely determined by the bulk phases alone.

C. Thermodynamic Consistency of Non-Uniqueness

The authors demonstrate that the mathematical non-uniqueness of particle trajectories at the interface (specifically in cases of slip and phase change) is physically consistent.

• They construct a specific example (Section 3, Case 4) where the velocity field satisfies mass/momentum conservation and the entropy inequality (Second Law of Thermodynamics), yet the kinematic ODE has infinitely many strong solutions.
• They show that the set-valued regularization (differential inclusion) correctly captures the physical reality that the interface acts as a "bridge" connecting the two phases, and the convex hull of velocities represents the admissible states.

D. Volume Conservation Conditions

The paper derives the precise conditions under which a two-phase flow is isochoric (volume-conserving). They prove that volume conservation requires:

1. The velocity field to be solenoidal ($\text{div } v = 0$) in both bulk phases.
2. No phase change ($\dot{m} = 0$) OR continuous density across the interface ($[[\rho]] = 0$).
   This highlights that for typical liquid-gas systems (where density jumps significantly), volume is not conserved if phase change occurs, even if the flow is incompressible in each phase.

4. Results

• Existence of Solutions: The kinematic differential inclusion admits local strong solutions for all initial values, even at the interface.
• Regularity of Flow Map: The set-valued flow map is shown to be locally Lipschitz continuous with respect to the Hausdorff metric, ensuring that the co-moving volumes evolve in a controlled manner despite the non-uniqueness of individual particle paths.
• Validation of Jump Conditions: The derived transport theorem naturally incorporates the standard jump conditions for mass and momentum (e.g., $[[\rho(v-v_\Sigma)\cdot n_\Sigma]] = 0$) without requiring ad-hoc assumptions.
• Counter-Intuitive Finding: In the presence of slip and phase change, the boundary of a co-moving volume can develop new edges or "arcs" emanating from a single point (as shown in Example 4.4 and Figure 3), a phenomenon impossible in classical single-phase flows.

5. Significance

• Theoretical Foundation: This work provides the necessary mathematical rigor to apply continuum mechanics balance laws (mass, momentum, energy) to sharp-interface two-phase models that include slip and phase change. Previously, such models were often used heuristically without a rigorous definition of the material volume.
• Numerical Implications: The generalized RTT is crucial for the Finite Volume Method (FVM) and other numerical schemes. It justifies the discretization of transport equations in multiphase flows where the interface moves and deforms, ensuring that numerical schemes respect the underlying physics of mass and momentum transfer.
• Handling Slip: By explicitly addressing interfacial slip (common in polymer flows and microfluidics), the paper extends the applicability of sharp-interface models beyond the standard "no-slip" assumption.
• Thermodynamic Consistency: The paper bridges the gap between mathematical non-uniqueness in ODEs and physical consistency, showing that the "fuzzy" nature of particle paths at a discontinuous interface is a feature, not a bug, of the sharp-interface approximation.

In summary, Bothe and Köhne successfully extend the Reynolds Transport Theorem to the complex, discontinuous regime of two-phase flows with phase change and slip, using differential inclusions to define co-moving volumes and a space-time divergence approach to derive the transport law. This lays the groundwork for more accurate and rigorous modeling of complex multiphase systems.