Rheological modeling with GENERIC and with the Onsager principle

This paper compares three frameworks for modeling complex fluid flows—local conservation laws, the GENERIC formalism, and the Onsager principle—by illustrating their application to isothermal and incompressible polymeric fluids.

The Gist

Imagine you are trying to predict how a bowl of spaghetti sauce (a "complex fluid") will flow when you stir it. It's not just like water; it has chunks, strings, and internal structures that stretch and twist. This is the world of Rheology.

The paper by Miroslav Grmela is essentially a debate between three different "rulebooks" or frameworks that scientists use to write the mathematical equations for how these fluids move. The author compares them to see which one tells the most complete and accurate story.

Here is the breakdown of the three frameworks, explained with simple analogies:

1. The "Traffic Cop" Framework (Balance Laws)

The Concept: This is the old-school approach based on the fundamental laws of physics: Mass, Momentum, and Energy must be conserved. Nothing is created or destroyed; it just moves around.

• The Analogy: Imagine a busy city intersection. The Traffic Cop (the laws of physics) only cares about the total number of cars entering and leaving the intersection. If 10 cars go in, 10 must come out.
• The Problem: This framework is great for simple fluids like water. But for complex fluids (like the spaghetti sauce), it hits a wall. It can tell you how the bulk of the sauce moves, but it doesn't know what happens to the pasta strands inside. The strands aren't "conserved" in the same way cars are; they stretch, break, and tangle. The Traffic Cop framework doesn't have a rulebook for the internal chaos of the strands.

2. The "Relaxation" Framework (GENERIC)

The Concept: This framework focuses on how a system naturally wants to settle down into a calm, peaceful state (equilibrium) when left alone. It combines the rules of motion (mechanics) with the rules of heat and disorder (thermodynamics).

• The Analogy: Imagine a chaotic dance floor. The "GENERIC" framework assumes that if you stop the music and stop pushing the dancers, they will eventually stop dancing and stand still in a calm, organized line.
• How it works: It looks at the whole system (the fluid + the internal structure) as one giant machine. It asks: "If we let this system run its course without outside interference, how does it naturally slow down and find its resting spot?"
• The Strength: It automatically ensures that the fluid behaves correctly according to the laws of thermodynamics. It guarantees that energy is conserved and entropy (disorder) increases, just like nature intended.
• The Weakness: It works best when the fluid is just sitting there or flowing in a standard way. It struggles a bit when you are actively pushing the fluid with weird, complex external forces that aren't just "stirring."

3. The "Minimizing Effort" Framework (Onsager Principle)

The Concept: This framework is based on the idea that nature is lazy. When a system is being pushed by an external force, it will choose the path that dissipates (wastes) the least amount of energy possible, or conversely, the path that maximizes the "efficiency" of the dissipation.

• The Analogy: Imagine you are walking through deep snow. You want to get to the other side. You have two choices: trample a straight line (high effort, high energy loss) or find a path where the snow is slightly packed down (less effort). The Onsager principle says the system will naturally "choose" the path of least resistance for the energy it's losing.
• How it works: It focuses specifically on the internal structure (the pasta strands). It asks: "Given that I am pushing this fluid, how will the internal strands rearrange themselves to handle that push most efficiently?"
• The Strength: It is very flexible. It is great for describing what happens when you actively drive the system with external forces.
• The Weakness: It doesn't automatically guarantee that the whole system follows the strict laws of thermodynamics unless you carefully build those rules in yourself.

The Author's Big Idea: "More Viewpoints, Merrier"

Grmela argues that you shouldn't pick just one rulebook. Instead, you should use them together like a toolkit.

• The "Two-Step" Dance: He suggests that the Onsager Principle is like the "fast step." It quickly figures out how the internal structure (the pasta) reacts to a push. Once that reaction is figured out, the GENERIC framework takes over as the "slow step," ensuring that the whole system (the sauce + the pasta) still obeys the grand laws of physics and energy conservation.

The "Extension and Reduction" Strategy:
Think of it like this:

1. Extension: Imagine you are studying a car engine. Instead of just looking at the engine, you imagine the engine is part of a giant spaceship. You study the whole spaceship to see how it naturally settles down.
2. Reduction: Once you understand the spaceship, you "zoom in" back to just the engine. Because you understood the big picture, you now know exactly how the engine behaves when you push it, without having to guess the rules.

Summary

• Balance Laws are the basic traffic rules (good for the big picture, bad for the details).
• GENERIC is the "natural settling" rule (great for ensuring physics is correct, but needs help with external pushes).
• Onsager is the "lazy path" rule (great for handling external pushes and internal details).

The Conclusion: The best way to model complex fluids is to let the Onsager principle handle the messy internal details and the external forces, and then feed that information into the GENERIC framework to ensure the whole system stays scientifically honest. By combining these perspectives, scientists can create much better models for everything from ketchup to blood flow.

Technical Summary

Here is a detailed technical summary of the paper "Rheological modeling with GENERIC and with the Onsager principle" by Miroslav Grmela.

1. Problem Statement

The paper addresses the challenge of mathematically modeling the flow of complex fluids (e.g., polymeric fluids, suspensions). While simple fluids (hydrodynamics) are well-described by local conservation laws (mass, momentum, energy), complex fluids possess internal structures (like polymer chains) that evolve on the same time scale as the flow.
The core problem is identifying a universal framework that:

1. Reproduces observed flow behavior.
2. Ensures compatibility with local conservation laws (mechanics).
3. Ensures compatibility with thermodynamics (approach to equilibrium).
4. Correctly couples the evolution of the internal structure with the hydrodynamic fields.

The author compares three distinct frameworks used to achieve this:

• Local Conservation Laws (Balance Laws): Based on mechanics.
• GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling): Based on the approach to thermodynamic equilibrium.
• Onsager Principle: Based on variational principles for driven systems and the reduction of detailed dynamics to coarser dynamics.

2. Methodology

The author employs a comparative theoretical analysis, using isothermal polymeric fluids as a specific case study to illustrate the differences and relationships between the frameworks.

A. The GENERIC Framework

The paper utilizes the GENERIC equation (Eq. 1), which describes the time evolution of a state variable $x$:
$$\dot{x} = L \Phi_x - \Xi_x$$

• $L$ (Poisson Bivector): Represents reversible, Hamiltonian kinematics (conservative dynamics).
• $\Xi$ (Dissipation Potential): Represents irreversible, dissipative dynamics.
• $\Phi$ (Thermodynamic Potential): A convex function (typically $-S + E^*E + N^*N$) acting as a Lyapunov function, ensuring the system approaches equilibrium ($\dot{\Phi} \leq 0$).
• Method: The author derives the specific Poisson bracket for polymeric fluids (involving mass $\rho$, momentum $u$, and conformation tensor $c$) and constructs the corresponding GENERIC equations. This formulation naturally includes both conservation laws and the assumption of local equilibrium.

B. The Onsager Principle (Dynamic and Static)

The paper reinterprets the GENERIC framework through the lens of the Onsager variational principle.

• Extended GENERIC: The system is split into hydrodynamic variables ($x$) and internal structure variables ($y$).
• Two-Stage Approach:
  1. Fast Stage (Dynamic Onsager): The internal structure $y$ evolves rapidly toward a quasi-steady state determined by the current hydrodynamic state $x$ and external forces. This is governed by minimizing a Rayleighian ($R = -\Upsilon + \langle y^*, F \rangle$), where $\Upsilon$ is the dissipation potential and $F$ is an external force.
  2. Slow Stage: The hydrodynamic variables $x$ evolve using the "slaved" internal structure from the first stage.
• Key Distinction: Unlike the Dynamic MaxEnt (GENERIC), which describes approach to thermodynamic equilibrium, the Dynamic Onsager principle describes the approach of a detailed dynamical theory to a coarser dynamical theory (less details). It can also handle externally driven systems that do not reach equilibrium.

C. Comparison via Polymeric Fluids

The author applies both frameworks to the same physical system (polymeric fluids with conformation tensor $c$).

• Balance Laws: Only provide the form of mass and momentum conservation ($\partial_t \rho + \nabla \cdot J_\rho = 0$). They fail to provide the evolution equation for the internal structure $c$ or the constitutive relation for the extra stress tensor without additional kinetic theory (e.g., Kirkwood theory).
• GENERIC: Provides a unified derivation where the stress tensor and internal evolution emerge simultaneously from the thermodynamic potential and dissipation potential.
• Onsager: Views the internal structure evolution as a variational problem driven by external forces (flow gradients) and dissipation, effectively deriving the stress tensor as a reaction to these forces.

3. Key Contributions

1. Unification of Frameworks: The paper establishes a rigorous mathematical link between the GENERIC formalism and the Onsager variational principle. It demonstrates that the Onsager principle can be viewed as a specific reduction or "two-stage" limit of the extended GENERIC dynamics.
2. Dynamic vs. Static Principles: The author distinguishes between:
   • Dynamic MaxEnt (GENERIC): Time evolution in state space toward thermodynamic equilibrium.
   • Dynamic Onsager: Time evolution in the space of vector fields, describing the reduction of a detailed theory to a coarser one (which may be driven and non-equilibrium).
3. Derivation of Stress Tensor: The paper shows how the extra stress tensor (crucial for rheology) arises naturally in GENERIC through the Poisson bracket structure and the anchor map, whereas in the Onsager framework, it is derived via the variational minimization of the Rayleighian.
4. Identification of Limitations:
   • Balance Laws: Insufficient for internal structure dynamics.
   • GENERIC: Currently limited to "imposed flows" as external forces; it struggles with arbitrary external forces that are not simple flow boundary conditions.
   • Onsager: More flexible for arbitrary external forces but requires separate specification of the coupling between internal structure and hydrodynamics.

4. Results

• Equivalence in Isothermal Polymeric Fluids: For isothermal polymeric fluids, the equations derived via GENERIC (Eqs. 16–19) and the two-stage Onsager approach yield consistent physical descriptions.
• Local Equilibrium: The GENERIC formulation inherently includes the assumption of local equilibrium (via the pressure term derived from the Poisson bracket), whereas standard balance laws treat this as an additional assumption.
• Non-Conservation of Internal Structure: The paper confirms that while mass and momentum are conserved (local conservation laws), the internal structure (conformation tensor) is not conserved; its evolution is driven by flow and dissipation.
• The "Anchor Map": The paper identifies the linear operator $K$ (anchor map) as the mathematical bridge connecting the external force acting on the internal structure to the resulting extra stress tensor (vector field) acting back on the fluid.

5. Significance

• Methodological Guidance: The paper provides a practical "rule of thumb" for rheologists:
  • If the problem involves imposed flows and requires a rigorous thermodynamic foundation, use GENERIC.
  • If the problem involves complex external forces or requires a variational approach to non-equilibrium steady states, use the Onsager principle.
  • The author suggests a hybrid approach: use GENERIC to derive constitutive relations (stress tensors) and Onsager to handle specific external driving forces.
• Theoretical Clarity: It clarifies the geometric and physical underpinnings of complex fluid modeling, moving beyond ad-hoc constitutive equations to a systematic derivation based on conservation laws and thermodynamic principles.
• Future Research: It highlights the need to extend the GENERIC framework to handle arbitrary external forces (beyond simple flow boundary conditions) to fully unify the modeling of complex fluids.

In summary, Grmela argues that while Balance Laws, GENERIC, and Onsager principles are distinct, they are deeply interconnected. GENERIC offers a comprehensive thermodynamic structure, while the Onsager principle offers a flexible variational tool for driven systems; together, they provide a robust foundation for modern rheological modeling.