Compressible and immiscible fluids with arbitrary density ratio

This paper presents a new thermodynamic-based theory for modeling compressible and immiscible fluids with arbitrarily high density ratios, addressing the limitations of traditional Navier-Stokes and Euler equations by deriving density evolution equations that properly account for true momentum evolution.

The Gist

Imagine you are trying to describe a dance between two very different partners: a heavy, slow-moving elephant (water) and a light, fast-moving feather (air). In the world of fluid physics, these two are "immiscible," meaning they don't mix like oil and water, but they do have a fuzzy boundary where they meet.

For a long time, scientists have struggled to write the "rules of the dance" (the equations) for systems where the density difference is huge, like 1,000 times heavier for the elephant than the feather. The old rules had a major flaw: they treated the elephant's weight as if it were a constant, unchanging number, even right at the fuzzy boundary where the elephant turns into a feather. This is like trying to describe a person turning into a ghost by saying, "They are still a solid human the whole time," which doesn't make sense.

Here is a simple breakdown of what this paper proposes to fix that problem.

1. The Problem with the Old Rules

The traditional way of calculating how fluids move (using the Navier-Stokes and Euler equations) relies on a shortcut called the Boussinesq approximation.

• The Analogy: Imagine you are pushing a shopping cart. If the cart is full of bricks, it's heavy. If it's empty, it's light. The old rules assumed that if you are pushing a cart that is partially full of bricks and partially empty air, the weight of the cart never changes as you push it. It just assumes the weight is a fixed average.
• The Flaw: In reality, as the cart moves and the bricks shift (diffusion), the weight changes. The old rules ignored the fact that the "momentum" (mass $\times$ speed) changes because the mass itself is changing at the boundary. They also assumed the air and water take up a perfectly predictable amount of space, ignoring that when they mix at the boundary, the space they occupy can actually wiggle and change due to pressure.

2. The New Approach: Energy Minimization

Instead of guessing how the density changes, the author starts from a fundamental principle: Nature always tries to use the least amount of energy possible.

• The Analogy: Think of a ball rolling down a hill. It doesn't care about the specific steps it takes; it just wants to get to the bottom (lowest energy). The author uses this "energy hill" concept to derive new rules for how the water and air interact.
• The Key Innovation: The author introduces a concept called "Excess Volume."
  • Imagine you have a bucket of water and a bucket of air. If you pour them together, you might expect the total volume to be exactly the sum of the two buckets. But at the microscopic level, when they meet, the molecules might pack together tighter or looser, creating a little bit of "extra" or "missing" space.
  • The old rules assumed this extra space was zero everywhere. This paper says, "No, that extra space exists, it changes from place to place, and it affects the density."

3. The New "Dance Rules" (The Results)

By accounting for this changing "extra space" and using energy minimization, the author derives a new set of equations that do three main things:

A. A New Way to Hear Sound (Sound Speed)
The paper shows that the speed of sound isn't just a random number; it comes directly from how the energy of the fluid changes as it gets squished.

• The Metaphor: Think of sound as a ripple in a crowd. The speed of that ripple depends on how tightly the people (molecules) are packed and how much energy they have. The new formula calculates this speed naturally, without needing to be told what it is beforehand. It even suggests that in a gas, the speed of sound is roughly the same as the average speed at which the gas molecules are bouncing around.

B. A New Rule for Pressure and Speed (Bernoulli's Law)
You've probably heard of Bernoulli's principle: "When a fluid moves faster, its pressure drops."

• The Twist: The old rule works great for water flowing in a pipe, but it breaks down when you have a huge jump in density (like water hitting air). The author creates a Generalized Bernoulli's Law.
• The Metaphor: Imagine a river flowing into a waterfall. The old rule says the energy stays the same. The new rule says, "Wait, as the water turns into mist (air), some energy is lost or transformed because the 'stuffiness' of the water is changing." The new equation accounts for this energy shift, making it accurate even when the fluid is changing its nature from heavy to light.

C. The "Bump" in the Density
This is perhaps the most visual result.

• The Old View: If you look at the boundary between water and air, the old models said the density would just slide smoothly from "heavy water" to "light air," like a ramp going down.
• The New View: The author's math predicts a bump. As you cross the boundary, the density actually goes up slightly before it drops down to the air level.
• The Metaphor: Imagine a crowd of people (water) trying to squeeze through a door into an empty hallway (air). As they squeeze through the doorway, they might pack together tighter for a split second before spreading out. The new theory predicts this "packing bump," which matches what advanced computer simulations (called Density Functional Theory) have already seen, but the old simple models missed.

Summary

This paper proposes a new way to write the laws of physics for fluids that are very different in weight (like water and air).

1. It stops pretending the weight is constant at the boundary.
2. It admits that the space molecules take up changes (excess volume).
3. It uses the principle of "lowest energy" to derive new rules that explain how sound travels, how pressure changes, and why the density at the water-air boundary actually has a small "hump" in it.

The author claims this new framework works for any mix of fluids, no matter how different their weights are, opening the door for more accurate computer simulations of things like rain falling, waves crashing, or bubbles rising.

Technical Summary

Technical Summary: Compressible and Immiscible Fluids with Arbitrary Density Ratios: Diffuse Interface Theory

Problem Statement
Current computational fluid dynamics (CFD) models struggle to accurately simulate immiscible fluid systems with high density ratios, such as water-air (ratio $\approx 1000$). Existing approaches rely on two primary methods, both of which possess significant limitations:

1. The Boussinesq Approximation: Assumes constant density ($\rho$) within the interface, utilizing the equation $\rho(du/dt) = -\nabla p + \dots$. This formulation fails to account for the "true" momentum evolution $d(\rho u)/dt$, neglecting density changes and the resulting momentum transfer. Furthermore, it cannot naturally generate gravity and buoyancy forces without ad-hoc additions, as these arise from density differences.
2. The Interpolation Method: Defines density as a linear function of volume concentration ($\rho = \rho_w \phi + \rho_a(1-\phi)$). This approach creates a fundamental conflict between incompressibility ($d\rho/dt = 0$) and diffusion ($d\phi/dt \neq 0$). Additionally, it assumes a uniform excess volume (often zero) across the domain, which contradicts physical reality where local excess volume depends on local pressure. Consequently, standard continuity equations and linear interpolation fail to capture non-monotonic density profiles observed in the water-air interface.

Methodology
The authors propose a novel theoretical framework grounded in the thermodynamic principle of energy minimization, distinct from the traditional derivation of Navier-Stokes and Euler equations. The methodology involves:

• Reformulation of Volume and Density: Instead of assuming zero excess volume, the model introduces a spatially varying excess volume ($v_e$) within the interface. This volume is redistributed between the fluid components, leading to partial densities ($\rho'_w, \rho'_a$) that are no longer constant but depend on local pressure and mixing. This necessitates two coupled evolution equations: one for volume concentration ($\phi$) and one for density ($\rho$).
• Dissipation-Conservation Principle: The authors establish a theorem linking the first law of thermodynamics (conservation) with the second law (dissipation). By defining the total system energy ($E$) as the sum of free energy ($F$), pressure energy ($P$), kinetic energy ($K$), and wall free energy ($W$), they derive kinetic equations by minimizing energy dissipation.
• Modified Gibbs-Duhem Relation: To resolve mathematical incompatibilities between spatial gradients and functional derivatives in existing models, a modified isothermal Gibbs-Duhem relation is derived. This explicitly accounts for velocity-dependent entropy and inertial work, leading to a rigorous definition of pressure that includes contributions from free energy and velocity entropy.
• Kinetic Energy Definition: The kinetic energy is formulated as $K = \int \rho u \cdot u \, d\Omega$ (linear momentum density $\zeta = \rho u$) rather than the conventional $\frac{1}{2}\rho u^2$. This distinction allows the separation of momentum balance (changing both $u$ and $\rho$) from velocity dissipation (changing only $u$).

Key Contributions
The derivation yields a generalized density evolution equation and three major theoretical advancements:

1. Generalized Bernoulli's Law: The standard Bernoulli principle ($p + \frac{1}{2}\rho u^2 = \text{const}$) is shown to be a special case. The generalized form accounts for velocity entropy changes and density evolution at the interface: $p + \rho u^2 = \text{constant}$ (under specific steady-state inviscid conditions), acknowledging that energy jumps occur across interfaces due to density differences.
2. Universal Sound Speed Formulation: A general expression for sound speed is derived based on the derivative of the free energy with respect to density: $c = \sqrt{d\Psi/d\rho}$. For ideal gases, this approximates the mean thermal velocity of molecules.
3. Generalized Equation of State (EOS): The theory demonstrates that the density evolution equation naturally generates an EOS for fluids (e.g., $p \sim \rho \ln \rho$ for ideal gases) without requiring a prescribed EOS, contrasting with standard diffuse-interface methods.

Results and Validation
The paper validates the theory through three case studies:

• Sound Propagation: The derived sound speed equation is consistent with perturbation analysis and correctly predicts the dispersion relation for sound waves in air.
• Ideal Gas EOS: In the steady state, the model reproduces the ideal gas law, confirming that the density evolution equation inherently captures thermodynamic state relationships.
• Water-Air Interface Dynamics: An analytical solution for a moving flat water-air interface in a tube demonstrates that the model predicts a non-monotonic density profile. Specifically, the density increases from the bulk water value to a maximum peak at the interface center before decreasing to the bulk gas value. This result aligns with Density Functional Theory (DFT) findings [11, 12] and contradicts the monotonic decrease predicted by linear interpolation methods. The model also correctly reproduces the Young-Laplace equation for curved interfaces.

Significance
The paper claims to provide a general framework for immiscible fluids with arbitrarily high density ratios, applicable to both compressible and incompressible regimes. By treating density evolution as a dynamic variable coupled with excess volume and viscous dissipation, the model resolves the contradictions inherent in the Boussinesq approximation and linear interpolation methods. The work suggests that viscous dissipation inherently alters density, challenging the strict incompressibility assumption in dissipative flows. The authors posit that this theory opens a new window for CFD, offering a physically consistent description of momentum and energy transfer at fluid-fluid interfaces where density ratios are extreme.