Second-gradient models for incompressible viscous fluids and associated cylindrical flows

This paper introduces a mathematically well-posed second-gradient framework for incompressible viscous fluids with pressure-dependent viscosity that ensures ellipticity of the governing pressure equation, and demonstrates that explicit solutions for steady cylindrical flows converge to classical Navier-Stokes results as characteristic length scales vanish.

The Gist

Imagine you are trying to predict how honey flows out of a jar, or how oil moves through a tiny crack in a machine part. For over a century, scientists have used a set of rules called the Navier-Stokes equations to do this. Think of these equations as a very reliable, old-fashioned map. They work perfectly for big rivers and wide pipes.

But, just like an old map might fail to show a new, narrow alleyway or a hidden shortcut, these classical rules start to break down in two specific situations:

1. Tiny scales: When the fluid is moving through microscopic channels (like in your bloodstream or a microchip).
2. High pressure: When the fluid is squeezed incredibly hard, changing how "thick" (viscous) it gets.

In these extreme cases, the old map says, "I don't know what's happening here."

This paper introduces a new, upgraded map called a "Second-Gradient Model." Here is how it works, explained simply:

1. The Problem with the Old Map: The "Invisible" Pressure

In the old model, fluid pressure is like a simple balloon: you squeeze it, and it pushes back evenly. But in these new, extreme scenarios, the fluid behaves more like a springy sponge. When you squeeze one part of it, the neighbors feel the squeeze too, not just because they are touching, but because of how the material is structured internally.

The previous attempts to model this "springy" behavior had a major flaw: they introduced a new variable called "hyperpressure." Imagine trying to drive a car where the steering wheel is connected to a mysterious, invisible lever that no one knows how to move. The model existed, but because no one knew how to control that invisible lever, the model couldn't make accurate predictions. It was mathematically "broken" in the middle.

2. The Solution: Giving the Invisible Lever a Name

The authors of this paper fixed the broken model by proposing a simple rule for that invisible lever. They said: "The hyperpressure is just a scaled version of how fast the regular pressure is changing."

Think of it like this: If the pressure is a hill, the hyperpressure is the slope of that hill. By linking them together, the model finally becomes "well-posed." This means it can now solve problems without getting stuck, and it guarantees that the math behaves nicely (it stays "elliptic," which is a fancy way of saying the solution is stable and predictable, even when the fluid gets very thick under high pressure).

3. The "Memory" of the Fluid

The new model adds something called an "internal length scale."

• Old Model: The fluid has no memory of its size. A drop of water and a bucket of water flow the same way if the speed is the same.
• New Model: The fluid "remembers" its size. It knows if it's flowing through a tiny straw or a giant pipe. This allows the model to predict weird behaviors that happen only in very small spaces, like the fluid not sticking perfectly to the walls or moving in a slightly different pattern than expected.

4. Testing the New Map: The Pipe and the Spinner

To prove their new model works, the authors tested it on two classic scenarios:

• The Pipe (Poiseuille Flow): Imagine water rushing through a pipe.

  • The Result: The new model predicts a slightly different speed profile than the old one, especially near the walls. But here is the magic: as the "internal length scale" (the size of the fluid's memory) shrinks to zero, the new model smoothly turns back into the old, trusted Navier-Stokes model. It's like a new GPS that defaults to the old map when you are driving on a highway, but switches to a detailed street view when you enter a narrow alley.

• The Spinner (Taylor-Couette Flow): Imagine a fluid between two spinning cylinders.

  • The Result: Again, the new model gives a more detailed picture of how the fluid spins and how the pressure builds up. Crucially, it shows that even when the fluid gets very thick due to pressure, the math doesn't explode or become impossible to solve.

The Big Picture

This paper is like giving engineers and scientists a universal remote control for fluid dynamics.

• The old remote only worked for "Standard Mode" (big pipes, low pressure).
• The new remote has a "Micro Mode" and a "High-Pressure Mode."
• Most importantly, the authors figured out how to program the "High-Pressure Mode" so it doesn't crash.

By solving the mystery of the "hyperpressure," they have created a tool that can help design better micro-machines, understand blood flow in tiny capillaries, and simulate fluids in extreme environments, all while ensuring that the math remains solid and reliable.

Technical Summary

1. Problem Statement

The paper addresses limitations in classical continuum mechanics regarding the modeling of incompressible viscous fluids in two specific regimes:

1. Small-length scale flows: Where classical Navier-Stokes (NS) models fail to capture size-dependent effects.
2. High-pressure flows: Where viscosity becomes pressure-dependent, a phenomenon observed experimentally (e.g., Bridgman's work) but mathematically problematic in classical formulations.

The authors focus on the Fried-Gurtin framework for second-gradient incompressible linear viscous fluids. While this framework introduces a third-order hyperstress tensor ($G$) to account for higher-order spatial derivatives of velocity, it suffers from a critical theoretical gap: the hyperpressure ($\pi$), a vector field appearing in the constitutive relation for $G$, remains indeterminate. Without a constitutive relation for $\pi$, the model lacks predictive capability, particularly for boundary conditions involving hypertractions. Furthermore, previous attempts to model pressure-dependent viscosity within classical mechanics often lead to ill-posed problems where the governing equations lose ellipticity.

2. Methodology

The authors develop a rigorous mathematical framework to resolve the indeterminacy of the hyperpressure and extend the model to pressure-dependent viscosities.

• Constitutive Closure for Hyperpressure: Building on arguments from prior literature (specifically [37]), the authors propose a specific constitutive relation linking the hyperpressure to the gradient of the mean pressure ($p$):
  $$\pi = \ell_1^2 \nabla p$$
  Here, $\ell_1$ is a characteristic length scale derived from the model's intrinsic parameters. This choice reduces the number of free constants and allows the pressure equation to be closed.

• Governing Equations: The model incorporates the balance of linear momentum with a non-standard inertial power balance (introducing a second length scale $\ell_0$) and the proposed hyperstress relation. The resulting momentum equation is:
  $$\rho \left( \dot{v} - \ell_0^2 \text{div}(\dot{\nabla}v) \right) = -\nabla(p - \ell_1^2 \Delta p) + \mu \Delta v - \mu \ell_1^2 \Delta^2 v + \rho b$$
  Taking the divergence of this equation yields a fourth-order elliptic equation for the pressure, in contrast to the second-order Poisson equation in classical NS.

• Pressure-Dependent Viscosity Extension: The authors extend the model to fluids where viscosity $\mu$ is a function of pressure, $\hat{\mu}(p)$. They demonstrate that the inclusion of the second-gradient term ($\ell_1^2 \Delta^2 p$) ensures the pressure equation remains elliptic regardless of the behavior of $\hat{\mu}(p)$, thereby resolving the well-posedness issues found in classical pressure-dependent viscosity models.

• Boundary Conditions: The study analyzes two types of adherence conditions:

  • Strong Adherence: $v = 0$ and $\partial v / \partial n = 0$ on the boundary.
  • Weak Adherence: $v = 0$ and the hypertraction $m = 0$ on the boundary.
    To uniquely determine the pressure in the fourth-order system, an additional boundary condition is proposed: $\partial p / \partial n = 0$ on the boundary.

3. Key Contributions

1. Resolution of Hyperpressure Indeterminacy: The paper provides a physically motivated constitutive relation ($\pi = \ell_1^2 \nabla p$) that closes the Fried-Gurtin model, making it mathematically well-posed and predictive.
2. Guaranteed Ellipticity for Pressure-Dependent Viscosity: The authors prove that for fluids with pressure-dependent viscosity, the second-gradient formulation guarantees the ellipticity of the pressure equation. This contrasts sharply with classical models where high pressure gradients can cause the equation to change type (become hyperbolic), leading to mathematical instability.
3. Explicit Analytical Solutions: The authors derive exact solutions for steady cylindrical flows (Poiseuille and Taylor-Couette) under both strong and weak adherence conditions. These solutions involve modified Bessel functions ($I_0, K_0, I_1, K_1$).
4. Asymptotic Convergence: Rigorous proofs are provided showing that as the characteristic length scales ($\ell_1, \ell_0$) tend to zero, the derived velocity and pressure profiles converge pointwise to the classical Navier-Stokes solutions.

4. Results

The paper applies the model to two canonical cylindrical flow problems:

• Poiseuille Flow (Steady Flow in a Tube):

  • Strong Adherence: The velocity profile includes a classical parabolic term plus a correction term involving $I_0(r/\ell_1)$. The discharge rate is modified by the length scale.
  • Weak Adherence: The solution satisfies the condition $G[n \otimes n] = 0$. The velocity profile differs from the strong adherence case but still converges to the parabolic profile as $\ell_1 \to 0$.
  • Convergence: Using asymptotic expansions of Bessel functions, the authors prove that the dimensionless velocity $u(\sigma)$ converges to $1-\sigma^2$ (the classical solution) as the dimensionless length scale $\lambda_1 \to 0$.

• Taylor-Couette Flow (Rotating Cylinder):

  • Strong Adherence: The velocity profile involves $I_1(r/\ell_1)$ and converges to the linear profile $v = \Omega r$ as $\ell_1 \to 0$.
  • Weak Adherence: Remarkably, for weak adherence, the classical linear solution $v = \Omega r$ satisfies the boundary conditions exactly, meaning the second-gradient effects vanish for this specific flow configuration under these conditions.
  • Pressure Field: The pressure is determined by a fourth-order ODE. Numerical solutions (using MATLAB's bvp4c) confirm that the dimensionless pressure field converges to the classical solution as length scales vanish.

5. Significance

This work is significant for both theoretical and applied fluid mechanics:

• Mathematical Rigor: It resolves a long-standing ambiguity in second-gradient fluid theories by providing a specific, physically grounded constitutive law for the hyperpressure, ensuring the system is well-posed.
• Physical Predictability: By guaranteeing ellipticity in pressure-dependent viscosity models, the framework allows for the stable simulation of high-pressure flows (e.g., in deep drilling or high-pressure lubrication) where classical models fail or become unstable.
• Bridging Scales: The model successfully bridges the gap between macroscopic continuum mechanics and microscopic scale effects, providing a consistent framework that recovers classical Navier-Stokes behavior in the limit of vanishing length scales.
• Practical Application: The explicit solutions for cylindrical flows provide benchmark cases for validating numerical codes and experimental data in micro-fluidics and high-pressure rheology.